Symmetry Methods in Physics in Memory of Professor Ya. A.Smorodinsky

International Workshop

SYMMETRY METHODS IN PHYSICS

IN MEMORY OF PROFESSOR YA. A. SMORODINSKY

Volume 2

Edited by A.N.Sissakian G.S.Pogosyan S.I.Vinitsky

Dubna 1994 NOTICE

PLEASE BE AWARE THAT THIS IS THE BEST REPRODUCTION POSSIBLE BASED UPON THE ORIGINAL DOCUMENT RECEIVED JOINT INSTITUTE FOR NUCLEAR RESEARCH

International Workshop

SYMMETRY METHODS IN PHYSICS

IN MEMORY OF PROFESSOR YA.A.SMORODINSKY

Dubna, Russia July 6-10 1993

Edited by A.N.Sissakian G.S.Pogosyan S.I.Vinitsky

Volume 2

Dubna 1994 Published by

Publishing Department Joint Institute For Nuclear Research Joliot Curie, 6 141980, Dubna, Moscow Region Russia

PROCEEDINGS OF THE INTERNATIONAL WORKSHOP ON SYMMETRY METHODS IN PHYSICS IN MEMORY OF PROFESSOR YA. A.SMORODINSKY

Photographs: By Yu.A.Tumanov

Volume II

A. J. Macfarlane Generalised oscillator systems and their parabosonic interpretation 319

L. G. Mardoyan, A. N. Sissakian, V. M. Ter-Antonyan, T. A. Ghatrchian Generalization of the Rayleigh formula to the model with ring-shaped potentials 326

S. Mashkevich Symmetries and quantum mechanical spectra of anyons 332

R. M. Mir-Kasimov The Snyders space-time quantization, Poincare group deformations and ultraviolet divergences 337

S. S. Moskaliuk, Yu. F. Smirnov Using reduce system for calculation of integrity bases of Lie group invariants 346

F. Miiller-Hoissen, A. Dimakis Differential forms and gauge theory on discrete sets and lattices 351

M. A. Mukhtarov On integrability of generalized toda lattice in quantum domain and self-duality equations for arbitrary semisimple algebra 358

A. F. Nikiforov Classification of q-polynomials as polynomial solutions of hypergeometric type difference equations 361

A. P. Nersessian BV-geometry on Kahlerian supermanifolds 369

A. V. Nesterov, A. G. Kosinov Generating invariants in many-cluster microscopic problems of nuclear physics 375

J. Nyiri Boson vacuum polarization in the field of a supercritical charge 378

V. S. Otchik On the two Coulomb centres problem in a spherical geometry 384 iv

V. Papoyan, G. Haroutyunian The conform connect/on between equations of GR and the Jordan-Brans- Dicke theory 389

V. Pervushin, V. Papoyan Quantum evolution of the Universe 396

A. D. Popov Symplcctic manifolds with symmetry and weakly G-invariont complex structures 404

Z. Popowicz " SUSY Lax pair in the Gelfsnd-Dickey hierarchy 409

P. N. Pyatov, A. P. Isaev Towards the classification of the differential calculi on g(,(n) 1,15

Y. P. Rybakov Maximally invariant configurations in the SU($) Skyrme model 4^3

V. L. Safonov

Symmetry and statistics of the elementary collective excitations in solids ...

R. M. Santilli Application of isosymmetrics/Q-operator-deformations to the cold fusion of elementary particles 433 V. I. Sanyuk

Algebraic and analytical features of (3+1) dimensional topological solitons . 443

P. Schaller Loop space and W-type algebras 450

W. Scherer, H. D. Doebner A nonlinear Schrddinger equation and some of its solutions 454

A. Schirrmacher Varieties on quantized spacetime 463

M. B. Sheftel' Symmetries, recursions and linearization for two-component systems of hydrodynamic type 471

N. A. Smirnova, Yu. P. Smirnov, Level clustering in the vibrational-rotational spectra of the icosahedral Hamil- tonian 475 V

Yu. F. Smirnov, A. Del Sol Mesa Orthogonal polynomials of discrete variable associated with quantum algebras su,(2) and su,(l, 1) 479

A. I. Solomon, R. J. McDermott General deformations of bosons and their coherent states 487

A. N. Sissakian, I. L. Solovtsov, O. Yu. Shevchenko Method of variational perturbation theory 494

Б. Sorace Inhomogeneous quantum groups in physics 501

V. Spiridonov Infinite soliton systems, quantum algebras, and Pain/eve equations 512

I. A. B. Strachan Infinite dimensional Lie algebras and the geometry of integrable systems ... 519

S. I. Sukhoruchkin Relationship in baryon mass spectrum as a possible reason of the fine structure of nuclear excitation 538

S. I. Sukhoruchkin Manifestation of electrodynamics parameters in energies of nuclear states and in particle masses .... 536

A. A. Suzko Geometric nonadiabatic phases and supersymmetry 544

V. M. Ter-Antonyan, A. N. Sissakian Matrix of finite translations in oscillator basis 55S

N. "V. Tho, V. I. Kuvshinov Geometry of group manifold and properties of chiral fields in vector parametrization of groups 556

L. Vinet, P. Letourneau Dynamical polynomial algebras in quantum mechanics 563

A. A. Vladimirov On quasitriangular Hopf algebras related to the Borel subalgebra of ah 574

R. F. Wehrhahn, A. O. Barut Geometric motion on the conformal group and its symmetry scattering 577 vi

P. Winternitz, G. Rideau Spherical Functions for the quantum group su(2), 58J

V. I.Yukalov, E. P. Yukalova, A. A. Shanenko Influence of colour symmetry on string tension 592

B. N. Zakhariev Fragments of reminiscences and exactly solvable nonrelalivistic quantum models 595

В. M. Zupnik Quantum deformations for the diagonal R-matrices 599 319

GENERALISED OSCILLATOR SYSTEMS AND THEIR PARABOSONIC INTERPRETATION A. J. Macfarlane Department of Applied Mathematics & Theoretical Physics University of Cambridge, Cambridge CB3 9EW, U.K.

1. Introduction In this work, we consider the Fock space descriptions of and some interesting properties of various bosonic oscillator systems. All are based on a single creation-annihilation pair. All are represented in a suitable Fock space of states |n) of form (a*)"|0), n = 0,1,2 • • with a ground state |0) such that o|0) = 0 and a number operator N such that N\n) = n|n). we pass from the quantum harmonic oscillator with its well-known commutation relations

[a,a'] = 1. to the study of a generalisation of it that we have elsewhere [1] called modification. This gives rise to the Calogero-Vasiliev oscillator [2],[3],[4] governed by

[0,0'] = l+2i/A", (1) where i/gR, and К = (—)N obeys

K = K\ K2 = 1, аК + К а = 0, а* К + Ка* = 0.

Below we review the construction of its Fock space !FU, for which 2e > —1 emerges as a condition sufficient to ensure the hermiticity properties implied by use of the dagger in a', and also the sti(l,1) and озр( 1|2) properties of A notable result, believed new here, is then explained. It states that for v = (p — l)/2, Tv coincides with the Fock space of a single paraboson of order p = 1,2* • •• An important consequence of this observation is that Cor Fock space work, one may use the bilinear commutation rule (1), with 1v — p+ 1, instead of the more awkward trilinear commutation rules [5] that characterises paraboson systems of order p. Then we go on to consider the q-deformation of the Calogero-Vasiliev oscillator. Setting out from the Fock space description of this, it is easy to follow the familiar method of replacing round brackets suitably by square brackets of the type

[*: r] = w

This leads to a system involving a pair b and ft1 governed by

44« _ 9±

the spaces, defined for v = (p — l)/2, that describe q-deformed parabosons of order p — 1,2 • • ••

Wc discuss also the relationship of T4V to representations of su(l, 1), and osp(l|2),. Wo discuss next some relationship between the content of the present paper and previous work. Firstly we mention our recent paper [1] in which a different approach was followed to the combination of q-deformation and (/-modification. There, rather than effecting a q-deformation of (1) as here, we considered tile simplest modification of the now familiar commutation relation [6] pi.I»] t ( aa - qa]a = q~N, (4) namely ae' — fja'a = g~N(l + 2fK). (5) The consequences of (5) - its Fock space, su(l, 1), and osp( 1|2), properties, ar. 1 hidden supersymmetry - show similarities with features of the present work, but have no parabose aspects. We ttesd the term Calogero-Vasiliev oscillator in relation to (1), because the latter author introduced it first explicitly [2] and gave its Fock space representation, and because it plays a crucial role in the modern progress in understanding [3],[4],[9] of the integrable many body model discovered by tlie former author [10]. Previous treatment of q-analogues of parabosc (and also of parafermi) oscillators appears in [11]. This paper also discsses their su(l,l), or sp(2R), and oap(1|2), properties, but not the relationship to ^-modification. We remark here that our approach gives rise to various interesting formulas not given explicitly in [11]. Wc mention here two works on general deformation schemcs, including parabose and para- fcrmi oscillators, namely the paper [12], which surveys earlier work, and the paper [13]. There is also the article [14] useful especially for su(l, 1), questions.Thc paper [15] contains information of q-''eformation of parabosc and parafermi operators, and mentions the relation-

ship of osp(l|2)? representations to the Fock space of parabose oscillators. Finally wc note a matter which attracted debate at the International Workshop at which the work described in this paper was presented. The matter concerns the understanding of systems of the type in focus here within the context of I.ie-admissibie algebras. An assessment may be sought on the basis of [16] and [17] and from references given in the former source.

2. The Calogero-Vasiliev Oscillator. We review here work on this system, partly to describe its relation to parabosons of order p, and partly to provide background for our discussion of q-deformation of the system later. In seeking Fock space description of a and a* subject to (1), we use the obvious facts that a and a' reduce and increase the quantum number n of the kets |n) by 1, i.e.- that

[Af,fl] = -a, [N, a'] = a'..

Because К = (—)N features in (1), it is convenient to treat even and odd kets |2rt) and |2n + 1) separately. Thus we seek to represent (1) by means of

2 (2n+ 1|«Ч2п) = (2ПИ2П-1) =9У , (6)

with (n\n) = 1 for n = 0,1,2 • • •. By taking diagonal matrix elements of (1) for states |2n) and |2n+ 1), one is led to difference equations which can be seen to possess the following solutions

p„=2n + 2i/ + l, 9„ = 2n. (7)

In view of the former we impose the restriction (2i/ + 1) > 0 or и > —1/2 on the modification parameter v of (1). It follows from (6) and (7) that

а*а|2п) = (2n + 2v + l)|2n), a*a|2n - 1) = 2n|2n - 1). (8) 321

Hence we deduce the operator identity

a'a = N + 1/(1 - K), (9) valid on the Fock space of the kets |n). Similarly

aa1 = N + 1+1/(1 +A'). (10)

Addition and subtraction of (9) and (10) now gives the expression

N = {a, af}/2 — (2i/ + l)/2 for the number operator of and the expected verification that (1) holds as an operator identity on Tv- Use of results of the type (9) and (10) is crucial to our methods here and elsewhere. The idea of adjoining К to the families of operators with whose algebras wc work offers valuable simplifications. The same applies to the more general use of exchange operators [31, [4], [9] in the study of Calogero systems. Turning to su(l, 1) or sp(2R) properties, we note that a7, c'2 and {a,a1} have commutation rules [а,а»а] = 2а', [а2,а»] = 2а = [а,{о,а'}] (И) that do not involve r or К. Then if is easy to show that

,2 2 A'+ = а /2, K. = -Ki = -a / 2, Кг = {a, a»}/4 obey an 5u(], 1) algebra, and that its Casimir operator

2 K = I<] + {K+, I<-}/2,

2 commutes with Kz, A'±. One may then use (1) to show that K takes the value

-3/16 + v(v - A)/4

on The fact that this gives the values —3/16 + u{u ± 1) on its even and odd subspaces can also be verified directly. Thus carry unitary irreducible representations of su( 1,1) with distinct eigenvalues of the Casimir operator of su(l,l). This generalises the situation well-known [18] for the simple harmonic oscillator (with v = 0 ) itself, 4nd one goes beyond as follows, recalling the work of [19] which drew attention to the relevance of o.sp-algebras to ordinary harmonic oscillator systems. To reach an algebra whose Casimir operator commutes with the entire Calogero-Vasiliev algebra of (1), i.e with a and a' as well as K, we evaluate [К2,а]. When wc do this calculation using only (11), i.e. without assuming any knowledge of [о, a*], we find that [К\а] =-[[а,а'Н/8, so that С = К2 + [а,а']/8 (12) commutes with a and hence af as well as K. In fact it is well- known [20j,[21j that а, а', К can serve as a set of generators of osp(l|2) and that C, given by (12), is its Casimir operator. It takes the value с = -3/16 + !/(" - Я)/4 + (1 + 2i/K)/8 on with the pieces that depend on К cancelling, so that с = —1/16 -Ц i/2/4 (13) 322

throughout Tv. Thus the Fock space of the Calogero-Vasiliev oscillator carries a single irreducible unitary representation of asp(l|2). To conclude the section, we describe briefly the connection of the work of this section to the

Calogero many body system. Let a0 and aj be a pair of ordinary harmonic oscillator variables and write

a0 = (я + ip)/i/2, a* - (i - ip)/y/2. Then obey(l), and define the Hamiltonian

И = {а,а*}/2 = (p2 + x2)/2 + 1/(1/ - A')/(2x2).

This Hamiltonian describes the relative motion of a two-particle Calogero system. It follows from (9) and (10) that on we have Л = 2N + 1 + 2и. Thus provides a complete set of eigenstatcs of // //|n) = (2n + 1 + 2i/)|n). n = 0,1,2 • • •.

3. The Parabose Oscillator of Order p. Parabose oscillators emerge as a serious quantum mechanical possibility from asking the question: what are the weakest commutation rules that can be imposed on an a, a* pair that make Iff, a] = -«, [Я, «••]=«•, follow from use of the Hamiltonian Я = {а,а'}/2. The answer is evident: we should impose only [{а,а'},а]= -а ={a\a2] (14)

and its adjoint. We observe immediately that these results have already been »ecn to hold in the Calogero-Vasiliev system, cf. (11). To obtain the Fock space representation of (И), see e.g. [5], it is necessary to supplement the specification а|0) = 0 of the ground state by the demand that it satisfy аа»|0)=р|0), p — 1,2,3 •••. (15) The definition N = {a,a'}/2+const. makes [/V,a] = —a follow from (11). Then the explicit form = {a,a'}/2-p/2, is in agreement with N|0) = 0, because of (15). It follows from this point that the completion of the description of the Fock space is contained in

<2n + l|a'|2n) = \/2n + p, {2n\a*\2n - 1) = л/2п.

This agrees with the results (6) and (7) for p = 2v + 1. The results given for the su(l,l) and osp( 112) properties of the Calogero- Vasiliev oscillator also reproduce, for и = (p —1)/2, results already known for the parabose oscillator of order p. Wc note in particular that (13) now reads

2 C=(p -2p)/16. (16)

The fact that Kat] = l + (p-l)iV, 323 holds in the Fock space of a parabose oscillator of order seems to be a new result here. Of course it docs not follow from the algebra (14) alone; it requires also the use of (15) and is true only as an operator identity on the Fock space. It fallows therefore that wc should have a natural and direct derivation of it in the context of the Green ansatz [22] for the parabosc oscillator of order p. We do; it is to be found in a forthcoming paper [23] on the algebraic description of parabose and parafermi systems based on a now explicit and technically convenient version of the Green ansatz.

4. The q-Deformed Calogero-Vasiliev Oscillator. This is a system, described in a Fock space by a pair b and 6', which have the nonvanishing matrix-elements

(2n + l|6'|2n) = V[2n + 2i/ + 1], n = 0,1,2 • • •, {2п\Ь^\2п - \) = n = 1,2,3' • •. (17)

This specification arises from the equations of section two by the insertion of square brackets in the sense of (2). We note that

[/V, 6] = —6, [N,b*] = b\ aiu built into our approach, and that (17) yields also

6b» = [yV + l+f(l + /0], b*b = [JV + i/(l - /<)]. (18)

It can be shown that (18) implies

и 1< 66» - = [l + 2и1<}ч^ -'' К (19)

Thus, although this result specifies the algebra with whose representation we work, we did not procccd by postulation of it and by construction thence of the representation: rather it simply emerged in the form displayed in (19) from our obvious starling point. We now deduce some algebraic consequences. Firstly we have

J л,+ + а [6 , Ь*| = Ь(9 " " ' + (20)

This notable result follows most easily from (18) via

6V = 6(66') = b[N + 1 + i/(l + /С)], , 2 , 6 6 = (Ь Ь)Ь = 6[УУ-1 + 1/(1 + Я)].

The fact that the trilinear commutation relations have come out so simply may bo considered to point towards an expected parabose interpretation of our analysis. Also, as v —> 0, (19) reduces to (4), and (20) reduces to its useful consequence

[aV] =«(<," + ,-")>

here written in terms of a rather than 6 for the q-deformed oscillator. If we put v = (p — l)/2 in our results, it reproduces findings of [11] for the q-deformation of the parabosc oscillator of order p. It can be said, as in the undeformed situation, that interpolates between the Fock spaces of these oscillators. Our work however offers easy access to more explicit formulas than were given in [11]. One such is the version of (20) that arises for v = (p— l)/2 and is valid for any integer p; it agrees with a result for p = 2 quoted in [11]. 324

Another arises by evaluating (19) separately in the even and odd subspaces of For I< = 1 and i/ = (p — l)/2, this yields

When p = 1 this is the usual q-deformed oscillator formula, but, for p > 1, contains a generalisation of it that looks very suitable for a parabose oscillator of higher order. For К = —1, on the other hand, and и = (p — l)/2, we get

6frt _ = [2 - ^±<"+"-4.

For p = 1, this is the familiar q-defonned oscillator result, while for higher integers p it reflects the fact that results for the even and odd subspaces often turn out to be of different appearance, a feature which can be avoided only by allowing К to appear explicitly. Wc can directly discuss the su[ 1,1), or sp(2R), and twp(l|2), properties of The definitions 0+= (9 + g-1)"1^2. Я- = -Я5„ 4B, = 2N + 2i/ + 1, arc the same as those used in the limit v —> 0 and lead to the same algebra. One proves

IB+,B_] = [2B,;2], in which the notation of (2) is used, most easily by considering actions of the two sides on the even and odd subspaces of separately. It requires very little more calculation to show that 2 2 ' В = + [B2 — l/2;2] , takes distinct values |(2f ± l)/4; 2]2, independent of n, on the subspaces. Thus BJ commutes with B,, but not with b or 6*. To find the Casimir operator C, of osp(l(2), which commutes with 6, one uses information from 2 calculations already done to compute (B ,6] on T4U± separately and castes the result into the form [B2,6] = -J{v)\K,b\- with 2/(0 = [(2* + l)/4;2]2 - [(2f - l)/4; 2]2. It follows that we may use the expression

С, = B2 + f(i>)I< - [1/2; 2]3/2

for the required Casimir operator. Here the last ^-independent term has been included so that our results take on the expected form in the limit q —* 0. Also one might reasonably ask for a more explicit alternative expression for the Casimir operator, one in which К did not appear explicitly. One can show however that, acting upon any state of whatever, that it takes the value V = (1(2" + l)/4;2]2 + l(2f - l)/4;2]2 - Ц/2;2]г)/2. When v = (p — l)/2, i.e. for the q-deformed paraboson of order p, this reduces to

2 2 2 c,P = ([p/4;2] + [(2 - p)/4;2] - [l/2;2] )/2.

Finally as q —» 1, i.e. for the undeformed paraboson of order p, we get

c, = (p2-2p)/16,

which coincides with the result (16) given above. 325

References

[1] T.Brzeziriski, I.L.Egusquiza and A.J.Macfarlane. Generalised Harmonic Oscillator Systems and Their Fock Space Description. Preprint, DAMTP 93-10,PAH-LPTHE 93-13.

[2] M.A.Vasilicv Int. J. Mod. Phys. AG:1115, 1991.

[3] L. Brink, T.H.Hanson and M.A.Vasiliev. Phys. Lett. B286:lli9, 1992.

[4] L. Brink, T.H.Hanson, S.Konstein and M.A.Vasiliev. The Calogero Model-Anyoiiic Rep- resentation, Fermionic Extension and Supersymmetry. Preprint, USITP-92-14, ITP-92-53, 1992.

[5] O.W.Greenberg and A.M.L.Messiah. Phys. Rev. 138B:T155,1965.

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[13] K.Odaka, T.Kishi and S.Kamefuchi. J. Phys. A: Math. Gen. 24:L591,1991.

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GENERALIZATION OF THE RAYLEIGH FORMULA TO THE MODEL WITH RING-SHAPED POTENTIALS L. G. Mardoyan1,2, A. N. Sissakian3, V. M. Ter-Antonyan1'2, T. A. Chatrchian1

1 Dep. of Physics, Yerevan State University 375049 Yerevan, Armenia 2 Idjevan College-University, Idjevan, Armenia 3 JlNfl Dun a, 141980 Russia

Abstract The motion of a quantum particle is investigated in the ring-shaped model when the "bare" potential is equal to zero. Factorized in the spherical and cylindrical coordinates bases of this model are derived. The generalization of the Raylcigh expression for the plane wave expansion in the spherical waves is derived.

Formulation of the Problem. Models with ring-sheped potentials are based on the Schrodinger operator with the potential part supplemented with the term Л/г2 sin2 6, where Л is a nonnegative model parameter and 0 is the angle between the axiz z and radius-vector of a particle. Thus, the ring-sheped models are models with a special axial symmetric addition to the "bare" potential. The most interesting are eases when "bare" potential is taken to be either a hydrogen atom (the Hartmann model [1-3]) or an isotropic oscillator (the Quesne model [4]). Recent intensive studies of these models have mainly dealt with bases and interbasis expansions in separable variables [5,6] and search for dynamic symmetries in the framework of ring-sheped potentials [7,8]. Note also that the Hartmann model has its origin in spectroscopic problems of the benzene molecule [3]. The aim of our paper is to investigate the free motion in the framework of ring-sheped potentials. We will derive wave functions of this model in terms of spherical and cylindrical coordinates and then will connect these bases by mens of an expansion that can be considering as a generalization of the Rayleigh expansion of the plane wave in spherical waves to the models with ring-sheped potentials.

General Information. An arbitrary model with the ring-shaped potential is described by the equation (ft = m = 1):

where t/(r) is the central-symmetric "bare" potential. It is known [9] that the solution of the equation (1) in spherical coordinates has the form

*(r,fl,W«) = R{r,6)Zlm(0,r,S)

2 where S = \/m + 2Л — |m|, R(r; 6) is the solution of the radial equation, Zim(0,

iZlm(e,;6)

AlZlm(0,Vi6) = (/ + 5)(/ + fi + l)Zlm(0,v>;e) 327

"flie explicit form of the Ztm(0,ip;6) function is given by expression

where is the normalization constant, and C"(x) are Gegenbauer polynomials: c«<*> = Ш^ТГ^ ++ ? 41)

Under the normalization condition

wc have

The representation for'the Zim(0,ip;S) function through the Rodrig's formula is valid too:

r ,„ / « (-l)^1"^""* f (21 + 2S + 1)Г« + |m| + 26 -Ir I) \ '

The information about the radial function li(r\6) depends on tile concrete form of the "bare" potential anr/is define of by the equation

where к =t \ДЁ.

The Bases. The'free motion in the ring-shaped model is described by the equation

LA us consider the solution of this equation in the sphcrical and cylindrical coordinates. The radial equation in the spherical coordinates has the following form

Thrt corresponding solution can be expressed via the Bessel function

ЛыМ) = £^ljl+s+i{kr)

Under the normalization codition

J Щ.,.т.(г, 0, ¥>; Л)ФЫт(г, в, S)dV = 2nS(k - к%,,6тт, 328

The asymptotics of the radial function Rki(S) for small and large г is given by • 'ie following relations:

D r-o 2Щ1 + 6 + 1),„, ,+t Я*'(Г'Й) — T(2f + 2fi 4- 2) ( )

ЯыМ) *sin[fcr-|(/ + 5)] These expressions go to the known results, if the parameter Ь is equal zero. So, we see that the spherical basis of the free motion in the ring-shaped model has the form [brk Фш(г,«,И<)= у— Jl+s+L(kr)Zln(0,v,6) (3) When <5=0, this result goes to the spherical wave. In the cylindrical coordinates the scheme of separation of the variables corresponds to the factorization

V2tt which leads to the radial equation

where ui = у/к2 — k*. The normalization condition is

Then for the regular solution of the equation (4) we have the following representation in terms of the /iessel function:

This solution has the asymptotic form

iWp; S) ± sin \uP - | (|m| + 5 - I) ]

The cylindrical basis is Ф »k.m{p,V,'\£) = In the following the notation are more convenient:

kz = кcos-jr, w = fcsin7, о < ц < тг in which

ФИт(р, v, z; S) = 7)e"«o. v*,», (5)

Expansion. Let us consider the expansion of the cylindrical basis over the spherical one. This expansion must have the form

= £ Wfc{6)4>kln(r,0,v-,i) (6) '=|m| Our purpose is to calculate the matrix The following steps must be executed for this purpose: 329

(a) substitute the expressions (3) and (5) into (6); (b) multiply both the parts of the expansion (6) by ip\<5), integrate over the solid angle and use the orthonormalization property of the ring-shaped function; (c) use tiie expansions

= Е-^птоттг

_ у» (ibcosi)' ~ ^ i! 1=0

(d) go to the spherical coordinates from the cylindrical ones in the left part of the expansion (6).

Instead of (6), doing the mentioned steps we arrive of the expansion

. (-ir(iisi..7)|m|+4+2" s!(|m| + 6 + 1),

(ih cocoss 7)' x £ * T) rlml+<42,4i Qlm (fi) 1=0 where

Ql?V) = j (cosO)'(sin W <5)rfn

Using the expression (2) and subsequently integrate by parts wc can be convinced that the n Q'3', (S) differs from zero only under the condition 2s + t + |m| - />0. Thus, all members of the scries contain r in nonncgativc power of r, so as in the limit, when r —» 0, we obtain

1 lan 2 V U tan 77)) * Н-2.У-/ /7) 1 ^ яs!(!ПгпH | + Л« ++ 1),(/-Н-2l),(/-H-2s)8)!! ' where if, / - H = 2»; p-N 1 = I l 2 J [bMzl, I _ |jn| = 2re + 1.

When I = ' - |m| - 2s the expression for the is easily integrated:

г а+ 2 <ЭЛ-|т|-2а( )-(-1) Г(2/ + 2.5 + 2)

x |т(2/ + 26+1){! - |т|)!Г(/ + |m| + 25 + 1) j

If wc substitute the last expression into (7) and use the formula 330 wc obtain

' _ Jr(21 + 26 + 1)Г(/ -h M + 26 + l)sin 1 * ^ \ fca-М)! 7J

(cos 7)'~lml(sin 7)1т1+* ^ ( I — |m| J-M-l. , .

The matrix W£(6) can also be expressed via the Gegenbauer polynomials. For this aim we should be use the expression which connects the hypergeometrical function of the argument x with the hypergeometrical function of the argumen2Fl{Q,/J,t 1 - zQ: + Д + 1 7i 1 Z)+ sf,(a, A7!«) = S)F(^|) ~ "

+ Г(7Г(Г)Г^Г7)(1 " - «.7 - Л7 + 1 - « - ft 1 -

arid the representation of the Gegenbauer polynomials over the hypergeometrical function

nv, л + n) ( n 1 n . Л

The final result has the form

и»,-r (w+1) x

x (2 sin 7)l",l+s * (cos 7) (8)

The expressions (6) and (8) define completely, in the free ring-shaped model the expansion of the cylindrical basis over spherical one. It may easily be checked that when m = 6 = 7 = 0 the expansion (6) turns into the known Rayleigh's expansion for the plane wave over the spherical waves:

1=0 *

Here

The expansion (6) can be reduced to the form known in the theory of special functions. To verify that, we substitute the spherical and cylindrical bases into (6) and expression (8) then, turn to the spherical coordinates. We obtain the expression:

(jfcr sin flsin 7)-|m|"{^H+s(/tr sin 0sin ">»<*<>" =

= v/2(Jtr)-l",l-«-^r + 6 + 0 x

i'-Hr(2|m| + 26 + 1)Г(/ + 6 + f )(l - M)i Г(Н + « + 1)Г(/+|т| + М + 1) X

* -W ^ (fcr)c):^^ (cos (coe 7) 331

Let's introduce new notes

y-kr, A = |m| + £ + i, n = / — |m|

So, our result turns into the expression

, co ^<:, (г^sin5sin^)i-•«Jл_^(г,sinffsin7)e^ ' • "'' =

л ( Л л = ч/2у~ Г(А) g i" (2Л"^ " + У ^ Л+л(у)Сп (со8 Р)Сп (cos 7) (3)

from the monograph [10]. Thus, firstly, we have convinced that our result is correct, secondly, we derived the connection of free the ring-shaped model with the concrete region of the theory of special functions. Remark additionally that for 7 = 0 (9) turns into the expression from the monograph [10] also:

Л л = Г(А) (|)~ f) i> + А)Л+лЫС„ (х)

Conclusion. We have obtained the generalization of the known Rayleigh's expansion for the case of ring-shaped potentials. Since that result can be used for the construction of the corresponding phase scattering theory with the ring-shaped potentials, this expansion acquires a principal significance.

References

[1] H.Hartmann. Theor.Chem. Acta 24, 201-206, 1972.

[2] H.Hartmann, R.Schuck, J.Radtke. Theor.Chem. /1с<а42, 1-3, 1976.

[3] H.Hartmann, R.Schuck. Inter. J. Quantum Chem. 18, 125-141, 1980.

[4] C.Quesne. J. Phys. A21, 3093-3103, 1988.

[5] I.V.Lutsenko, G.S.Pogosyan, A.N.Sissakian, V.M.Ter- Antony an. Theor.Math.Phys., 83, 419-427, 1990.

[6] A.N.Sissakian, I.V.Lutsenko, L.G.Mardoyan, G.S.Pogosyan. J1NR Commun., P2-89-814, Dubna, 1989.

[7] C.Gerry. Phys. Lett. A118. 445-447, 1986.

[8] M.Kibler, P.Winternitz. J. Phys. A20, 4097-4108, 1987.

[9] L.G.Mardoyan, A.N.Sissakian, V.M.Ter-Antonyan, T.A.Chatrchian. Preprint JINR, P2- 92-511, Dubna, 1992.

[10] H.Bateman, A.Erdelyi. Higher Transcendeal Functions v.S, New York, Toronto, London. 1953. 332

SYMMETRIES AND QUANTUM MECHANICAL SPECTRA OF ANYONS Stefan V. Mashkevich

N.N.Bogolyubov institute for Theoretical Physics, 252143 Kiev, Ukraine

Abstract A system of non-interacting anyone (particles obeying fractional statistics) in a harmonic potential is considered. The general symmetry properties of tlm system are analyzed, among which supersymmctry plays an essential role. It is shown that knowledge of these properties allows one to calculate exactly the numbers of states of the three-anyon system which interpolate between bosonic and fermionic states with given energies and angular momenta.

It is well known that in two-dimensi jnal space, particles with fractional (intermediate) statistics, or anyons [?, ?] are ailowed to exist. Their defining property is that their wave function transforms as V .Ф =cxp(iV6)4' (1) where Vjk is the transformation of coordinates which corresponds to an anticlockwise interchange of particles j and k, and 8 is the statistical parameter, which can take any real value. Since S = 0 obviously corresponds to bosons and 8 = 1 to fermions, there is a continuous interpolation between these two cases for intermediate values of 5. For fractional 6, P and T symmetries are evidently broken, since interchange in clockwise and anticlockwise directions leads to different phase factors. However, since the Hamiltonian is P and T symmetric, the spectrum is invariant with respect to the change 8 —> —8. Further, by virtue of the definition (??), the transformation <5 —» (5 + 2 in fact changes nothing. Therefore one can restrict oneself to considering the interval 8 6 [0,1]. The main difficulty of anyon problems is that an anyonic wave function determined by (??) is multivalued and has to contain factors of the form {zj — г*)', where Zj = Xj + iijj is the complex coordinate of j-th particle. Such factor cannot be expanded as a linear combination of products of one-particle functions, and this means that the standard procedure of constructing bosonic (fcrmionic) functions as (anti)symmetrized such products cannot be generalized on anyons. Put it another way, a multi-anyon problem is essentially many-particle even if there is rio interaction in the sence that the Hamiltonian is a sum of one-particle ones. By virtue of the aforesaid, it is natural that only the two-anyon problem is exactly solvable — if the potential allows separation of the center-of-mass motion [?, ?] . The three-anyon problem is already non-trivial, but at the same time yet sufficiently simple to allow more or less detailed treatment. Therefore it has received considerable attention, having been studied within different approaches. Most of these approaches were Conner'ed with perturbative [?]-[?] or numerical [?, ?] methods. The only exact results available were the analytic expressions for energies and wave functions for a subset of states [?] and the symmetry of the third virial coefficient [?, ?]: a3(8) = angular momentum. Thus the qualitative picture of how the spectrum interpolates between the bosonic 333 and the fermionic ones becomes completely clear, and the only still unknown point concerns the exact S dependences of energy for a number of states. We consider the system of three anyons in a harmonic potential, the Hamiltonian being

l я = , н, = -ща- + -zjZ- (2) j=i

(m = 1 , w = 1 ; dj -л щ , dj = g|v) . The problem is to find those solutions of the Schrodinger equation HФ = ЕФ which satisfy (??). The operator of angular momentum is L = J2Li • L> = z'd> ~ z'i8'>- (3) j=i

For 6 = 0 (bosons) and 6 = 1 (fermions), the exact solution is available. In what follows it will be sufficient to consider the relative motion, since the center-of-mass motion is unaffected by change of statistics. The multiplicities of bosonic and fermionic states with energy E and angular momentum L in the relative motion spectrum are given by

2 2 S3(b/f)(£, L) = ±(E - L ) + ~ [d(E + l,3)d(L,3) + d(E - l,3)d(£,3)

-d(E,Z)d(L - 1,3) - d{E, 3)d(L + 1,3)] ± ir(£ - L,4)r(B + L,4), (4)

(the upper/lower sign refers to B/F) under the conditions |L| < E and r(E - L, 2) = 0. Here r[a,b) = a mod b is the remainder of division of a by b, and

( 1 ifr(o,6) = 0 d(a,6)={ . (5) | 0 otherwise

As S changes, energies and angular momenta of stationary states also change; so each of these states is characterized by continuous functions E(S) and L(6). It follows directly from the interchange conditions (??) that for N anyons L(S) = L(0) + in our case Ц1) = Z,(0) +3. As for energy, it has been established [?, ?, ?] that the difference £(1) - £(0) can take Tour values: +3, +1,-1,— 3. In the first and the last cases the functions E(S) arc linear, and the wave functions are known exactly- ]?], in the other two ones those functions are non-linear and not known exactly. Thus, a (E, L) bosonic state interpolates to a (E + n, L + 3) fermionic state and belongs to one of the four classes according to the value of n which can equal ±3 and ±1. (It is implicit of course that in the subspace of states with the same (£, L) one chooses the "correct" ones in the same way as in perturbation theory for degenerate levels.) Let us denote the numbers of states in each of the classes by rn(E, L). To determine these numbers, one needs four equatins for each (E,L). Two equations arc provided by the "law of conservation of states" at the bosonic and fermionic points; they read, respectively,

3 3 r+ {E,L) + r+\E,L) + r~\E, L) + r- (E,L)=g3B(E,L), (6)

f+3(£ - 3, L - 3) + r+1(£ - 1, L - 3)

3 + f~\E + 1, L - 3) + Г (Е + 3, L - 3) = g3F{E, L), (7)

Two more equations follow from the known symmetry properties of the spectrum. First, in perturbation theory it is easy to establish [?, ?] that at Fermi statistics, two states with ш

opposite angular momenta have opposite values of slopes, that is, derivatives (dE/dS)g=j. Since the linear behavior of the (+3) and (-3) states is exact, this implies the equality

r+3(E - 3, L - 3) = r~3(E + 3,-L - 3) (8)

Second, there is thesupersymmetry property. Namely, it was pointed out by Sen (?) that there exists an operator Q which annihilates some of the linear states but acting on a nonlinear state with statistical parameter S always produces another nonlinear state with same energy and with statistical parameter I + S. For this operator one has [A, Q] — 2Q so that a state coming from (E, L) at 6 = 0 to (E 4- 1, L + 3) at S = 1 turns under Q into that coming from (E,L + 2) at 6 = 1 to (E + 1, L + 5) at S = 2. Now, parity transformation turns the latter into a state corning from (E + 1, — L — 5) at S = 0 to (E, -L — 2) at £ = 1. Therefore the last equation is

f+,(E,L) = f-\E + \,-L-5). (9)

The four equations (??)-(??) and the expression (??) for <73в and gsF would suffice to calculate f"(E, L). At the same time, in order to further simplify the consideration it is useful to take into account the property of S0(2,l) symmetry and its related concept of "towers" [?, ?]. It has been shown [7] that there exists a S0(2,l) algebra formed by the Hamiltonian H and the couple of operators K+ and K. for which [//,A'±] = ±2K± and [L, h'±] = 0 takes place and which never produce unphysicul ('singular) states if acting on physical states; Л'_ annihilates some of the states while K+ does not. This leads to the following picture: All states fall into "towers" so that two consecutive members of one tower have energies differing by 2 and angular

momenta equal for all &. Mounting a tower is provided by A'+ and descending by A'_, and each tower has its lowest or "bottom" state — the one which is annihilated by A'_. Denoting by //'(/i, L) the number of bottom stales with the same characteristics as in f"(E, L) , one has the following obvious equality

6n(£, i) = r"[E, L) — f"(E - 2, L) . (10)

Correspondingly

• (П)

Thus, instead of counting (and finding) all states it is sufficient to count (and find) only bottom states. Listed below are the exact formulas for bn(E, L) obtained from Eqs. (??-??):

+3 re+ii i (E,L)- 6 £±i] + [i] + i-«w,

for 2; (12)

E' \\L ~ L'\ l-r(E,2)l , r(E',6) 6+'(£,£) = 1 . 6 4 ' 2 4

for • - 2

where Е- = E - 3r(£,2), С = 2^-2 + r(£, 2) (M)

and the formulas for fn(E, L) obtained from these by applying (??): 335

L2 + 6L + 5 1 -r(£,2) d(L, 3) 12 4 3

forO

-E2 + 6EL - Ы? + 12Д - 12L d(L, 3) - d{E, 3) г+3(£,L) = (15) 48 + 3

d(E-L- 2,4) + [l-2r(fi,2)]

for £=2 < L < E - 2;

£? + 6Д/, + 9Z>a + 16E + 48L + 48 r(E-L,4) d(E - 1,3) 48 + 8 + 3

for =f* < L < -1,

Ег + 6EL — 15L2 + 16.fi — 72/, — 96 r(£-L,4) d{E - 1,3) + 48 8 3 • (16) for -1 < L <

£'-2£Z, + £2-4£ + 4Z, r(£-L,4) 16 + 8

for £=5 <£<£-6.

In these formulas, E and L should be of the same parity, i.e. r(E — L, 2) = 0 ; otherwise, as well as if L does not fall in any of the ranges specified in the formulas, the corresponding multiplicities vanish. We do not write down the formulas for Ь~г,г~г and b~3, f-3 as they follow immediately from the obtained ones upon applying (??) and (??). Since the states under consideration are solutions of the Schrodinger equation, the obtained results should in principle follow directly from that equation. However, it is at present unknown how they could be derived that way. We will only comment that the general behaviour of the functions fn(E.L) agrees with semiclassical considerations. Formulas (??) and (??) show that "in average" greater L correspond to greater n. That is, (+3) states are at most those with L close to the maximal L = E — 2, for (+1) states L is typically lower, and so on. Consider now the semiclassical interpretation. There are three relative vectors, each of them rotating either anticlockwise or clockwise; the corresponding relative angular momentum is positive or negative, respectively. Making particles to be anyons means, in this language, adding £ to each of the relative angular momenta; so one such momentum contributes in E(S) with +<5 if positive and with — 6 if negative. The slope then would be n = 3 — 2s with s the number of relative vectors rotating clockwise; clearly, the more is L, the less "in average" is s. For a more detailed discussion, see ref.[?]. I would like to recall once more the great influence exerted on me by contacts with the outstanding physicist and outstanding man who was Yakov Abramovich Smorodinsky. I am pleased to thank G.M.Zinovjev for his attention to my work and constant support, and Diptiman Sen for stimulating discussions. 336

References

[1] J.M.Leinaas, J.Myrheim, Nuavo dm. 37B, 1 (1977).

[2] F.Wilczek, Phys.Rev.Lett. 49, 957 (1982).

[3] C.Chou, Phys.Pcv. D 44, 2533 (1991); Erratum: Phys.Rev. D 45, 1433 (1992).

[4] D.Sen, Nucl.Phys. B360, 397 (1991).

[5] A.Khare, J.McCabe, Phys.Lett. B269, 330 (1991).

[6] C.Chou, L.Hua, G.Amelino-Camelia, Phys.Lett. B286, 329 (1992).

[7] J.McCabe, S.Ouvry, Phys.Leii. B260, 113 (1991).

[8] M.Sporre, J.J.M.Verbaarschot, I.Zahed, Nucl.Phys. B389, 645 (1993).

[9] M.Sporre, J.J.M.Verbaarschot, I.Zahed, Phys.Rcv.Letl. 67, 1813 (1991).

[10] M.V.N.Murthy, J.Law, M.Brack, R.K.Bhaduri, Phys.Rev.Lett. 67, 1817 (1991); Phys.Rev. В 45, 4289 (1992).

[11] Y.-S.Wu, Phys.Rev.Lett. 53, 111 (1984).

[12] G.Dunne, A.Lerda, S.Sciuto, C-A.Trugenberger, Phys.Letl. B277, 474 (1992).

[13] D.Sen, Phys.Rev.Lett. 68, 2977 (1992).

[14] S.V.Mashkevich, Phys.Lett. B295, 233 (1992).

[15] S.V.Mashkevich, Kiev Institute for Theoretical Physics preprint ITP-93-32E (hep-th # 9306094) (1993); submitted to Phys.Rev. D.

[16] F.Illuminati, F.Ravndal, J.Aa.Ruud, Phys.Lett. A161, 323 (1992). 337

THE SNYDERS SPACE - TIME QUANTIZATION, POINCARE GROUP DEFORMATIONS AND ULTRAVIOLET DIVERGENCES. R. M. Mir - Kasimov Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia

Abstract

The quantum theory is considered in which momenta belong to the space of constant nonzero curvature. The conjugated configurational space is connected with the momentum space by the Fourier expansion in matrix elements of the group of motions of this space. The generators of the translations in the corresponding quantized configurational space are differential - difference operators and can be considered as the generators of the q— deformations of corresponding groups. In 3 - dimensional ease (corresponding to relativistic quantum mechanics) the most illustrative example is the relativistic oscillator f which coincides with well - known q- oscillator, q = r.~" '/'t™c*, In four dimensional case wc consider the Quantum Field Fheory in which 4 - momenta off the mass shell belong to the momentum space of constant nonzero curvature. In this ease the deformed character of the translations leads to radical modification of the singularities of the field - theoretical functions. As a result, the S - matrix elements do not contain the non - integrablc expressions.

1. Introduction. We consider the quantum field theory (QFT) in which the 4 - momenta belong to the space of constant curvature. This idea was expressed many years ago in (1-3], where noncommuting space - time coordinates x^ were intruduced. W.Pauli indicated [2] that these operators are

in fact the generators M4lt of the pseudo - euclidean rotations in 5 - dimensional momentum space or the generalized "shifts" of the de Sitter momentum space. This approach is maybe the oldest example of applying ideas of noncommutative geometry in physics. In papers [4 - 12] the attention was concentrated on the curved momentum space. In [7,8] a new configurational representation based on the corresponding Fourier expansion was introduced. In papers [11,12] the concept of a gauge field in the theory with curved momentum space was considered. The configurational £ - representation is canonically conjugated to the de Sitter momentum space. The geometry of this new configurational space at small distances differs from the pseudo - euclidean geometry. In particular the Poincare group generators are realized here as finite - difference operators [10]. This aspect of the ( - representation connects it closely with the theory of quantum groups [13 - 20]. The essential point here is that the finite - difference "derivatives" do not satisfy the usual Leibnitz rule, but the generalyzed one [13,17]. In the very important case of the q - oscillator the rising and lovering operators are finite - difference operators [13,20]. The noncommuting matrix elements of the basic representation of the quantum GL, (2, C) group are realized by the finite - difference operators. On the other hand, as it was stressed in [17], the quantum differential calculus is of principal importance for applying the deformation theory in physics. 338

The remarkable property of the £ - space is its "causal structure". This space splits into two irreducible regions: the time - like one (continuous), where an additional invariant, the sign of the "quantized" time, exists and the space - like (discrete) one. However in the ( - space there docs not exist the analog of the light cone, i.e. the surface, which in the usual Minkovsky space splits these two regions and on which all singularities of the QFT are concentrated. Using the natural generalizations of the causality and locality conditions in the ( - space a perturbation theory based on the extension off the mass shell in the de Sitter p - space was developed. The S - matrix constructed in this way obeys all axioms of the QFT. In fact the hypothesis of the non - euclidean geometry of the p - space is an additional axiom. In usual QFT it is accepted without sayng that the mass shell p2 - mJ = 0 (1) is embedded in the flat pscudo - euclidean p - space. In our case the 4 - momenta belong to the de Sitter space 2 pMp« _ -MVp = -AfV (2) where/: = 0,1,2,3, M b a universal constant, the fundamental mass. In what follows we put h = c = M = 1 (3)

The mass shell (1) can be embedded in the curved space (2) as well. The quadratic form (2) can also be written as Pt,pi = ffKtpV = -l, (4) K, L = 0,1,2,3,4;<ж.£, = diag( 1, -1, -1, -1,-1). In section 4 we show that in £ - space the problem of multiplication of the singular field - theoretic functions is modified in a major way in comparison with usual QFT. In the { - space we have the only singularity, the simple pole. Together with a simple rule for contour integration in £ space, which follows from geometrical considerations,this leads to finite expressions for S - matrix elements. We stress, that this modification of the structure of singularities, and absence of the ultraviolet divergences, is direct consequence of the deformed (finite - difference) character of the momentum operators in ( - space.

2. The configurational £ - representation. We introduce the generalized "interval" - <7 in the case of the de Sitter p - space as an eigenvalue of the Casimir operator S of the group of motions of this space. This is the SO(4,1) - group. We have

= (5)

where G>, = + j G = de.l\\G^\\ (6) 1 - p' and MKL = i(vK~-PL-^) (7)

are the generators of the de Sitter group. The two series of unitary, irreducible representations of the group SO(4,1) exist, which correspond to the following eigenvalues of the operator S (5):

S < f | p >= s < f | p >= a(a + 3) < £ | p > (8) 339

1. The continuous A - series a = tA - -!, 0 < Л < oo. (9)

2. The discrete L -series

In the flat limit the Л - series turns into the time - like region of the usual Minkowsky z - space (a2 > 0), the L - series into the space - like region {a2 < 0. In the A - series the additional discrete invariant (the "sign of time") also does exist. We stress that there is no analog of the light cone in the ( - space, or in other words, there is a gap between the two parts of the spectrum (9) and (10). The functions < £ | p > play the role of the "plane waves" in the theory with the curved momentum space. Let us consider them in detail in the case of the continuous 'time - like" region. We can show that in this case

where 5 - vectors Ni belong to the 5 - cone:

9klNkNl = 0 (12)

The subscript "+" in (11) means that we should treat the "plane wave" < £ | p > as a generalized function. Later on, when analysing the S - matrix elements we shall'sce that in the QFT with the curved p - space the ultraviolet divergences are absent and the problem of regularization of the field - theoretical objects (propagators, D - functions, etc...) does not arise. It looks like the problem of regularization in our approach is transferred from the dynamical quantities to the kinematical quantities, the plane vawes < £ | p >. It is important to stress that here the solution of this problem is a direct consequence of the q - deformations of the displacement operators and the requirement of the proper flat limit (M oo). Each of the poles of tile cone corresponding to positive and negative signs of the time

component N0 > 0 and No < 0 is transitive under de Sitter group transformations. We conclude that in the "time - like" A - series an additional discrete invariant, the sign of the

"time" N0, exist. The quantities < { | p > are kernels of the Fourier transformations on the de Sitter space, or the Gelfand - Graev transformation. We call the "point" of the configurational f - space the set of 4 variables £ = (л,лд, (13) where N„ is the four - dimensional part of the five - dimensional isotropic vector belonging to the "contour" Г that crosses all generatrices of the 5 — cone (12) We choose the contour Г N\=l, (14)

hence is the unit time - like 4 - vector:

N2 - if = 1 (15)

In the flat limit (p4 и 1, ph

< £ | p >= ev v ' e'^i = e ^ (16)

The dimensional units are restored in (16) , xM = AJV^ 340

3. The deformed generators of translations. The q - plane structures. An important fact is that in the £ - space the generators of translations p„ are realized as the differential - difference ( deformed ) operators. Let us consider the operators

fa =— coscosliz^h A (4e Aa» —— PiрЛ) —— -——[ coscosh hA Д si sin nU 0jN vco cos s ' K+^v cos On cos ipfi Э sinyjy д V-fe.

sinhA d0N sinh A sin On дрн I '

Pi = sinh&smONCostp — pA — -.—cosh Д sin Of, cos <рцтгг N \ / + 0Д

cos 0N cos удг д sin tpN д \

sinhA d0N sinh Д sin 0N dipn I * '

P2 = sinh AsinCjvsini^jv (e~vs — pA — —^—r I cosh Д sinfljvsin^Jr- V / [a +1) у 0Д

cosOaicos<рн д cosд \ (}g) 1 sinhA flfl/y sinh As\n0N8ipN) '

p3 = sinh Д cos — рЛ — .—L__ [ cosh Д cos0N-Д- v / (

sin Ov д \ о , v

- <20>

( ( Ld , 3 . , д Д>> Л , , P4 = + __ smh _ _ __Jj_c „ j (21)

In (21) Afj denotes the Laplace - Beltrami operator on the hyperboloid, A,0N,tpN, are hyper- spherycal coordinates of the unit 4 -vector N

N0 = cosh Д, 77 = sinh Д- m rn = (sinfljv cos ipN, sin On sin v>iv, c°s Олг) (22)

It is easily seen that (17 - 21) obey in the operator form the de Sitter condition (2). We can easily be convinced of that the Poincare algebra /50(3,1) is deformed in the £ - space. Taking as components of the position operator the quantities

*» = (23)

we come to the relation between the 4 - angular momentum operators Jut, and Lorentz group

generators Mu„ J= Xj.p,, - = M„„ (24)

• 9 •Л,2 = X\PI - x2pi = (25)

Now we are going to demonstrate the connection between the momentum operators (17 - 21) with the Manin - like q - plane structures [14]. We stress that this connection, which is trivial from the mathematical point of view, has a profound physical meaning. A simple 341 example showing the connection between finite - difference operators and the quantum plane structure is given if we realize the q - plane non - commuting coordinates by operators

x = у = ex° (26)

It is easily seen that the operators (27) obey the basic q - plane condition [14).

xy = qyx (27)

where q = ел (28) The set of linear transformations of the q - plane (27) is the quantum group G£,(2,C). Now let us consider the q - plane structures connected with the momentum operators (17 - 21) in the if -space. We consider the quantities

д д xi = cosh ij = sinh , x = cosh A

A = ^ (30)

where m is a fixed mass m «£ M.

The operators of finite - difference differentiation Xi and x2 obey the following Leibnitz rule

д д д д д sinh -jfofM = si"h ' cosh + cosh q^K0)' sinh ^sM (31)

It is remarkable that the same Leibnitz rule is valid for the q - derivative introduced by the following definition

a(q-«г1) We have = dJ{a)-cM°) + c,fH- (33) where operator c, is given by с,До-, n„\ ) =- /Ы +- ЛтМ (34,,„).

Let us note that in the finite - difference calculus the "second Leibnitz rule" for cosh ^ - derivative exists:

cosh f (a)-g(o) = cosh ^-/(

Operators i], z2, х3, obey the relations

Х1Х2 = 9+13Г1 + Xj

X1X3 = q+x 3X2 + Ц-ХцХ2 Z1X4 = q+XjXi + q-x 3X2 X3X4 = q+x 4X2 + д_ХзХ) (36)

where ?+ = g + g = cosh A, = 4 g = sinh A (37) 342

Taking into account the identity яХ-ч1~ 1 (38) we obtain the inverse relations

= + 4-х2X4 I3I2 = + 9-X1X4

X4X1 = q+xii4 + 7_T2I3

XtXi = q+Xi 14 + q-.x 1X3 (39) As a consequense of eqs. (36 - 39) the similar q - plane structure for the momentum operators pi, exists. Let us write the pi. in the form

l ) 5) PL = a l x, +Qi xs (40)

The coefficients are differential operators in Д, 0N, depending also on a. An essential

point is that operators commute with x3 and x4. Using (29,36 - 40) we have

PL.*;% = Ч+XSPL + q-x^kL,

Pl.x* =

ki.X3 = q+хзкц + q_x4pt,

fcbx4 = q+x4kL + 9_х3Рл, (41)

where the operator fci = afx, + Q^U (42) was introduced. After adding and substracting the relations (41) and passing to the operators

*(£.,±) = PL ± ^и У±=х3±X4. (43) we obtain the q - plane relations

Х(/,.±)У± = 92/±X(i,±), i(L.±)J/T = 9~'v7xlli±) (44)

4. The Field Theory We indicate briefly here the principal elements of the QFT in the quantized £ - space. Our main goal is to show the absence of ultraviolet divergences in this approach. Let us consider the scalar field ф with mass m. The free equation has the form

2(cosh p - p4)

where cosh/J = УГТт5 (46) In the flat limit eq.(47) turns into the Klein - Gordon equation. Applying the Fourier transformation we come to the scalar field in the configurational £ - space: m = (27)37* / < * 1 p > m dn> (47) On the mass shell the field (p) coincides with the scalar field of standard theory. Then we can

define the operator of 4 - momentum Pfl in a usual way:

ф{р)е~'^и'г" = e'^Xp) (48) 343

Combining (47) and (48) we arrive at the field with bilocal dependence on x and f

iP ФЛО = c ' е^ф(р) dttp (49)

It is important that f is a translation-invariant variable:

= (50)

This provides the translational invariance of the theory. The Pauli - Jordan commutation function and its frequency parts have the form

£>({) = i [«•'(€), «"(0)] = J < t I P > £(po)«(2p4 -2m4)dnp (51)

DW = ^(Й)5 / < * 1 P > 0(±ро)й(2р4 ~ 2m*)d{1P (52) The D(±)(4) - function vanishes in the discrete space - like region of the £ - space. This fact demonstrates the "causal structure" of the approach considered. Each step of calculating the integrals (51) and (52) has an analog in the conventional QFT, which we follow as close as possible. For example upon integrating the "angular" part in (52) in the continuous part of the £ - space we get D+(£) in the form

(53) where 0 i ¥>(^o.A) = ^T^° (-.V4m4 + mcosh(x-A))+ dX (54)

The operator p has the form

. . . ( .» Рг\ coshA , « 8 „ ,

In the flat limit p->2iVIsinhA^ (56) where A = xj - x2 and

(JVe, Л) -»

+ U(-\)Ka (шуГЛ) (57)

We have

D^(x) = ~^(x0,X) (58)

After differentiation we come to the explicit expression for the positive frequency part of the Pauli - Jordan function:

D«\x) = i-^xoWAJ-^OfA)^)^'1»^^)

+ в(-х0)Н^ (mx/A)] (mV^A) (59) 344

'ГЬе "worst" singularity in this expression is

^ФоЖА) that arises as a result of the differentiation in eq.(58) Let lis stress that the differential operator Jj comes from the standard realisation of the momentum operators as the infinitesimal (differential) translation operators of Poincare group. In the £ - space the momer' ч operators are deformed (are finite - difference oprators) and as a consequence of this fact the character of the singularities changes completely. To show this, let us return to eq. (53). Instead of the differentiation jj we have here the deformed momentum operator p which expresses the tp(N0,a) in terms of its values ip(N0,a ± 1) at neigbouring points a ± 1. There are no derivatives of step 0 functions giving singular terms in conventional QFT. As in the usual theory, we can complete the calculations and express D, D±, Dc and other functions in terms of Legendre functions. For example,

e{N )Г(-<7 - 2) + 0(-JV )r(<7 + 1) 2;(27T)V* 0 o ^Г© (60)

This example is generic in the sense that the only singularities we encounter are poles of the Г - function. Taking into account the spectrum of a (cf. section 2) we see that the only input can come from the point a = 1. On the other hand in the perturbation theory [10] we have the higher degrees of Z)(±> - functions. This gives poles of higher orders, i.e. integrable singularities.

References

[1] H. Snyder, Phys. Rev.71 (1947 ) 38; ibid72 (1947) 68.

[2] W. Pauli, unpublished, see footnote in the first Snyder's paper.

[3] C. N. Yang, Phys Rev.72 (1947) 874.

[4] I. E. Tamm, Collection of Scientific Worifcs, ( Nauka Publishers, Moscow, 1975).

[5] Yu. A. Golfand, Sou. Phys. JETP 10 (1959) 356; ibid 16 (1963) 184; ibid 17 (1964) 848.

[6] V. G. Kadyshevsky, Sov. Phys. JETP 14 (1961) 1340; Sov. Phys. Doclady 7 (1963) 235; ibid7 (1963) 1031; in book Problems of theoretical Physics, dedicated to the memory of I. E. Tamm (Nauka Publishers, Moscow, 1972).

[7] 11. M. Mir - Kasimov, Sov. Phys. JETP 19 (1964) 605; ibid 19 (1964) 757; ibid 22 (1966) 629.

[8] V.G. Kadyshevsky, R. M. Mir- Kasimov and N. B. Skachkov, Nuavo Cimenlo 55 A (1968) 233.

[9] A. Donkov, V. G. Kadyshevsky, M. Mateev and R. M. Mir - Kasimov, Proceedings of V. A. Steklov Mathematical Institute, CXXXVI, (1975), 85.

[10] R. M. Mir - Kasimov, Preprint (Dubna, JINR), E2 - 11893 (1978); 345

[11] V. G. Kadyshevsky, AW. Phys.В 141 (1978) 477. V. G. Kadyshevsky and D. V. Fursacv, Theor. and Math. Phys 83 (1990) 197;

[12] R. M. Mir - Kasimov, Phys. Lett. B, 259, (1991), 79.

[13] R. M. Mir - Kasimov, J. Phys. A24 (1991) 4283; E. D. Kagramanov, R. M. Mir - Kasimov and Sh. M. Nagiyev, J. Math. Phys.31 (1990) 1733.

[14] Yu. 1. Manin, Les publications du Centre ie Recherches Mathimatiques, University de Montreal,(1988).

[15] L. D. Faddeev, N. Yu. Reshetikhir. and L. A. Takhtajan, Algebra i Analisl (1989) 178.

[16] D.Levi, P.Winternitz, Journ.Math.Phys., 34, (1993), 3713-3730.

[17] J. Wess and B. Zumino, Preprint (CERN - TH - 5697/90), (1989); Preprint (KA - THEP - 1990 - 22), (1990).

[18] L.Castellani, Phxjs. Lett. B279, (1990), 291.

[19] J.Lukierski, A.Novicki, H.Ruegg, V.N.Tolstoy, Phys. Lett. B264, (1991); 331.

[20] A. J. Macfarlane, J. Phys. /422 (1989) 4581; L. C. Biedenharn ibid 22 (1989) L873.

[21] M.Arik and A.Saracoglu, Bogasici University, Istanbul, Preprint, (1993); E.Celeghini, R.Giachetti, E.Sorace and M.Tarlini, Jo urn. Math. Phys.,31,(1991), 2548 - 2551; E.Celeghini, M.Rasetti and G.Vitiello, Phys.Rev.Lett.fiO, (1991),2056 - 2059; V.I.Manko, G.Marmo, S.Solimeno and F.Zaccaria, Phys.Lett.Al76,(1993),173 - 175; M. Chaichian, P. P. Kulish and J. Lukierski, Phys. Lett. 5237 (1990) 401. R.Floreanini, V.P.Spiridonov, L.Vinet Comm. Math. Phys.,137,(1991),149; A.l.Solomon, and J.Katriel, Journ.Phys.A., 23, ( 1990), LI209 - L1212; A.delSol Mesa, M.Moshinsky, Yu.F.Smirnov, The contribution at this Symposium. 346

USING REDUCE SYSTEM FOR CALCULATION OF INTEGRITY BASES OF LIE GROUP INVARIANTS S. S. Moskaliuk

Institute of Theoretical Physics of Ukrainian Academy of Sciences, Metrologicheskaja str., 14®, 252143, Kiev, Ukraine Yu. F. Smirnov

Institute for Nuclear Physics, Moscow State University, Moscow, Russia

Abstract The algorithm and corresponding REDUCE-program for the modelling of the >ationa] integrity basis (RIB) of invariants or covariants of Lie compact groups is proposed. In the algorithm, the Molein's generation function and its generalizations are used, which give a possibility to determine the composition and structure of RIB. As a test example, the RIB of a representation of the complete orthogonal group of transformation is constructed in the three-dimensional space for an arbitrary number of vectors and symmetrical second- order tensors.

Let us give some operator set {ij}, а = 1,2, ...,[7] forming the representation basis 7 of the group G, ([7] and a are dimension and an index respectively of the representation row 7). The main problem solved in this article is an algorithm modelling and its realization in the REDUCE system for finding general expressions for all independent invariants and covariants of the G group which can be set up from the degrees of the operators Some information is given from the classical theory of invariants, which is necessary for a further insight [1]. Statement. Proceeding from the operators {tj}, it is always possible to construct a set of homogeneous polinomia.1 invariants In+k of the group G called the integrity basis of invariants and characterized by the following attributes: a) the first N invariants, called basic ones, are algebraically independent; b) any polynomial invariant I may be represented in the form

/ = Po + In+IPI + ... + rN+KPK, (1)

where {Я;} are polynomials composed only of the basic invariants N = [7], with i 6 [l,....,Jb] and К > 0. The invariants /лг+1,...,/лг+к, called the auxiliary ones, enter into expression (1) linearly (i.e. in power not higher than in the first one). Let us call the degree of a polynomial I, in the operators tj the order of invariant /,-. It is evident that the set of all homogeneous polynomial operators of the degree fI, constructed from Q, forms the basis of some, generally speaking, reduced representation Dn of the group G. This basis may be expanded over irreducible representations Or of this group

(2) г

Here, п(Г,7,Г1) are the coefficients of the expansion of the symmetric degree 7° in the series over irreducible representations of the group G. Molein's function Ф(Г,7,Л) is a generating 347 function for the multiplicities п(Г, 7, П) in the expansion (2), i.e. the coefficients of its expansion in Teylor's series over the auxilliary parameter Л

П Ф(Г,7,А) = 5>(Г,7,П)А (3) n give the quantity of linear independent covariants of the kind Г and power fi which may be composed from the operators tj. Molein's Theorem [2]. The generating function for a number of linearly independent covariants of the kind Г and power fi of a finite group G to the continuous group G, respectively, are defined by the expressions

where xr(s) ^Xr(C)^ is a character of the group element g (of the class C) in irreducible repre-

sentation Г; Dr(g) is a matrix of the element g (of the class C) in the representation

Г, E is a unit matrix, |G| is an order of the group, Vc = J dC is the volume of group.

Examples. I) In the group of rotation SO(3) a class is defined by the value of the rotation angle в (0 < 0 < 2jt) around some axis, and thus the character of the irreducible representation DL depends on this angle в and is given by the known formula

х(б) = £ = 2 = e'e- (6)

Let the volume element in the form dC — sin^^dO, then

Vc = ± J*" sin'tdO = I. (7)

As representative of the class С it is convenient to choose the rotation by the angle в around the axis Z (the axis of quantization). In this case the matrix D1 has the diagonal form with e в 1 the eigen-values = e' = 2, /i2 = 1, /13 = е~' = г" . As a result, expression (5) in the given problem becomes

(8)

The integral over в from 0 to 2ir is the integral over the circle of the unit radius with the center at the point (0,0) in the plane of the complex z

1 ЫР)= ~! 4z* (l-j>)(a-p)(l-jw)- W

The latter is reduced to the sum of residues at the points z = 0 and z = p which are located (at p < 1) inside the unit circle and are simple poles. 348

Therefore, in-the case L — 0 wc finally obtain

Consequently, the integrity basis of invariants, constructed from the vector {/,, reduces to one invariant of the second order UiUi which is the Kazimir operator L2. 11) In the previous example, we have discussed the invariant constructed from the vector values (7 = D1). Analogously, one can calculate tho generating function Фо(Л) for the invariants composed of the tensor components of the second rank = ±2, ±1,0), i.e. 7 = D2. In 2 e 2 this case, the matrix D has the diagonal form with the eigen-values: fit = c" = z , /i2 = 6 l -2 2 e'° = z, ft3 = 1, /x4 = e' — z~ , /<5 = е '" = г' and the generating function 4>o(d), like (8), is determined by the integral over the group (on 0, 0 < в < 2?r):

£ ( dz_ (1 - г)2 (1 _ rf)(l - rf,)(l - i)(l _ «feajfl - *)'

that at d < 1 equals the sum of residues inside the unit circle in the poles z = d and г

Ф.М = V «as f Zfll^m _} =

• = f (1 ~ d)2d (1 - ^/df-Jd 2(1 - d) [(1 -

(i + Vd)2(-Vd) 1 (12) (1 + d\/3)(I - + d)2Vd (1-)•

Consequently, the integrity basis of invariants composed of the components of one symmetric tensor of the second rank Q has the form

IrQ = (Q)«, (13)

2 where the matrices Ц^Ц run the set ||Q0||, ||(?0|| , ||f?;#. Ill) Analogously, one can obtain the generating function for the modelling of the integrity basis of invariants composed of an arbitrary number of vectors and symmetric tensors of the second rank [3-5]. The results of examination of this generating function at computers are given in the next table of the integrity basis invariants under the orthogonal group 0(3) for m symmetric tensors and p vcctors which consists of: a) basic invariants including only the symmetric tensors; b) p sets of basic invariants obtained by I he substitution of each of the p vectors instead of Ui in the set of basic invariants including symmetric tensors and one vector; c) p(p — 1 )/2 of sets of basic invariants obtained by the substitution of all possible combinations of two different vectors instead of (/,, (/,' in the set of basic invariants including symmetric tensors and two different vectors.

In the tables given below: ZN are the matrices Z^ is the product a of the matrices all the indices satisfy the conditions K,L, M, N,P,Q = 1,2, ...,m; К < L < M < N < P < Q. A. Basic elements including only the symmetric tensors

= (a = 1,2,...), (14) 349 where the matrices QA run the set

1) ZK, Z\, Zfc

2) ZKZL, Z\ZL, ZlZK, Z^Zl, 2 3) ZKZbZM, ZIZlZm, ZIZMZK, ZhZKZL, zKzizb, ZtZlZfc, ZMZ KZl,

4) ZKZLZMZN, ZKZLZNZM, ZX-Z^ZmZN, Z]ZmZnZk, ZI,ZnZKZL, 2 Z}/ZKZlZm, ZJ^ZLZHZM, Z}ZnZuZk, Z%ZmZKZI„ Z mZkZlZH,

Z*ZIZmZn, z%z*MzLzN, Z\ZIZLZM, ZlZ\,ZKZN, Z\Z%ZKZM,

ZhZj/ZicZt., ZI;ZLZKZMZn, ZIZkZmZNZk, Z^ZMZNZKZtj, ZmZNZKZLZK, ZKZKZLZKZM', 5) ZKZUZMZNZp, ZKZLZNZPZM, ZKZLZPZMZN, ZKZMZNZLZP, ZKZmZLZPZN, ZKZNZLZMZP, Z^ZCZMZ^ZP, Z*ZMZNZPZK,

ZhZfiZpZifZi, Z^ZpZuZ^Znf, ZpZf

ZpZkZMZLZN, Z%ZlZpZnZm, ZIZPZ^ZMZK, ZpZ^Z^ZuZi,, Z^ZMZKZiZP, Z^ZMZKZLZP, Z^ZKZIZPZS', 6) ZkZmZqZpZlZn, Zk%n%mZlZqZp, ZkZnZmZqZj Zp, 2>кZNZQZI,ZMZp, ZKZnZQZMZLZP, ZKZPZLZNZMZQ, ZkZPZMZLZNZQ, ZKZPZMZNZLZQ, ZKZPZNZI,ZMZQ, ZkZPZN Zm ZI, ZQ . B. Basic elements including only the symmetric tensors and one vector

UIUI and (<$>),,{/,-, (0=1,2,...), (15)

where the matrices Q^ run the set

1) ZKl Zh

2) ZKZL, Z*KZL, ZKZl,Z\Zl, ^

3) ZKZlZm, ZMZKZI, Z^-ZLZM, ZkZIZM, ZHZ^Z^,, ZmZ]-Zl, 2 Z lZmZK, ZlZkZIf, Z>KZIZM, Z^ZhZi, ZJZHJZK, Z%ZLZkZM, ZKZIZMZL, ZMZkZLZM; 4) ZkZLZMZN, ZMZKZ^ZN, ZMZNZKZL, ZKZMZNZL, ZNZKZMZL, ZKZNZLZM, ZKZLZMZN, ZKZIZMZN, ZkZLZ^ZN, ZKZLZMZJ,,

ZNZKZlZM, Z\ZmZnZk, Z^ZKZ^ZU, Z^ZMZ^Z},, Z^Z^Z^ZN,

ZKZIZnZM, ZmZlZkZJ,, ZnZkZ[,Zm, ZlZmZ^,Zk, Z^Z^Z^Z^,

ZNZ1ZmZk, ZMZNZIZK- 5) ZpZIZNZKZM, ZPZKZNZMZL, ZL,ZPZKZNZM, ZLZMZPZKZN,

ZMZlZpZKZN, ZKZPZlZNZm, ZhZrZMZLZfj, ZKZpZmZ^ZI.,

ZKZpZNZLZM, ZkZpZnZmZl. C. Basic elements including only the symmetric tensors and two vectors

L UM, UI{Q^)..U } + U[{QF)IJVJ, (/3= 1,2,...), (16)

where the matrices QJ,2' are given in (15) and

WV; - (7 = 1-2, -.), (17)

where the matrices Q'3' run the set 350

1) ZKZL, Zl-ZL, ZIZK, Zl-Zl, ZIZLZK, ZlZKZK, ZlZKZ,„ Z\Z\Zk, Z\Z\Zk, Z\ZIZiA 2) ZKZLZ„, Zt,ZMZn, ZmZkZl, Z\ZlZM, ZfZMZK, Z\fZKZu Z\ZMZLl ZM'ZlZK, ZIZKZM, Z^Z'^ZM, Z^ZIZK, ZKZ%,ZL, ZmZIZK, ZkZLZMZK, ZIZMZ^ZK, Zl,ZKZKZ[„ Z^ZKZLZM, Zl-ZlZM,r ZlZl,ZK,rZlZl.ZM, ZIZ\,ZL, Zl,Zl-ZL-

3) ZkZI.ZMZN, ZKZlZNZM, ZKZMZNZL, ZLZh-ZMZN, Z^ZkZNZm, ZMZKZLZN, ZiZ^ZmZy, ZKZ^ZNZL, ZMZ^ZLZki ZNZJZKZM, 2 ZmZ};ZnZ{,s ZKZ^ZIZM, ZNZIZMZK, ZI,Z mZKZn, ZNZ^ZIZM, ZKZIZMZN, ZIZIJZNZk, ZUZ%ZK%L\ 4) ZI,ZNZKZMZP, ZKZMZLZNZP, ZKZLZmZPZN, ZKZLZNZMZP,

ZkZmZPZLZH- In conclusion, it shoud be noted that in the process of realization of the algorithm described above the REDUCE-program is created which allows one to calculate all pletliysms of the given representation ol a continuous or finite group G, to research the structure of an enveloping algebra of a semi-simple Lie group G, to construct an integrity basis for the branching coefficients under the reduction of all irreducible representations of the Lie group G on its Lie subgroups /У, and also to decompose tensor products of two arbitrary irreducible representations of the Lie group into the direct sum.

References

[1] Kuznuthov G.I., Moskaliuk S.S., Smirnov Y.F., Shelcst B.K Graphic theory of representations of ortogonal and unitary groups. - Kiev: Naukova durnka, 1992.--288 p. (in Russian).

[2] Molcin T. t)ber die invarianten der liner Substitutions Gruppe //Berliner Sitzungsbcrichtc. - 1898. - 11. - S.1152-1156.

[3] Smith G.F. On isotropic integrity bases // Arch.Rat.Mccli.Anal. - 1963. - 12, N5. - P.420-425.

[4] Vcrlan A.F., Moskaliuk S.S. Mathematical modelling of continuous dynamical systems. - Kiev: Naiik.Dumka, 1988.-288p. (in Russian)

[5| Gromov N.A., Moskaliuk S.S., Smirnov Yu.F. Trees, contractions and analytical prolon- gations methods in representation theory of Lie groups and quantum algebras. - Kiev: Nauk.Duinka, 1993.-300 p. (in Russian) 351

DIFFERENTIAL FORMS AND GAUGE THEORY ON DISCRETE SETS AND LATTICES Folkert Miiller- Hoissen

Instilut fur Theoretische Physik, Bunsenstr. 9, D-37073 Gotti л gen, Germany and Aristophanes Dimakis Department of Mathematics, University of Crete, GR-71409 Irak lion, Greece

Abstract

We discuss generalizations of the notions of differential forms and gauge theory to discrete sets, discrete groups and lattices.

1. Introduction Concepts of differential geometry like differential forms and gauge theory have been extended to (in general noncommutative) algebras and, in particular, to discrete spaces [], 2, 3]. This considerably extends the possibilities to build physical models. For example, attaching to each space-time point a finite (two-point) space - in the spirit of Kaluza-Klein theory - led to a reformulation of the standard model of elementary particle physics in which Higgs fields attain a nice geometric interpretation [4]. Such models may be regarded as a step towards a theory of discrete space-time. From different perspectives corresponding theories have been suggested and developed by many authors (see [5] for example). This paper gives a brief introduction to some basic concepts of differential calculus on algebras, concentrating on the algebra of functions on a discrete set. The basic rules of differential calculus and gauge theory on an associative algebra are recalled in section 2 and specified for the case of a discrete set in section 3. A discrete set can always be supplied with a group structure which allows us to define analogues of Maurer-Cartan forms (section 4, see also [2]). The example of a finite set of N elements with the group structure Zp/ is discussed in section 5 where contact is made with Connes' two-point space gauge theory model. Furthermore, in section 6 we discuss the relation between the universal differential calculus on the ЛГ-point set and differential calculus on a 1-dimensional (periodic) lattice (as considered in [6, 7]).

2. Universal differential calculus and gauge fields on an algebra Let A be an associative complex algebra with a unit element I. With A one associates the universal differential algebra ('universal differential envelope') fi(.A) in the following way. An exterior derivative operator d maps the elements of A =: fl°(*4) into (formal) differentials which span the space П'(«4) of 1-forms as a bimodule over A. It maps furthermore r-forms into (r+ l)-forms, i.e. d : ПГ(Л) -» fir+I(.A), and satisfies

dl 0 (2.1) cP 0 (2.2) d(wu>') вы ш' + (—Vfuidw' (Leibniz rule) (2.3) 352 where ui and cj' arc r- and r'- forms, respectively. SI (A) denotes the space of all forms. Let ф he an clement of V := A" which transforms according to ф i-» U ф where U is an element of a n x n matrix group with entries in A. Then

1)ф := <1ф + А ф (2.4) with a gauge potential l -form A has the same transformation properly as ф if and only if A transforms as

A' = f/ AU~' -dUU~l . (2.5)

This implies that the field strength 2-form

F = dA + AA (2.6) transforms according to F •-+ U /'' l/~l. An involution * oil A can he extended [4] to the differential algebra via

(/i dfi••• dfrY = V" be given where V* denotes the dual module and (иф)* = V'W. If U is unitary, i.e. U1 = (/"', then ф* transforms as ф* i-> фШ~1 and the assumption (Оф)' = implies

Л1 = —A (2.8) so that A has to be anti-llermitinn. Although we are dealing with a considerable generalization of the notion of differential forms on a manifold, the basic formulae of gauge theory remain unchanged. A more subtle problem is to define an analogue of an integral for the generalized differential forms (in order to construct action functional) [1].

3. Differential forms and gauge fields on discrete sets Let M. be a discrete set. We enumerate the elements of M as ro* where the index к takes values in a subset X of the set of integers Ж with 0 6 X. A denotes the algebra of C-valucd functions on M with the usual pointwise multiplication of functions. A is a commutative, associative and unital complex algebra. With each € M we associate a function au 6 A such that xji,(m<) = by. The functions x\t satisfy the identities

£ Xt, = 1 (where I(rri) = 1 Vm € M) (3.1) к

xk xt = S„ X( . (3.2)

As a consequence of these two identities, every function f € A has an expansion /=£/("«*)**• (3-3) к The functions xt thus span A linearly over C. We choose the function 1 in (3.1) as the unit element of A which has to satisfy (2.1) (alternatively one may think of adding an extra unit element to the algebra of functions on Л4, cf [1]). (3.1) then implies

5] dxk = 0 (3.4) к 353

so that the differentials dxk are linearly dependent. The differentials dik with к ф 0 are linearly independent, however, and

df = £ dxk = Yj ~ /("">)! (3-5) к кф О shows that df = 0 if and only if / takes the same value at each element of M- (3.2) and (3.3) imply

/И = /Ии V/€.4 (3.6) which leads to the identity

fdxk = f(mk)dxk-djxt . (3.7)

The z* are real functions, i.e.,

(xk)' = xk . (3.8)

This extends to an involution of the differential algebra via (2.7). Different choices of involutions are also of interest, see section 6. Let us write a gauge potential 1-form И on a discrete set M as

A = ^2dxkAk (3.9) к with functions Ak. Inserting this expression in (2.5) and using (3.7), we find

dx dU (1 + £ x„ Ak) = Yj k W(mk) Ak - A'k U]. (3.10) к к This equation is satisfied when

xk Ак = -1 (3.11) к x A'k = U(mk)AkU- . (3.12)

Because of (3.4) the coefficients Ak in (3.9) are not uniquely determined by the left hand side of (3.9). The corresponding freedom is fixed, however, by the condition (3.11). One easily checks that (3.11) is gauge-invariant. Using (3.7) and (2.5), we have

U dxk = dxk U(mk) -dUxk = dxk U{mk) -U Axk + A' xk U(mk) (3.13) which shows that

Dxk := dxk + A xk (3.14) transforms covariantly,

1 D'xk = U DxkU(mk)' . (3.15)

For the covariant derivative of a field ф an M (which transforms according to ф i-t иф) one finds

= £ Охкф(тк). (3.16) 354

Using the Leibniz rule for d, (3.6), (3.11) and (3.1), one can express Dtp as a right-form,

00 = VкФ<1хк (3.17) Mo where

Xt v* Ф ••= Y, lM"to) Ф(,т0) - At[mk) ф(тк)]. (3.18) t

4. Differential forms and gauge fields on a discrete group Every discrete set can be given a group structure. Let G be a discrete group with group multiplication {g,g') >—» gg' and unit element e. Right and left action of G on the algebra A of functions on G are then given by (Rgf)(g') := f(g'g) and {Lgf)(g') := f(gg'), respectively. One finds that the 1-forms dx := Yl '' Vs"1 (4Л) я'

х = are left-invariant, i.e. LgniOs = я"в'я~' The 1-forms Os arc in this sense analogues of left-invariant Maurer-Carlan forms on a Lie group (see also [2]). They satisfy l llg.Og = flads,(j|1 with ad3.(ff) = g'gg'~ and the identity

= ° (4.2) s as a consequence of (3.4). Instead of (3.7) we now have

fO, = BgRsJ~6s,cdS (4.3) and in particular dj = [Or,f\- Furthermore,

df = /• (1-4) 9

Acting with d on (4.1) leads to

do, = (4-5) er

which resemble Maurer-Cartan equations. The involution * acts on 6g as follows,

(«.)' = • (4.6)

We may also introduce n'jM-invariarit Maurer-Cartan forms which satisfy similar relations.

Let us write a connection 1-form A in terms of the 1-forms B3I

Л = (4.7) 9

with functions Pr The condition (3.11) now translates into the simpler equation

Pc = -1 (4.8) 355

(where the 1 stands for 1 times the unit matrix of the gauge group). Under a gauge transformation,

1 Рд = (Яви)Р1и- . (4.9)

The field strength of A is

41 p = Eft.' + • ( °) M*

Using (4.7) and (4.6) the condition (2.8) translates into

= (Vg 6 G). (4.11)

S. Th? case of a finite set In this section we consider a finite set M of N elements. The index set I is chosen as {0,1,..,, N — 1}. M can be equipped with the group structure of Zn- Let us introduce the function

ЛГ-1 (5.1) k=a where g € € is a primitive jV-th root of unity, i.e. qN = 1. Like the ц, к = 0,..., N — 1, the set of functions y°,... also span the algebra A of functions on M. Any function / on M can be written as

/ = Х>*Л , (5.2) fc=0 <=o

Its differential is then given by

# = (5.3) k=0

Note that the differentials dy, dy1,... are linearly independent although yk depends algebraically on y. One can 'correct' this by imposing an additional condition on the universal differential algebra (see section 6). For the Maurer-Cartan forms introduced in section 4 we obtain (in the

case G — SZN under consideration)

and

к Увк=ч 0ку (* = l,...,yv-l). (5.5)

For each к ф 0 we thus have the algebra of a 'quantum plane'. The finite-dimensional representations (for q a root of unity) [8] induce corresponding matrix representations of the universal differential algebra [3]. 356

Let us consider the example of a two-point space in more detail. In this case we have q = — 1 and j/J = I which implies

ydy = —dyy. (5.6)

The field strength of an anti-Hermitian connection A now takes the form J , F=(6,) [P1 P1- 1] (5.7) where (4.8) and (4.11) have been used. The two-form (0i)J = —\{dy)2 commutes with all / e A so that the transformation law F' = UFU-1 is shared by the coefficient function of F. One can therefore build a gauge-invariant Lagrangian,

4 2 С := Tr(F' F) = (0,) Tr[P\PX - I] (5.8) where Tr denotes the ordinary matrix trace. In order to construct an action, a kind of integral is nieded, a trace tr acting on forms. Using the properties which tr has in a representation of the differential algebra (cf [3, 4]) one finds

S := trC = tr(flj) 2Тг[Ф*Ф - l]2 (5.9) where we have set Ф := P\(m0) and used /'i(mi) = Ф' which follows from (4.11). The constant tr(flj) plays the role of a coupling constant. When the formalism is extended to M x where M is a manifold, Ф becomes a field on M and (5.9) the usual Iliggs potential, a crucial observation made in [4].

6. From the universal differential calculus on a finite set to a calculus on a periodic lattice So far we have only used the general rules of differential calculus. It leaves us with the freedom to impose additional conditions to reduce the 'dimension' of the calculus, i.e. the number of linearly independent differentials. Let us consider again the example of an JV-point set considered in section 5. The relation

ydy = qdyy. (6.1)

(of which (5.6) is a special case) leads to a consistent differential calculus on the algebra generated by у [9]. Using the Leibniz rule and (6.1) we find n-l dyn = £ Ут dy = H, dy y-1 (6.2) msO

N N l N where [n], := (1 - 2 the condition (6.1) is not consistent with the *-involution for which y' = y-1. It is consistent, however, with another

involution defined by x*k = хл>-* for which у* = y. For N — 2 both involutions coincide. Applying d to (6.1) leads to (dj/)2 = 0. After the reduction, the differential of a function / is given by

df = dyMLiM . (6.3)

which involves the so-called q-derivative. The reduction iias led us from the universal (/V — 1)- dimensional differential calculus to a 1-dimensional calculus (on a 'q-lattice'). 357

In [6, 7] a certain deformation of the ordinary differential algebra on a manifold has been considered. In one dimension, the deformation can be expressed in the form

[X,dX] = dXa (6.4) where X is a coordinate function on R and a is a positive real constant. It turned out that this differential calculus could be restricted to a lattice with spacing a. In terms of the new coordinate у = qz^a with q 6 С , q not a root of unity, the commutation relation (6.4) is transformed into (6.1) (see [6] for details). If 7 is an TV-th root of unity, we are considering a closed (periodic) lattice of N points instead of a lattice on the real line. Field theories were formulated with the higher-dimensioned generalization of (6.4) in [7]. An integral is naturally associated with this differential calculus. The action of lattice gauge theory can then be recovered from the usual form of the Yang-Mills action [7]. Besides the 'complete' reduction to a 1-dimensional differential calculus, there are reductions of the universal differential calculus to higher-dimensional calculi (provided that N is large enough) [3]. It is interesting that in this way theories formulated on various discrete structures and lattices may be viewed as arising from a universal theory, i.e. a theory formulated on the universal differential algebra. Acknowledgment. A. D. and F. M.-H. are grateful to the Hc.acus-Foundation and the Universitatsbund Gottingen, respectively, for financial support.

References

[1] A. Connes Publ. I.H.E.S. 62 (1986) 257; R. Coquereaux J. Geom. Phys. 6 (1989) 425

[2] A. Sitarz, preprints TPJU-7/92, TPJU-18/92, Phys. Lett. 5 308 (1993) 311

[3] A. Dimakis and F. MuIler-IIoissen, preprint GOET-TP 33/93

[4] A. Connes, in The Interface of Mathematics and Particle Physics eds, D. Quillen, G. Segal and S. Tsou (Oxford, Oxford University Press, 1990), p. 9; A. Connes and J. Lott Nucl. Phys. В (Proc. Suppl.) 18 (1990) 29

[5] V. Ambarzumian and D. Iwanenko Z. Physik 64 (1930) 563; A. Ruark Phys. Rev. 37 (1931) 315; H. Snyder Phys. Rev. 71 (1947) 38; D. Finkelstein Phys. Rev. 184 (1969) 1261; R. Feynman Int. J. Theor. Phys. 21 (1982) 467; H. Yamamoto Phys. Rev. 30 (1984) 1727, 32 (1985) 2659; L. Bombelli, J. Lee, D. Meyer and R. Sorkin Phys. Rev. Lett. 59 (1987) 521; G. 4 Hooft Nucl. Phys. В 342 (1990) 471

[6] A. Dimakis and F. Muller-Hoissen Phys. Lett. В 295 (1992) 242

[7] A. Dimakis, F. Muller-Hoissen and T. Striker Phys. Lett. В 300 (1993) 141, J. phys. A 26 (1993) 1927

[8] A. Morris Quart. J. Math. Oxford 18 (1967) 7

[9] Yu. Manin, Bonn preprint MPI/91-60 358

ON INTEGRABIHTY OF GENERALIZED TODA LATTICE IN QUANTUM DOMAIN AND SELF-DUALITY EQUATIONS FOR ARBITRARY SEMISIMPLE ALGEBRA M. A. Mukhtarov1 Laboratory of Theoretical Physics, JINR, Dubna, Russia E-tnail:murad@thsunl .jinr.dubna.su

Abstract The generalization of the t'Hooft's At solution of self dual Yang-Mills fields for every semisimple Lie algora is found. It is shown on the example of two-dimensional generalized Toda lattice that semisimple Lie algebras uf the classical problem reduce in quantum domain to corresponding quantum algebras.

1.Self-dual equations are the systems of equations for the parameters of a group element G considering as the functions of four independent arguments z,z,y,y (see for example [1])

1 (ОгС-')г + ((?гС- )=0 or (G->G,)-+(G->Gy)- = 0 , (i)

where Gt = d,G. The system of equations (1) can be partially solved

GrC"1 = /„ , GjG"' = -f,

where the element / takes values in the algebra of corresponding group. System of equations on / has the following form Лг+ /»» + !/»,/.) = 0 (2) As a starting point we take the self-dual equation in form (2) which will be solved by interactions introducing a small parameter q at the quadratic term in (2). For zero approximation we have & + У&-0 with the solution f°= JF°Wd\ = J F°(y + Xz,z-\y,X)d\

Here F° is an arbitrary function of its arguments. The terms of the first approximation are subjected to the equation

Л. + & + [/?,/?] =0

and its solution has the form fl = k lF*Wdx

lOn leave of absencc from Institute of Mathematics and Mechanics, 370602 Baku, F.Agaev str.9, Azerbaijan 359

Continuing the reduction process we find that

^-orhi./'-w"

satisfy the system of equations for the terms of n-th approximation

Лг + fb-+ £[/,. /Г'"°] = 0 QSO

Multiplying Fn (A) on gnand introducing the notation

a=0

we find that F satisfy the following singular integral-differential equation with the Cauchy type kcrnal ) dA) (3)

Thus [2], we see that every solution of (3 ) with the initial condition

F |»=c= F°(y + Xz , z — \y,

leads to the following solution of self-dual equations (2)

'-J/*/F{X, gi)dX\,=l

In [3] it is shown that the solution of nonlinear equation (3) can be found from the linear equation

*(A) = Ita„(.n(l + /^), (4)

in terms of which the solution of self-dual equation is expressed as follows

/ = J 0(A)dA

The integration of (4) is equivalent to the following homogeneous Riemann problem [4]

exp (2rrif° (A)) G* = G"

Here G± are boundary values of the function G taking values in a group. It are analytic, respectively, inside and outside the integration contour. Chosing the special boundary conditions we solve this Riemann problem and obtain the wide class of solutions including the known t'Hoofts solution

(

2. In quantum case let's consider the generalized Toda lattice

which can be obtained from self-duality equations by symmetry reduction [5]. Calculation of Heizenberg operators can be done on the basis of finite half 5-matrix and operators exp(—X;) are polinomials over g2. The same expression for Heizenberg operators can be obtained from the classical formular [6]

exp (—Xi) =< г I exp £ M+MZ1 exp £ /i^jj | i > , where y>f (z±) arc the asymptotic field operatos, M± satisfy the equations of S-matrix type

± M±,I± = M±L± (

[x*, х-}=, [л,, xf]=±Kijxf where К is the Kartan matrix of a simple algebra, t is a numerical parameter. When t —» 0 the quantum relations turn into the corresponding formulars of the simple Lie algebra and the quantum solution of generalised Toda lattice to the classical one.

References

[1] Yu.I. Manin (ed.), Geometrical ideas in physics, Mir. Publ..1983,Moscow.

[2] A.N. Lcznov, M.A. Mukhtarov, J.Malh.Phys., 28, 1987.

[3] A.N. Leznov, M.A. Mukhtarov, Prepr. 1HEP 87-90, 1987, Serpukhov.

[4] A.N. Leznov, M.A. Mukhtarov, Prepr. ICTP 163, 1990, Trieste, Italy.

[5] A.N. Leznov, M.V. Saveliev, Group methods for integration of nonlinear dynamical sbstems, Birkhauser-Verlag Publ., Basel, 1992.

[6] A.N. Leznov, M.A. Mukhtarov, Teor.Mai.Fiz.,71, N1, 1987. 361

CLASSIFICATION OF q-POLYNOMIALS AS POLYNOMIAL SOLUTIONS OF HYPERGEOMETRIC TYPE DIFFERENCE EQUATIONS A. F. Nikiforov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences 12504?, Miusskaya Square 4, Moscow, Russia

Abstract Рог the most general case of hypergeomctric type difference equations on uniform and nonuniform lattices the explicit expression of polynomial solutions in terms of generalized 9-hypergeometric series is obtained from the difference analog of the Aodrigues formula. From this expression, the formulas for all particular cases are derived by an appropriate choice of parameters and by taking various limits. Basic hypergcoinctric series arc also used. The consideration of these polynomials gives us the classification of corresponding ^-polynomials in accordance with the values of parameters q,/i of the lattice x(s) = ci9* + ciq~' + сз = c\(q' + q'''1*) + С3, = ci/cj, and with the number of zeroes of the function <7(3) that is the coefficient before the senior difference derivative in the equation of hypergeomctric type. All earlier introduced q-polynomials are included in our scheme of classification. This report is a brief version of the article which will be published in journal "Integral Transforms and Special Junctions", Gordon and Breach Science publishers, N3, 1993. The work is fulfilled under financial support of Fundamental Research Fund of Russia (N 93-012-352).

1. The difference equation of hypergeometric type. The Rodrigues formula. In different fields of physics and mathematics the wide use is made of quantities which are determined on a discrete set of argument values. As an example, we may refer to Clebsch- Gordan and Racah coefficients. It has been proved that many of such quantities may be described with the aid of so called classical orthogonal polynomials of a discrete variable which satisfy the difference equation of hypergeomctric type proposed in [1] (sec also [2], §12, §13):

o-[x(s)] + Ay(s) = 0. (1) Ax{s - 1) Vi(s)j 2 "|Aa:(s) Vi(i)

Here &f(s) = f(s + l)-f(s), Vf(s) = f(s) - f(s - 1), (2)

l u x(S) = c1q' + cIq- + c3 = ci(q' + q-" ') + c3 (q = (3)

A general theory of polynomial solutions of Eq.(l) was first constructed in [1] by generalizing the method applied earlier to differentia! equations of hypergeometric type [4,2]. For certain values of A = A„ we can obtain the expression for polynomial solutions yr.(s) = j/n[(x(s)] {y„(x) is a polynomial of a degree n in x) in the form of the difference analog of the fiodrigues formula

*.(«> = ^п'Ы*)], (5) where p(s) satisfies the equation

V V<™> f(s) Z

к " =x(s + -)\ pn{s) =p(s + n)JJ(r(s + i), B„ = const. i=i

2. The representation of polynomial solutions in the form of generalized q - hypergeometric series. Using the Watson and Sears transformations let us reduce the Rodrigues formula to the form [3, p.141]

A \n ( 2 Bn Sl + 32 + + s + x Ciq-»HK ) ^ * fW'i + +

V Sl + S2 + /t,Sl +s3+/l,s1 + s4+fl I ' Here we introduced the generalized q-hypergeometric series

_ f ai,aj,...,a?+i I \

ь„ь2,...,ьР M~ (a|,) (a+ (8) к=0 Ш1М/' * - Er'

When q 1 the expression (a|g)* transforms into (a)t. Therefore it is obvious that when q —> 1 the 9-hypergeometric function (8) transforms into the hypergeometric function p+iFp(z). For the polynomial yn(s) the series contains only (n +1) first terms for which к = 0,1,..., n, since di = —гг. From (7) it is seen that with the given lattice function x(s) determined by

parameters q and (i the form of polynomial yn{s) depends only on the zeros s,- in the function

[7], Stieltjes and Wigert [8,9], Pollaczek [10], Karlin and McGnnpf Ul], Al-Salam and Carlitz [12], and the Askey-Wilson polynomials [13-17], introduced by means of the basic hypergeometric series (see also [18-21]). These different polynomials can be obtained from (7) or from formulas that are equivalent to (7) by an appropriate choice of parameters q, (i, s; and by taking some limits. For example, the Askey-Wilson polynomials can be obtained from (7) by setting is x(s) - cos в, i.e. q' = e , c\ = c2 = 1/2 (/i = 0).

3. The scheme of classification of polynomial solutions. 3.1 The classical case q = 1. 3.1.1. The expression for y„(s) on the lattice (3a) (cj = c\q~u^K2, c-i = /ici), corresponding to q = 1, when the number of zeroes of cr(s) is equal four, may be obtained from (7) by taking the limit when q-* 1:

•«1 +SI + FI,SI + S3+ FL,S\ +S4 + M

Particular case of formula (9) is the formula for Racah polynomials ui,a,e,[i(s)] [2, pp. 160,161]

at ji = 1, C! = 1, c3 = 0, = a, s2 = — b s3 = fJ — a, s4 = b + a. On the Scheme enclosed below the polynomials, defined by equations (7), (9), and the Racah polynomials arc situated on the upper horizontal level (line), which corresponds to four zeroes of the function o(s). 3.1.2. In order to derive from (9) limit cases, when the number of zeroes of using the limit transition —» oo, where A = — l/s<:

у (s) = (-l)"^(s, + 52 + p) n(si + S3 + n c;n •i1

when ц = 1, ci = 1 [2, §13]. In the Scheme enclosed the considered polynomial families are situated on the second, third, fourth and fifth horizontal lines (counting from above) to the right from vertical line drawn downwards from the rectangle, which corresponds to polynomials

(9) of general form at q = 1, where ц and st (i = 1,2,3,4) are given numbers.

3.1.3. When q = 1, it remains to consider the limit transitions at ft —• oo. If Ci = 1 /р, сг = 1,

c3 = 0, then function x(s) coincides in the limit with function x{s) = a. When x(s) = a, function

zeroes 5j and s2 of function <т(я) and two zeroes Si and 32 of the polynomial

of the second degree with equal leading coefficients: cr(s) = Д(я — st)(s — s2), c(s) + r(s) = A(s — -Si)(s — si)- This formula can be derived at p —» oo from general case of quadratic lattice (3a), if one chooses constants A = A[p) and a,- = Si(p) in formulas

3.2.1. The expression for yn(s) with q ф 1, when ft is finite and the number of zeroes of a(f) is equal four, gives us formula (7). Particular case of formula (7) is for example, Askey- Wilson polynomials. In the Scheme enclosed Askey-Wilson polynomials are situated in the first horizontal line to the right of most general case. 3.2.2. The lines drawn downwards and to the left from the fundamental rectangle bring us to the rectangles which correspond to the cases when function 0. It is clear that (7) enables us without difficulties to fulfil the limit transitions at q -* 1, because lim,_i фя(я) = л, lim,_i(a|g)/c = (a)*. On the other hand w hen q is fixed (q ф 1), it is natural to rewrite (7) in terms of basic hyperhcometric scries [21]

ai,a2,...,ar 1 _ rfp б.л,...,^'9'2] -

r+ ..у [/ ntnt(t-lW3l''~ ' { ~ f^{buq)(b2,q)k...(bp,q)t{q,q)k[ " I '

— a where (a,q)k = П«=о(1 ?')> (a, 0 In fact, instead of (7) we obtain the equivalent formula

yn{s) =

« [ (7a) TVrq't + 0 we can consider the corresponding cases without difficulties. For example, at 2 s — s we q*< —* 0, A — — l/i/>,(54),a(s) = q'/ . П?=1 0ч(- <) obtain the dual q-Ilahn polynomials All these cases are considered in [22]. 3.2.3. When q ф 1 it remains to consider the limit transition at д —» oo. The case of —t 0

brings to lattice function of the form i(s) = C\q' + c3 and the case of q" —» 0 brings to x(s) =

c\q~' + c3. Let us consider only the first limit transition at q~" —» 0, because the limit transition

q" —> 0 reduccs to first if one substitutes l/q for q and C| for c2 [22]. In the most general ease the functions it(s) and

expressions can be derived from the general case of nonuniform lattice x(s) = c^q* + c2q~' + С3

at q-" -» 0, if si(fi) = Si, s2(ft) = s2, -s3(p) = -3i - H, Mf) = ' A[fi) =

к4д-д-(а1+п+«+й)/2Д Да a result one derives the following expressions for y„(s): / -n _»i+«j-ii -ij+ti-1 „31-3 4 9 + Упм = Dn зч>2 ( ' ' ;<Г" ' 365

а When 5, = 0, з2 = N + a, Si = -ft - 1, s2 = N - 1, A = —1/к , с, = 1/к, с3 = -l/к, this formula coincides at q —» 1 with that for Hahn polynomials. That is why it is natural to call the polynomials derived at the given values of parameters 17-Hahn polynomials. There are also possible other cases when polynomials a(s) and a(s) + r(s)Sx(s — i) may have degree which is less than two, and cases when the number of finite zeroes is less than degree of polynomial. Using the corresponding limit transitions it is possible to derive twelwc expressions for yn(s) in terms of basic hyperheometric series [22]. The considered families of polynomials are situated in the right lower corner of the Scheme, in the rectangles connected by lines with rectangle corresponding to the fundamental case at ц = oo when q ф\. The rectangles situated in the third lino from above correspond to the case of two finite zeroes of the function

References

[1] A.F.Nikiforov, V.B.Uvarov: Classical Orthogonal Polynomials of a Discrete Variable on Nonuniform Lattices; Preprint No.17, Keldysh Institute Appl. Math., Moscow (1983) [in Russian],

[2] A.F.Nikiforov, V.B.Uvarov: Special Functions of Mathematical Physics (Nauka, Moscow, 1984; Birkhauser Verlag, Basel, Boston 1988).

[3] A.F.Nikiforov, S.K.Suslov, V.B.Uvarov: Classical Orthogonal Polynomials of a Discrete Variable, (Springer-Verlag, Springer Series in Computational Physics, Berlin, Heidelberg, New York, 1991).

[4] A.F.Nikiforov, V.B.Uvarov: About one new approach to the construction of the theory of special functions, Math. Sbornik, 98(140), N4(12), (1975) [in Russian].

[5] A.A.Markov: On Some Applications of Algebraic Continued Fractions (Thesis, St. Peters- burg 1884) [in Russian].

[6] P.L.Tchebychef: Sur ^interpolation des valeurs equidistantcs (1875), in liuvres, Т.П. (Chelsea, New York, 1961) pp.219-242.

[7] L.J.Rogers: Proc. London Math. Soc. 24, 171, 337 (1893); 25, 318 (1894); 26, 15 (1895)

[8] T.J.Stielljes: Recherches sur les fractions continues (1894-1895), in Oeuveres, T.2. (No- ordhofT, Groningen 1918) pp.398-566.

[9] S.HfyerJ:Arkiv for Matem.,Astron.och Fysik 17, 18 (1922-1923)

[10] F.Potlaczek: Compt.Rend.de l'Acad.Sc. Paris 230, 1563 (1950)

[11] S.Kariin, J.McGregor: Trans. Amer. Math. Soc. 85, 489 (1957) Збб

SCHEME Classification of polynomial solutions difference equations of hypergeometric type

t = l,fl = l 4 = Si =a,S2 = -b,S3 = 0-a,S( = b + a Si(l,2- 3,4) Racah

(M — oo)

4 = 1,ц S,(1,2,3)

Dual Hahn

Si = O.Sj = N Si =0 ,Si = W + a q = = oo 4 = Up St = -I,S2 = JV — I S, = -/?-!, Si = N- s,(i,2),si(i,2) $(1,2) Chebyshev Hahn (r(s) —const)

Si =0 0 = 1,/i = oo Si =0 q = 1,/i = oo S, : q = lp Si = —7 5,(1),S,(l) S,=N S((1),S((0) Sill) Meixner Kravchuk Charlier (T(S) =const) ij = 1, pt = oo S,(0).5((l) S,(0) (No new polynomials) (r(«)=conat) 387

z(s) = cos в (?J = e")i ц = 0 x(s) = + = 9. Я 1 Sill,2,3,4) 4 =a,4 •b, q" •b'l = — — fc,5a = /7 — a,S| : b + a Askey-Wilson q-Racah

(M —

S,(l,2,3)

Dual q-Hahn JVl ЛГ2

Si = 0, S3 = M + a ?, /j = oo 2 1 2 0 Si{ 1,2) Si = -0-l,S, = N-l а(1,2),а(1,2) S,(l,2) S,(l) Л(1,2) 5,(0) q-Hahn Al-Salam —Carlitz NS Г N4 ЛГБ J J 2 2 2 2 2 1 1 2 SiW S.-(l) S.(l) Si(l) 5,(0) Si(l) 5.(0) 5i(l) £(1,2) q-Meixner q-Charlier cj^Kravchuk ЛГ8 1 N9 JV10 I ЛГЦ i ^^ N12 4,1' 2 2 2 2 2 1 1 2 1 2 0 2 5,(0) S.(0) 5,(1) Si(0) S,(0) S.(0) S,(0) Si( 0) 5j(l) S.( 0) 5i(0) 5i(0) 3,(1,2) Stidtjes- -Wigert 368

[12] W.A.AI-Saiam, L.Carlitz: Math. Nachr. 30, 47 (1965)

[13] R.Askcy, J.A.Wilson: A set of orthogonal polynomials that generalize the Racah coefficients or Gj-symbols, SIAM J. Math. Anal. 10, N5, 1008 (1979).

[14] J.A. Wilson: Hypergeometric series, recurrence relations and some new orthogonal polynomials,Thesis,Univ.of Wisconsin,Madison, (1978).

[15] R.Askcy, J.A.Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.- Memoirs Amer. Math. Soc. 319 (1985).

[16] C.E.Andrews, R.Askey: Classical Orthogonal Polynomials, in Polynomes orthogonaux ct applications, Lecture Notes in Mathematics, Vol 1171 (Springer-Verlag, Berlin 1985) pp.36-62.

[17] J.A.Wilson: Some hypergeometric orthogonal polynomials. - SIAM. J. Math. Anal, 11, N4, 690 (1980).

[18] W.A.Al-Salom, M.E.II.Ismail: J.Math. Anal. Appl. 55, 125 (1976).

[19] W.llahn: Ubcr Orthogonal I'olynome, die 7-Diffcrenzen- gleichungen Genugen.- Math. Nachr.2, 4 (1949).

[20] W.llahn: Uber Orthogonal polynome, die Linearen Functionalgleichungen Genugen, in Polynomes orthogonaux et applications, Lecture Notes in Mathematics, Vol 1171 (Springer, Berlin, Heidelberg 1985).

[21] G.Gaspcr, M.Rahman: Basic Hypergcomctric Series, (Cambridge Univ. Press, 1990).

[22] A.F.Nikiforov, V.B.Uvarov: Polinomial solutions of hypergcometric type difference equations and their classification, Integral Transforms and Special Functions, Gordon and Breach, N3, 1993 (to be published).

[23] T.H.Koomwindcr: Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials, Centrum voor Wiskunde en Informatica, Department of Pure Mathematics, Report PM-R8703, June 1987. 369

BV-GEOMETRY ON KAHLERIAN SUPERMANIFOLDS A. P. Nersessian

Laboratory of Theoretical Physics, JINR

Abstract The basic objects of BV-formalism -antibrackets and operator Д on Kahlerian supermanifolds are constructed. The connection between the solutions of В V master equation and characteristic classes of basic manifold is established. 1. Introduction Batalin-Vilkovisky formalism (BV- formalism) is the most general method of quantization of gauge field theories [1]. In Witten's paper [2]it is demonstrated the necessity of an investigation of its geometry for the construction of the background independent string field theory. The realization of this program was started in [3], It is known that BV-formn!ism use the unusual structures - an odd Poisson brackets (antibrackets) and so-called "operator Д". Since the goal of BV- formalism is to solve the master equation which is formulated by the use of antibrackets and the operator Д then it is important to understand the geometry of supermanifolds provided by the odd symplectic structure. In [4]-[6] the odd symplectic structure and operator Д were constructed on the supermanifolds associated with the tangent bundle of Kahlerian manifolds ( these supermanifolds are naturally provided by both even and odd Kahlerian structures). Thus, on such supcrmanifolds the BV-formalism can be considered. In the present paper we'll study some properties of BV-formalism and its "master equation" on the Kahlerian supermanifolds. We will show that the "partition function" in BV- formalism is independent with respect to the choose of measure and defined by Daulbault characters of the basic manifold. For the simplicity we assume that the basic manifold is finite-dimensional one. Of course, such case has no straight operator interpretation . However, it clarifies the geometric meaning of BV master equation and can be used for the developing of more realistic cases. The paper organized by the following way: In Section 2 we propose the invariant definitions and basic properties of antibrackets and operator Д on supermanifolds and formulate the main theorem of BV-formalism. In Section 3 we give the definition of an odd Kahlerian structure on the supermanifold and construct the corresponding antibrackets and operator Д on it. We consider the BV- master equation and connect its solutions with the characteristic classes of basic Kahlerian manifold.

2. BV-Formalism In this section we propose the basic structures of BV-formalism and its main theorem. Odd Poisson bracket (odd bracket, antibracket) of functions f and g on the supermanifold M is the bi-linear differential operation 370

which satisfy the conditions

P({/,flh) = p(/) + p(fl) + l (grading condition) (2.2) {/,

(_l)W/)+i)(p(A)+i){;i{ffiA}1}] + cyci.pcrm.(/>S,/l) = 0 (Jacobi id.). (2.4) Here xA are the local coordinates of Л4, -jj^x and denotes right and left derivatives correspondingly. If M. has an equal number of even and odd coordinates then the odd bracket can be nondegenerate one. Then one can corresponds to it the odd symplectic structure. Locally nondegenerate antibrackcts can be transformed to the canonical form :

where {х',в{) are appropriate local coordinates (Darboux coordinates) (p(x') = 0, p(0,) = 1),

dRxA p{i) = ^«(iHBer-gp-. (2.7)

On such supcrmanifold one can invariantly defines a second order differential operator, so-called " operator A", which plays the essential role in BV-formalism [7]. Its action on function f(x,0) is the divergence of the Hamiltonian vector field D/ = §^к{хА, /{1,0)}^ with the volume form dv:

(2.8)

where is the Lie derivative along D^ . In coordinate form: 1 f)H ^-jJEaM*4''».)- (2-9) Operator Д has no analogue with even symplectic structures. The oddness of the antibracket (2.1) leads to the nontrivial grading of Д, and the "anti- symmetricity" condition (2.3)to dependence of second derivatives. From the Leibnitz rule and Jacobi identity (2.4)it follows, correspondingly

(-l)p0"{/,sh = A(/S) - /49 - (-1 )pW(4/)<7, (2.Ю) MLah = U^gh + (-1Г<1,+,{А/,5},. (2.11) The density transformation rule (2.7) implies for the operator Д the following transformation rule under canonical transformations:

Д'/ = Д/ + ^{ logj,/},, (2.12)

where J is the Jacobian of the canonical transformation of the odd bracket, Ы is the operator Д in new coordinates. 371

Let us write also the following useful expressions:

д 1(g) = /'ЫД9 + ~ГШч,д}и (2.13) where f(g) is an even complete function, and g is an even function on Л4. However, the operator Д in BV-formalism has to satisfy the nilpotency condition

Л* = 0, (2.14) which is violated for arbitrary p[x, 0). For example, if the symplectic structure is canonical one, then (2.14) holds if p(x,9) satisfies the equation

A""Y/P = 0. (2.15)

If the nilpotency condition for operator Д holds, then it can be reduced to the canonical form [8]

1=1 It was shown in [8], that an odd symplectic structure can be defined only on the supermanifolds associated with cotangent bundle of manifolds , i. e. on the supermanifolds which can be covered by local coordinates systems (x\ ®i) connected by transition functions:

- Л дх' x'=i'(i) (a), = (b) (2.17) i=i where (2.17a) are transition functions of the basic manifold. Obviously, if on the basic manifold the symplectic or Riemannian structures are defined, then we can describe the odd symplectic structure in terms of coordinates associated with coordinates of tangent bundle of basic manifold. Batalin-Vilkovisky formalism is based on the following theorem [1], [8]: Theorem. If M is the supermanifold provided with the odd symplectic structure and with the nilpotent operator Д and L is the lagrangian submanifold on it, then the integral

JL HdX (where d\ is the induced volume form on L, H = exp(S)) is independent with respect to smooth deformations of L if Я satisfies the master equation AH = 0. This integral vanishes if H = ДR.

Thus, the aim of BV-formalism is to study the even cohomologies of the operator Д.

3. Master Equation on Kahlerian Supermanifolds As we noticed in the previous Section, the odd symplectic structure can be constructed on the supermanifolds associated with the tangent bundle of any symplectic or/and Riemannian manifold. Thus, we can construct the odd symplectic structure on the supermanifold associated with the tangent bundle of Kahlerian manifold. Moreover, we can construct on this supermanifold the even symplectic structure and, as this supermanifold is complex one, also the even and odd Kahlerian structures. Let M be a complex supermanifold and zA are local complex coordinates on M. A symplectic structure Q" , where к = 0(1) if the symplectic structure is even(odd), on M is called Kahlerian if in local coordinates zA it takes the following form:

A B +i K A B ГГ = -i(-iyl 1M ^« 1g ABdz Л dz , (3.1) 372 where k 9 aB = рШ =рА+рв + к. Then, there exists a local real even (odd) function A'"(z, z) (Kahlerian potential), such that

dL dn

The following Poisson bracket corresponds tofi"

{/,*>. = > - (-D^'^^'^ll) • (3.3) where !liB9ec = 4 - 9iB = (-1)<^>+»><»<в)+»>,,?*. It satisfies the conditions of reality and "antisymmctricity"

{/,*}« = {/,»}«, {/,ff}K = -(-l)W)+")(i""+,"{5,/}» (3.4) and the Jacobi identity :

(-1 )МЛ+«)МЧ+«>{/| {g, + cycl.perm.(/,S,M = 0. (3.5)

Let Л4 is associated with the tangent bundle TM of the Kahlerian manifold M, and zA = (a/",a") are the local coordinates on it,

ш" =wa{w) (a),5» = £g

(here and further a = 1 ,...ЛГ, N = dim^A/) . Let

be a Kahlerian metric on M, К is its Kahlerian potential. Then the local functions

1 Л'о(ш,ш,о-,ст) = Л'(ш,й) + /'(1511(И,Й)Л ), (3.8)

Л,(ш,ш,о-Г' / - , <х-ч) = дК(ы,й>) . дК(ш,й>) „ (3.9)

(where с is an arbitrary complex constant, F(x) is a real function such, that F'(0) ф 0) correctly define on ЛЛ the even and the odd symplcctic structures (this is not the most general form of Kahlerian potentials on such supermanifolds [4]): 0 b c П = -i(L „gbedw° Л du + iW^guDo* Л Da') (3.10) О1 = -^(н^'ЛШ'-Ш'лД/) (3.11) where Da" = da° + Г?У dwe ,

b 1 b b Li = t a + iF'{r)Rlia

a b It is easy to see that in the coordinates (w ,0a = igaid ), in which M is associated with T'M, the odd Poisson bracket takes the canonical form. The operator Д corresponded to (3.13) lakes the form (i dL i dL\ 1

It is nilpotent if VaP = 0, (3.16) or, in other words, if p is a characteristic class of M:

P = p{r,ri,r2,:-,rN) where c d r = rt = tr(ikeia a )\ k = \,...N (3.17) correspond to Chcrn classcs of M. In this case

(3-18)

Because on Kahlerian supermanifold, considered above, exists the even Kahlerian structure (3.10), we can choose on it the natural density which is invariant under even canonical transformations:

„(«.«.a,*) = -/ВегП(0)лй = (3-19)

which, obviously, satisfy the equation (3.16). The function F can be obtained from the symmetries of considered supermanifold, for example, from the Hamiltonian reduction [5]. In the case, if basic Manifold is complex projective space СЯ(ЛГ), F = log(l + x), thus />=(l + r)2. Obviously , V" J^r corresponds to the operator of the covariant divergency S = *d* on M with some effective weight. Then, the even solution of "BV master equation" (even cohomologies of operator Д)

Де5 = 0

in this case is following

; = _L + + c.c.) j ,

where S0 is an arbitrary even function on M and //Jj((M) is 2&-th Daulbault cohomology class of M (which coincide with the Chern cohomology class on Kahlerian manifold). Therefore, if Lagrangian submanifold Lt is (k.2N — fc)fl-dimensional one, and

k w k dX = ^d yd - V,

then the "partition function"

s k 2N k ZW= f e d\= f (cII?(M) + c.c.) d yd - 4. (3.20) JLk JLk '374 is p- independent and is defined by the Лг-th characteristic number , bj, arbitrary constants and boundary conditions of Lk, where 6* is the fc-th Betti number of M. Thus, can be nontrivial only on the Lagrangian sub(super)manifolds with an even-dimentional basic manifold. Of course, in our consideration we maximally simplified our problem considering -he finite- dimensional Kahlerian manifold. For the obtaining of more correct results it is necessary to consider the presented constructions at the level of operators symbols. However, wc wish that presented results give some intuitive, "geometrical" explosion of BV master equation.

References

[1] Bataliii I.A.,Vilkovisky G.A. - Phys.Lett., 102B (1981), 27 ;Phys.Rcv., D28 (1983), 2567

<2] Witt

MI E. -Phys. Rev. D46 (1992), 5446 l ach B.-Nucl. Phys. B390 (1993) ,33

• fiian A.P. - Theur Math, Phys., 96 (1993) No. 1 , 140

Khudatci-dian O.M., Nersessian A.P. - J. Mat':. . . ys., 1993 , No. 11 (to appear);

[6] Khudavwrdian O.M., Nersessi i .-> P. - Mod. Phys. Lett. A8 (1993 ) 2377; Aoyania S., Vandoren S. - Preprint KUL-TF 93/15; Aoyama S. - Preprint KIJL-TF 93/16

[7] Kiiudavcrdian O.M. - J. Math. Phys., 32 (1991), 1934 ;

[8] Schwarz Л. - Comm. Math. Phys.155 (1993) 249 '375

GENERATING INVARIANTS IN MANY-CLUSTER MICROSCOPIC PROBLEMS OF NUCLEAR PHYSICS A. V. Nesterov, A. G. Kosinov

Bogolybov Institute for Theoretical Physics, Ukrainian Academy of Sciences, 252143 Kiev-143, Ukraine E-mail: [email protected]

Schrodinger and later Glauber gave rise to using the generating invariants, GI (generating functions, generalized coherent states). The most exact definition of them was given by Perelo- mov [1]. The possibility of employing GI for the study of collective and cluster modes in light nuclei was discussed in [2]. However, in that paper the point at issue were, in the main, the problems of two clusters, whereas the goal of the present paper is to discuss the question of employing the GI's for solving the many-cluster problems, where the GI's are used for constructing such oscillator basis which provides the possibility of description of the clustcr wave functions in terms of the oscillatory shell model, while the functions of the cluster's relative motion can be expanded in the oscillator functions. Proceeding to many-cluster problems, we keep, on the whole, the traditional order of the approach under consideration: • Constructing the GI's;

• Calculating the matrix elements of operators in question between the GI's - the generating matrix elements;

• Obtaining the recurrence relations between functions of the employed basis. We illustrate all the above-mentioned with the simplest example. We are concerned in the three-particles problems and consider the matrix elements of the unit operator only. The GI's will be constructed of the one-particle orbitals

|i>=exp{-ir? + 2Riri-R?} (1) where n is the one-particle coordinate (divided by the oscillator radius) of t-th (i = 1,..., A) nucleon, and Я,- is the dimensionless generator parameter of a cluster. After averaging, converting into the Jacobi coordinates and antisymmetrization we obtain, in the three-cluster case, the matrix element of the unit operator as a sum of terms of the following type

I - exp{Q,,XiYi + o12XiY2 + £»2iX2Yi + a22X2Y2} where the вц are constants, X!, X2, (Yb Y2) are the Jacobi coordinates which specify the relative position of clusters centers of mass for the left-handed ( right-handed ) functions. We henceforth employ them as the Bargmann variables [3]. In order to obtain the overlap integrals

< NJ^NZLI] LM j N[LL-}H2H; LM > (2) between basis functions one should find coefficients of the expansion

I = У] < Щ11-ПГ12; LM | H^H^'TLM > '376

where summation is performed over all allowable values of indices, Xj, Xj, Y], Y2 are the moduli of generator vectors, and ^^'"''(flu) are the bipolar spherical harmonics. The values (2) are in fact none other than the known Talmi- Moshinsky-Smirnov coefficients, their explicit form is rather complicated. However, employing the method of generating matrix elements, we provide ourselves with ample opportunity of obtaining different recurrent relations for

< nili',nili; LM | nt?i; n2/2; LM > We henceforth omit the indices L and JVf in recurrent relations as well as, in general, those indices of matrix elements in the right-hand sides which coincide with corresponding indices of a matrix elements in the left-hand sides. For example, the relation:

< n1/i;na/3 | nJwhili >= 1 {a?, < n, - 1 | n, - 1 > + 2n[2(m + /,) + 1]

,I+, +L + < 74 - 1 | n, - 1 > -2(-l) ' ana12V/(2J, + 1)(2J2 + 1) x

„ r(ofi+i,0r/,+i,0 Г + 1 /2 + 1 L "1 x * Ною Ч010 \ J2 1 /

x < n, - 1 I n, - 1,/, + l;n2 - l,i2 + 1 > +

+ { 1 Ь J 1 f } < П, - 1 I /, - 1; ft, - 1 J, + 1 > +

where C,*" , are the Clebsch-Gordan coefficients, and < > are the 6j symbols, can 2 1 Зл 35 J6 J be derived by applying the operator Д, = д /дх\ to both sides of the expansion (3). It is obvious that, applying the operators Д2, Atl Д2, we may derive three similar relations more. Other recurrent relations between overlap integrals with the same vaiye of the total orbital momentum can be derived by using other scalar operators, for example V, V2, ViV2, or any other. If we wish to interrelate matrix elements with different values of the total orbital momentum then we must use operators which are not scalar. For example employing operator Vb we obtain

< "l'l! n2li',LM I "i'iI n2ij; LM >=

= + I, + 1) { ^ L_ j L~ 1 I }]" x

L x {(-1)"+'»^Т[2(П1 + /,) + 3] { h _ j £ ~ \ I } x x -

x < J,-1;L-11 я» — 1,Гг+ l;i — 1 > - '377 - (-о-^^иДЧЬ x < i, - 1; i - 1 | n2 - 1, Г2 + 1; I - 1 > -

l l - ^{C-A J2 l}]} (5)

All necessary matrix elements can be derived by using the above-mentioned procedure. More- over, if we convert from the basis of two unbounded oscillator to the hyperspherical Bargmann variables, then we obtain matrix elements between the /{'-harmonic oscillator basis. Concrete realization of the procedure may be related to the study of the bound state characteristics of light nuclei, the binary reactions with allowance for clusterizing the nucleus - target within the framework of RGM algebraic version, and also the reactions with several structural particles in the output channel, with the implication involved of the "democratic" scattering in this last case.

References

[1] A.M.Perelomov - Comment. Math. Phys., 1972, v.26, p.222.

[2] G.F.Filippov, V.S.Vasilevsky, L.L.Chopovsky - Phys. Elementary Particles and Atomic nuclei, 1984, v.15, p.1338.

[3] V.Bargmann - Commun. Pure. Appl. Math., 1961, v.14, p.187. '378

BOSON VACUUM POLARIZATION IN THE FIELD OF A SUPERCRITICAL CHARGE J. Nyiri

ЯМА'/, Department of Theoretical Physics, Central Research Institute for Physics, H-1525, Budapest 114, P. О. B. 49, Hungary

Abstract The behaviour of the boson vacuum in the presence of a heavy supercritical charge is investigated. It is shown that the supercritical charge is totally screened in the limit when its radius is much smaller than the Compton wave length of the bosons in the vacuum.

1. Introduction The theory of the supercharged nucleus was initiated by the work of Pomeranchuk and Smorodinsky [1] in 1945. The considered phenomenon is the following: if there exists a nucleus

Nz with a charge Z > Z„ (where the theoretical value is Zcr = 137, practically Z„ ~ 180), then this nucleus will decay into an atom and a positron:

+ Nz -» Az_, +e The resulting atom can be either stable or unstable. If it is unstable, it decays again - upto a situation, when the total charge of the atomic state A%-„ becomes sufficiently small. This is quite a peculiar thing: it means, that the nucleus behaves like a resonance. In other words, a nucleus Nz with Z > 137 is unstable, and causes" falling into the center" in the region of

space between the radius r0 and the Compton wave length of the electron. (There are many papers devoted to the theory of the supercharged nucleus, see e.g. the review [2] ). Considering particles in an external field of a source sufficiently strong, i.e. with a charge Z > Z„, we will observe besides free states also a spectrum of bound states for both fermions and bosons.

In the fermionic case, due to the Pauli principle, each bound state corresponds to one or two electrons, i.e. to an atom with Z — 1 or Z — 2. In the bosonic case one can put onto a single level an arbitrary number of particles. When Z is growing, the binding energy increases and the level comes close to zero. For fermions, even when passing zero, nothing really new happens. Although the atom with Z — 1 or Z - 2 becomes lighter, than the nucleus,' the latter can not decay into an atom and a positron until the level reaches —m and the nucleus becomes unstable. In the bosonic case, however, instability arises as soon as the level passes zero, since we can put onto the first negative level a number of particles sufficient to obtain a large negative energy. '379

The theory of the supercharged nucleus turned out to be a very powerful tool 111 theoretical physics. As it was shown by Gribov, the revealed mechanism provides a unique possibility to bind light particles in a small region in space, and on the basis of this phenomenon the theory of quark confinement was developpcd [3-5]. As soon as the gluons in QCD are also "charged" particles, and since the anti-screening of the charge in QCD leads, generally speaking, to the appearance of supercritical charges, the changes of the boson vacuum in the field of a supercritical charge might turn out to be significant for the understanding of the structure of QCD. In the following we will consider the simplest case, namely: the structure of the vacuum state of non-interacting charged scalar

particles with masses rn in the field of a supercritical charge of the radius r0 provided that mr0 « 1 i.e. that the Compton wave length of the scalar particles is much larger than the charge radius. We will show, that under these conditions the supercritical charge will be entirely

screened in the limit mr0 —» 0. While in the fermionic case in the presence of a charge Z the stationary states correspond to atomic states with a charge Z — N where N is large enough to fulfil Z — N < Z„, in the case of bosons the ground state is a coherent state of a large number of bosons with zero energy, which corresponds to a definite classical field ip(r) . Besides, there exists another interesting phenomenon for bosons: if Z is sufficiently large, bound states of opposite sign (i.e. antiparticles) appear.

1 1 л — KV1

This, i.e. that the field with Zc binds not only particles of negative charge —e but also those of c, happens because the charge density of bosons in an external field A is not positive definite. Indeed,

p = ф(и> — А)ф

is always negative, if close to the source (with small ш and large A values). When ш —» 0, the two states approach each other. At a critical Z„ they coincide, and a new vacuum state appears due to the condensation of a large number of bosons on the level ш — 0. The charge of these bosons changes the total charge of the system which will be determined by the minimum of energy of the state consisting of the heavy nucleus and an undefined number of bosons on the level ш = 0. The problem of a supercritical charge in the boson vacuum was, to our knowledge, first considered by A. B. Migdal [6]. He discovered, that the instability of the boson vacuum differs significantly from that in the fermion vacuum in the sense, that here a stable solution can occur only if one takes into account the interaction of bosons (e.g. by introducing in the Lagrangian a term of the type Atp4 ). This interaction restricts the charge of the bosons around the supercritical charge in such a way, that the total charge of the system becomes equal to the critical one. As it was shown [7], the introduction of an additional interaction is not necessary. In order to obtain a stable solution it turns out to be sufficient to take into account in a self-consistent way the influence of the screening charge of the condensate on the wave function of the particles in the condensate. In this case, however, the charge of the system becomes zero in the limit mro —» 0. '380

Since Aif'1 type interactions always exist for charged scalar particles, the right answer has '

to he determined by the relation between mr0 and A. Independently of this relation, however, the density of the screening charge decreases at distances r much larger than r0 but less than £ as a power and not exponentially as the fermionic screening density.

2. The instability of the vacuum of scalar particles in the field of a supercritical charge.

Consider the interaction between charged scalar particles described by the field ip(x) and

the electromagnetic field Au(x) having a source j"'[x). The corresponding Lagrangian can be written in the form

£(i) = -^ЫхЖЛ*) + (V^)-V^ - mW - Av(x)jf{x) (1)

where

= - iAu4> (2)

The equations for /1,, and ip are

~ e2j?(x) - leVV^ - (3)

and

VV + mV = 0, (4)

respectively. Assuming, that e2 <£ 1 , but j"' is proportional to a large number Z so that Ze1 ~ 1, the vacuum polarization (the last term in (3)) can be neglected in the case of a stable vacuum. If the source is considered to be a static one, and we choose A„ = (A 0,0,0), wc obtain the usual equation

2 2 - d A = e 0p"' (5)

for A(x) and, respectively,

Mr) = Ц-, (6)

if r > r0, where r0 is the radius of the external charge, and a = ^ .

The stationary zero point fluctuations фш(г) of the scalar particles are defined by the equation

|(а>-М)2 + д2-т2Ыг)=0 (7)

In case of r > го this equation is the usual one for the motion of a particle in the Coulomb

field. The finiteness of the radius of the charge leads to the decrease of гЛ(г), if r < r0 , and thus the region r < ro does not create any ambiguities. The eigenstates of this equation have a continuous spectrum ш < —m and w > m, and, for small Zct values, discrete Coulomb-type bound states which at ш values close to m condense towards the point w = m (Fig. 1). These discrete states describe physical states of atomic type and give no contribution to the zero-point '381 vacuum fluctuations, which are determined by states belonging to the continuous spectrum. 0)

-Wl

As Za is increasing, the eigenvalues situated at ш close to m are moving to the left, and the deepest u> state reaches the point u> = 0 at a critical Z„a. However, before reaching this value of Za, a bound state arises with a charge of the opposite sign corresponding to a state with и near —m , which moves to the right. At Za = Z„a the two poles coine to the point w — 0, and at Z > Z„ the poles move to the complex plane. Such a behaviour of the singularities which differs significantly from that in the Dirac equation (where no second pole arises, while the first pole moves to the unphysical sheet at и = —m) is formally due to the fact, that for ш —> 0 the Klein - Gordon equation (7) docs not depend on the sign of the field A. From the described picture of the motion of singularities it is clear, that at Z —> Z„ there exists an infinite number of states with the same energy, since the states ш+ and u>_ can be filled up by an arbitrary number of particles. The move of the poles to the complex plane at Z > Z„ expresses the instability of the perturbation theory vacuum, since a solution of the equation (7) appears which increases in time. This solution which corresponds to ш = ш\ + iuii at wj > 0 has rather interesting features. As soon as it is of the form

iw 4>M, t) = e- Wr), (8) by calculating the total energy of the field tp

H = J <Рг[до<р'доф + (),

and its total charge

Q = - J oV - (Vo¥>)V] (10) we obtain

2 3 2 2 5 s H = e ^'' J d r[(M + m - /l )Mr)| + a,rt(r)ft¥»(r)] 00

Q = -2e2^' Jd3r[^+A(r)]Mr)\2 (12)

increasing in time, which would contradict the conservation of the energy arid of the charge, if only H and Q are not equal to zero. Hence, the instability occuring at Z > Z„ corresponds to the growth in time of the field and hence the charge of the growing scalar field will screen the external charge. In this situation it is natural to expect, '382

that with the slow increase of the external positive chatgc the positive charge of the scalar field will move towards infinity, while the remaining negative charge will screen the external chargc at small distances at least to such an extent that the total chargc of the system becomes less than critical. Consequently, the stationary state will correspond to the static scalar field which satisfies the equation

(Э3-М2-т2)р(г) = 0 (13) and the field A will be determined by the equation

- в2A = е2рм1(г) - A(rV(r) (14) where we substituted ip by \/2ei^. The set of equations (13), (14) is analogous to the Fermi-Thomas equation for the definition of the ^elf-consistent field in the atom and of the electron wave function in this field. The smallness of the ch?.rge с of the particle does not play a role any more (c2 disappeared from the equation) since it is compensated by an arbitrary number of particles in the condensate.

3. The solution of the equation for the stationary state.

The equations (13), (14) contain three parameters: Za, ra and m. We will suppose, that mr0 -С 1 and consider first Za near Zcra. As it will be seen, Z^a equals Under these conditions

o(-) A(r) = -lai (15) r

with a slowly changing a. If r/r0 = 1, we haveor = Za. Introducing (15) in (13) and neglecting 2 m in the region r0 < r < i, we obtain

гг

The quasi-classical solution of this equation in the region a(r) > J is

(17) (а2 — 1/4) Ц„ г

where S is to be defined by matching with the region r <; r<>. If 6(r) did not decrease with the increase of г, ф(г) would have nodes in those points r„ for which

\Ja2 — 1/4— + i = nir. (18) J TO This would mean, that ф is not a ground state and there exist solutions with complex ы values. Consequently, a2(r) has to become less than 1/4 before the left hand side of Eq. (18) becomes jr. If at r = rj we have a3 = then for r > r, the function ф can be written in a quasi-classical way as a sum of two terms

ф ~ ae*+ + be*- (19)

where '383

X± = l/2(nг ± f Vl/4- «2—. (20)

In order to obtain a solution V = j which decreases at г —» oo, only one of the exponents in (19), namely e*-, has to be different from zero. The quasi-classical condition for the decrease of the solution requires, that

If at r —» oo a2 —» const. < j,

V-(t-) (22) i.e. the decrease is slower than J. This, however, contradicts the equation (14) for the field A, which has no solutions of the form const./г in this case. Thus we can conclude, that o(r) has to go to zero at г —> oo, i.e. there has to be a total screening of the supercritical charge in the limit mro —> 0. If this happens, we have for r —> oo

ч»=4- (2з> The equation for a will be of the form

rJ<3?Q = ab2 (24) with the decreasing solution

^W+l/4-1/2 (25)

If i^(r) is given, С and Г| can be defined in terms of Za and r0 from the equation for a. The quantity b2 has to be determined from the equation (21) which provides us with the decreasing solution for v>(r). The same b2 defines the normalization of tp1 i.e. the density of the condensate.

References

[1] 1. Pomeranchuk, Ya. Smorodinsky, Journ. Fiz. USSR 9 (1945) 97

[2] V. Popov, preprint ITEP - 169 (1980)

[3] V. Gribov, Physica Scripta T15 (1987) 164

[4] V. Gribov, Lund preprint LU TP 91-7 (1991)

[5] V. Gribov, Orsay preprint (1992)

[6] A. B. Migdal, ZhETF 61 (1971) 2209

[7] V. Gribov, J. Nyiri, Lund preprint LU TP 91-15 (1991) '384

ON THE TWO COULOMB CENTRES PROBLEM IN A SPHERICAL GEOMETRY V. S. Otchik

Institute of Physics, Belarus' Acadcmy of Sciences, Minsk, Belarus

Since the first work of Schrodinger [1] on the quantum mechanical Kepler (Coulomb) problem on a three-dimensional sphere S3, a number of papers have appeared which are devoted to different aspects of this problem (see e.g. [2]-[9]). Recently, its solutions have found application to describing of spectra in quarkonium physics [10]. This stimulates interest to investigation of others quantum mechanical problems on the sphere, among which is the two-centre Kepler (Coulomb) problem. It is well known that the separation of variables in the Schrodinger equation with potential generated by two Coulomb centres in Euclidean space is closely related to the existence of conserved Rungc—Lenz vector in the corresponding one-centrc problem. An analogue of this vector for S3 was found in papers [2]-[4]. It turns out that in the two-centre problem on S3 also exist conserved operator (in addition to the component of angular momentum), and Schrodinger equation admits separation of variables [11]. In what follows, these results are summarized briefly, and solutions of separated equations are presented in the form of series of hypergeometric functions. The connection between series with different domains of convergence is also found. The Schrodinger equation on the sphere S3 can be written in the form [4]

2 Hij> = Еф, II = H0 + U = -(\/4R )Mk,Mk, + U, (1)

where

2 s Л/и = - г/—, /.•,/=1,2,3,4, R = xkxk = x + x = (ii,x2,x3),

and xk are Cartesian coordinates of four-dimensional Euclidean space into which the sphere S3 is embedded. We use units such that h = m = e = 1 (m and e denote mass and charge of the particle). If the particle moves in the field of a Coulomb charge Z located at the point °x = (0,0,0, R), the potential U is given by

U = -Ях„/Я|х|. (2)

Then with Hamiltonian (1) commutes vector operator [2]-[4]

A = 2^(LXN~NXL) + ZR' (3)

where д 9 L =-гх x-, Niv , =-.(*„--xg-)•/ 3 . .

An obvious generalisation of the equation (2) for a charge Z\ placed at an arbitrary point гх of the three-dimensional sphere S3 is

Ux = -ZxCxx)fRy/R< - ('хх)г, (4)

where (xxx) denotes scalar product of four-dimensional vectors 'x and x. '112

Now, let the particle move in the field of two Coulomb charges Zx and Z2 placed at the points lx = (0,0, —к, k') and 2x = (0,0, к, k'), respectively. Then one can show (see [11]) that with the Hamiltonian

4 2 2 // = Ha + {/, + f/2, U2 = -Z2Cxx)/.Ky/R - ( xx) (5) commutes the following operator:

2 2 2 2 2 2 , A = A-' L - k (N - X + Г ) + 2Hkk'(Zlx'3/\x \ - Z2x'37lx"|), (6) where x3 = k'x3 + kx4, x3 = k'x3-kx4, x' - (xhx2,x3), x" = (х,,х2,хз). Thus, like the case of flat space, we have two operators commuting with the Hamiltonian: operator L3 and operator Л (6). Therefore one can expect tb4 the Schrodinger equation with potential generated by two Coulomb charges allows separation of variables. A straightforward examination shows that the separation is possible in only one of the six orthogonal coordinate systems on S3 found by Olevskii [12]. We use for separable coordinates notations sec [13] in which the point on S3 is given by

x/R = (cnacn/3cos 0,cnacn/3sin 0,snadn/?, dnasn/3), -К

Here the .lacoby elliptic functions in the variables о and /3 have modulus к and k' respectively. The real and imaginary periods of the functions with modulus к are 4A* and 4 A*' respectively. The Hamiltonian (5) expressed in terms of coordinates a, /3, ф has the form H i (-LJL A , J_JL 2R2(k2cn2a + к'2сп2РУспс,даС"ада + cn РдрС"Рдр> 1 д2 ksnadnaZ. + k>snPdnPZ+ -_„,„ 2Я2сп2асп2/? дф2 R(k2cn2o к'2сп2Р) ' * 2 ± " W Substituting into the Schrodinger eqation with Hamiltonian (8) expression ф = u(a)v(P)eim* one obtains ordinary differential equations

(—4~cna4- ~ —7- -2ftZ-fcsnadna + 2fl2fc2£cn2a +A)u = 0, (9) cna da act cn'a

{^blpCnl3i - 5jf ~ 2 W'sn/Jdn/? + 2Й}кггЕст?р - X)v = 0. (10)

Here A is eigenvalue of the conserved operator Л (6). The change of variables _ (i + l/7)(fc'sno — dna) (1 + iy){

т/ У2 2 1/2 и = ? %-1Г (у-аГ-/,

m m 2 T 2 2 v = z >\z - l) ' (z - a) +g, r± = I(l+ (2 ER ± 2iRZ+ + l)" ) (13) yield for / and g Ileun equations [14] m+1 m+1 Ay P £L + f + + + ~ f = о (14) dy2 V У S-l y-ajdy y{y-l)(y-a)

m + 1 m + 1 fl + ( 4. , 49 , Bz-g _ (15) 2 + + + dz ^\ z 2 - 1 2-a) dz z(z- 1 )(z - af ~ ' '386

where

p = -iRZ. + (m + l)(rna + a.) + \/4ikk',q = iRZ+ + (m + l)(mo + т+) + A/4 ikk',

A = ( 1. Then it can be easily checked by (11) that for —К < a < К y(a) lies in the domain of convergence. Therefore, one can obtain solution of equation (9) bounded for —К < a < К by substituting in (12) solution of the equation (14) in the form of series of hypergeometric polynomials

00 /= XI CriFA™-I-r,m + i + r+l;m+l;y), (16) r=m—I where / is a поп negative integer. The coefficients c, are determined by the three term recurrence relations

Qrcr+, + 0rc, + 1rCr-l = 0, (lr - m + 1Щ1 + l)(t, + 2 - 2(7-) - iRZ-I °r 2(2/,+ 3)

2 2 where lr = I + r and A' = A/(fc' — к ). Introducing the continued fractions

R = — = L = — = ~a* cr_ 1 pr + nrRr+l' ' c,+1 Pr + Ъ RT-I one obtains a transcendental equation

R,L r-. = 1, (17)

which determines values of separation constant A for which the series (16) is convergent. For small inter-centre separation which corresponds to large values of 7 one can ohtain from (17) an approximate expression

2 l г 2 5 -2ER (m + / + / - 1) + 1(1 + l)(m - 3( - 31 + 2)] + 0(1/1''). (18)

Now we proceed to find the bounded solutions of the equation (10). It follows from (11) that, as runs between limits of its variation —A'' and А'', z(j3) describes the path beginning at z = 1. going beyond the singularity z = a, and ending at z = 0. This path can be continuously deformed into one consisting of three parts, two of which lie inside the ellipse of convergence, and the third is a closed loop around z = a. The solution of the equation (10) bounded at /3 = K' (z = 0) has the form

rss-eo

.Pr" = r(fr+m + l)r(-!/r + m)2Fi(Kr + m + l,-!/r + m;m-t-l;z), (19)

where vT — v + r, and v is an auxiliary parameter. Recurrence relations for d" are

агЧ+1+/ЗгЧ+7Я-1=0, '387

„ _ {Vr + m + 1)[(кг + l)(iy + 2 - 2т+) -f iRZ+] ar = 2(2i/t + 3)

The parameter и is determined from the condition of convergence of series (19)

Q Df Г U 1 nv — 7f TV dr ~ r

° ~ ' Г №t + ' ' ~ JUl ~ К + 7r Дг-1 '

The solution of the equation (10) bounded at /3 = —A'' (z = 1) is 00 т 2 2 T+ «V = г / (г - 1Г' (г - a) £ = Яг"(1 - г). (20) r=-oo

The solutions of the equation (10) which are multiplied by constant numbers, when a closed cirquit is described around the singularity z = a, can be presented in the form

rs-oo

P? = + T+ + г_)Г(—i/r - 1 + r+ + r_)I>r + 1 + T+ - r.)

хГ(-1/г + T+ - r_)2F!(i/r + r+ + r_, -i/r - 1 + T+ + r_; 2r+; 1 - 2); (21)

«-•г=г-(г- - £ Г= —ее

= 2^1 К + 1 - T+ + T_, i/r - г+ + т_;2 - 2r+; 1 - г). (22)

Coefficients rf" and J" are related by the equation

,-„ I>, + 2 - r+ - т_)Г(—ty + 1 - r+ - r_)r(fr + m + 1)Г(-1Л- + m)

(-1)'ГК + 1+т+-г.)Г(-1УР + т+-г_)

The series in (19), (20) converge inside the above described ellipse i;i the z-plane, and scries in (21), (22) inside the similar one in the г-plane. In order to find connection between solutions of the equation (10) with different domains of convergence, we consider further solutions of this equation defined by = (e-'X " (—l)mu'i")/2 cos27ri/r(m + I); (23) Щ = Г(2г+ - \)й'{

2 - [Г(1 - 2r+)sinjr(f - тг - r_) sin ir(f - r+ + г.)/* ]^. (24) Solutions (23), (24) can also be presented as series of hypergeometric functions, a.ttd by comparing these series one can establish the following relation:

u2 =

(25) ~ 2cosTri/r(-i/ - 1 + r+ + г_)Г(—1/ + r+ - r_)

X r(2f + 2r + 2)(—г)! (£ Г(-2^ -2г)н) ' '388

The obtained relations enable us to describe behaviour of solutions of equations (10) and (15) in all the complex plane of г . For bound states in the problem, the solution u" must go over into the solution u',", after encircling the singularity z = a in the negative direction. This condition determines energy levels of the particle. For small inter-centre separation we can obtain an approximate expression for the n-th level

0 _ WZ,Z2lP[l{l+\)-'im>] f ZI _ " ny2[2l — 1)(2/ + 1)(2/ + 3)/(i + 1) \ n2 Ft2) 4 +0(l/7 ), / > 0; (26)

= £o + _ (S _ g) + 0( 1/У), I = 0.

Here is the n-th energy level of the part'.rlc in the field of the point charge Z+ = Z\ + Z2.

References

[1] li.Schrodinger. l'roc. Roy. Irish Acad., 1940,v. A46, 9.

[2] P.W.lliggs. J. Pliys., 1979,v. A12, 309.

[3] H.I.Leemon. J. Phys., 1979, v. A12, 483.

[4] Yu.A.Kurochkin, V.S.Otchik. Dokl. AN BSSR, 1979, v. 23, 987.

[5] A.A.Bogush, V.S.Otchik, V.M.Rcd'kov. lzv. AN BSSR, 1983, N3, 5f>.

[6] A.O.Barut, R.Wilson. Phys. Lett., 1985,v. A110, 351.

[7] A.O.Barut, A.Inomata, G.Junker. J. Phys., 1987, v. A20, 6211..

[8] S.I.Vinit.sky, L.G.Mardoyan, G.S.Pogosyan, A.N.Sissaklan, T.A.Stri/.h. Yad. Fiz., 1993, v. 56, N3, 69.

[9] V.N.Pervushin, G.S.Pogosyan, A.N.Sissakian, S.I.Vinitsky. Preprint E2-92-582 J1NR, Dubna, 199i.'.

[10] A.A.Izmest'ev. Yad. Fiz., 1990, v. 52, 1697.

[11] V.S.Otchik. Dokl. AN BSSR, 1991 v. 35, 420.

[12] M.P.Olevskii. Mat. sb., 1950, v. 27(69), 379.

[13] E.G.Kalnins, W.Miller, P.Winternitz. SIAM J. Appl. Math., 1976, v. 30, 630.

[14] A.Erdelyi et, al. Higher transcendental functions, v. 3, —N.Y.:McGraw-Hill, 1953.

[151 N.Svartholm. Math. Ann., 1939, v. 116, 413.

[16] A.Erdelyi. Quart. J. Math., 1949, v. 15, 62. '389

THE CONFORM CONNECTION BETWEEN EQUATIONS OF GR AND THE JORDAN-BRANS-DICKE THEORY V. Papoyan and G. Haroutyunian

Yerevan State Uniiersity, Armenia

1. Introduction Kaluza [1] has sucseeded in the achievement of formal unification of Maxwell theory and General Relativity (GR). The essence of the unification is the following : a five dimensional Riemannian metric is being introduced

f flit + AjAk | Ai \ gAB = ! (1)

\ Ak | gss )

А, В = 0,1,2,3,5; t,fc = 0,1,2,3 If we rcduce the field equations of the theory G® = 0 corresponding to (1) to 4 - dimensional ones (G® - being the Einstein tensor of 5 - dimensional space), then in the case when all variables are independent from x5, and also assuming that 355 = const, we'll obtain Einstein-Maxwell set of equations.

Considering the transformational properties of для, Jordan [2] has noticed, that jSs is actually a scalar and he has modified Kaluza anzats so, that

gik + yAiAk I уА{ BAB = I I j (2) yAk I У where у — y(x) is a scalar function, which is conventionally known as a gravitational scalar, and which in accordance with the Dirac's hypothesis of "desrepiting" universe [3] is choosen in a way, that in weak gravitatinal fields у —• ya ~ l/G0 (where Ga is the Newtonian gravitational constant). If we reduce into 4 dimensionality the equation G® = 0, Corresponding to the

metrics (2), keeping the condition of independence of all variables on x5, we obtain equations differ from GR equations, namely

yR? = 8 - T6*) + V.V-C— (3) О " <, у

it = ^ (4) Here the energy-momentum tensor of the matter and nongravitational fields, ( - the dimensionless constant of generalized theory of grawity (GTG). In space infinity,

2(2-0 ... '390

It is easy to notice, that at у = y0 and (-100 the GTG equations coincidcs with GR equations.Brans and Dicke [4] used the another physical reason leads to tensor-scalar theory of gravity with the same field equation, therefore this theory is called sometimes "Jordan- Hrauso-Dickc theory". At present work the conform conncction of GR and GTG equations is considered, and in the second part the possibity of mutual generation of exact solutions of GR and GTG for axcsyminetric stationary ease is given which is illustrated by a concrete example. As a result a new statical solution of Gl{ with a charge is obtained. In special spherically symmetric case this solution is coinciding with the known solution of Reissner-Nordstrom.

2. The conform correspondense of the equations of GR and GTG Let in one and the same manifold set two the 4-dimensional Riemannian structures are given in conform correspondense according to the equation

Ы*) = е1<г •№•*(*) (6)

As it is known, Ricci tensors of conforinly corresponding Riemannian spaces V4 and V4 are ccniiicctrcl with equation [5] ,

Rk _ R k _ 2[ V (Tk _ a ak + 6i a.ak j _ 6k Viai (7)

when: V, is symbol of the covariant differentiation, and the dash over the indices is meaning that the value belongs to (in particular (...)* = jw(.. )i )• The conform transformat'ons (7) are establishing correspondence between 11 functions дц(х), a(x) and 10 values

1. Let us assume, that да(х) is satisfying the GR equations

R? = 0 (8)

and g-,k(x) to the vacuum equations of GTG

Vts* = 0 (10)

substituting (8) and (9) into (7) we obtain

^J- ~ С ^г = 2l~ + e

Comparing (1) and (2) one can notice that the connection of is simple enough. Let us assume, that the connectin has the following form

e2" = yn (12)

Using (12) let us exclude

k l ,„ n V.y* _n(«-f2)-2C y{y n(n-l) cky,y (n-1) — 6f_ (i3)

In order to satisfy (10) let us convolute on i = k, which will give

3(n-l)J = 3-2C (14)

It is not difficult to obtain the condition of integrability of (13).

da)

This condition may be rewriten in a different way

yP-C-lki = 0 (16)

p where C lkt is the Weyl's conform tensor. Thus we can formulated: Statement X . Let g,t(i) and y(x) are the solutions of GTG vacuum equations and

У г -сш = 0 then n 2 Sifc(x) = у gik{x); 3(n - l) = 3 - 2( will be the solution of the corresponding problem of GR.

2. Now let us assume that ffn(x) and y(x) are satisfied the GTG equations and gn(x) to the vacuum equations of GR. If me permit

e2' = у"

it is easy to prove. Statement 2. Let gn(x) be the solution of vacuum equations of GR, then, if any solution of the equation

Vj(7t + 2(!n±l) 0 m x ' with the condition of integrability

p °v • C ,kt = 0 is found, then

Ы*) = " 9ik(x) ; у = e2o/ra ; 3(m + l)2 = 3 - 2f

will be the solution of the corresponding problem of GTG.

3. Using (7) it is not difficult to demonstrate, that if we accept in (6)

then, after the conform transformations of GTG equations (31 in the presence of matter and and nongravitational fields will coincide formally with EListein equations '392

Gi = SirGo • (ft + fv*) (19)

moreover an additional source in the form of the tensor of "energy-momentum" of a scalar field

v (20) 16ttG0 ' 2 be present and instead of equation (4) we shall have

= (21)

£ Thus, it is enough to find о^ by given T, then to find the corresponding value т( , and later make a transition to the solution of Einstein equations (19). ( Let us emphasize, that in such an approach the motion equations arc distorted by the presence of a scalar term, because here Vt(7'/ + т*) = 0). Let us mention also, that in cases, when the diagonal elements of t* are having nonzero values only, the presence in (19) of the term connected with scalar field, could be taken into account by the introduction of effective "energy densities" ec and the "pressure" pc of the scalar field, which are satisfying to the ultimately severe equation of state ec = pc . For the sake of completeness let us present a less known result, the proof of which is simple (see e.g. [2]): Theorem (Schuking) 11 elecrovacuum equations of GTG in V4 space are equivalent to 10 equations

H 3-2 С гаЛ V-= ^2 • ^у 2 (22)

in the conformly corresponding space V4 with a metrics

glk = V9ik

One can show, that if /i,t = Vi^fc (Ф ~ is an arbitrary scalar) , then, as a consequence of contracted Bianchi's identity we shall have ViV* = 0 . This means in (22) an equation for the scalar potential is also present.

3. Axially symmetric stationary case Stationarity and axial symmetry arc allowing such a selection of coordinates, that (a:0, x1, i2, x3) = (/, x', x2, y>) and the vacuum equations of GTG arc admiting to introduce Weyl canonical coordinates p(x\ x2), 2(1', x2) [6].

2 2 2 ds = ds —ds n 2 2o 2 2 2 2 da , = e <'"> ( dt - q{z, p) dV ) - p c~ ° d

Let us transform (23) conformly

ds2 = 7,С-"/3) ds2, p = \/3~—2f (24) '393

and compose a matrix from the components of metrics tensor

The independent components of matrix equation

V^-'Vff) = 0 (20)

are the GTG equations, which determined gu and

/,.' = a''

The GTG equations, determining джж = gpp , will look like

(In fob = lTr(f.' /Л

(In = + (28)

Formally the equations (26) and (28) look exactly as the equations of the corresponding GR. problem (see e.g. [7, 8]). Thus, the axisymmetric stationary GTG problem after conform transformations is formulated in a way, that, at first, for its solution the methods of Backlund transformations [9, 10], or the inverse scaterring methods could be used, and, secondly could be formulated the following

Statement 3. If the set of у, e2°, c2", q is the vacuum solution of stationary axially symmetric problem of GTG, then the corresponding GR problem has the following solution

2 2 2 9ti = ye "] Sw = -VP *' "', 9tv = Ч\ 9гг = g„ = i/ew/<[>3-2< (29)

here Ф is determined by known у according to

• JO"*).-^. = ^ (») And, vice versa, by the known solution of GR, using (29) one can find the solution of the corresponding GTG problem, if preliminary determine у = y(z. p) as a solution of

V (^j = 0 (31)

and Ф from (30).

4. New exact solution of GTG with charge Recently the authors had found a Reissner-Nordstrom type solution in GTG frame [11] '394

/R±ROY'4 v = у°{7ГЛъ) where a, 17 arc in the integration constants, whereas

i + ^ + (32)

m, Q is the mass and the charge respectevely. At £ -t oo (a = 0, i) = 1) Eq. (11) is being transformed into Reisner-Nordsrom solution of GR. Let us use Statement 3, then we r-n obtain GR solution in the following form:

2 2 ds = de - J/- (dS + dp ) + ? v];

2 2 Й=Л(г+ + г_); r± = (г ± к) + p ;

(33) к n m ' П.+ к'

p _2r0 + m f 2r„ - m f R - кУ"\ 2n [ 2ro + m \R+ к ) J ' The electric field strength is determined by the expression

k2Q /Д-«\" /И? - (34) F2(R2 — к?) \R + к) V ) The solution posesses quadrupol momentum

(35) о n

and at n = ! is coinciding with Reissner-Nordstrom solution.

Aknowledgement The authors are grateful to G.Sahakian, N.Chernikov, to the partisipants of the seminars on "The theory of spacetime aiid gravitation" LTPH of JINR and the Chair of Theoretical Physics -Л Yerevan University for stimulating discussions. This work was supported in part by Mayer Foundaion Grant, awarded by thi. American Physical Society. References

[1] Th.Kaluza,Zum Unitatsproblem der physik,SiU. Prcus. Akad. WW, S.966, 1921.

[2] P.Jordan,Schwerkrafi und IYeltall, Vieweg und Sohn, Braunschweig, 1955.

[3] P.A.M.Dirac,/>roc.floy.Soc. A165, 199, 1938.

[4] C.Brans, R.H.Dicke,PAye.fie». 124, 952, 1961.

[5] A.Petrov,A new Methods in General Relativity "Nauka",Moscow, 1966.

[6] V. Papoyan ,/ЫгорЛу«. and Sp. Sci. 124, 335, 1986.

[7] V.Belinsky, V.Zakharov,JETP, 77, 3, 1979.

[8] G.Alexeev, Trudi MIAN, USSR, 176, 211, 1987.

[9] G.Ncugebauer, D.Kramer,Gen.Rel.Grav., 13, 195, 1981

[10] D.Kramer, et al. Exact Solutions of the Einstein Field Equations, Berlin, 1980.

[11] G.Haroutyunian, V.Papoyau, Astrophysica, 21, 587, 1984. '396

QUANTUM EVOLUTION OF THE UNIVERSE

V. Pervushin1 and V. Papoyan2

'Joint Institute for Nuclear Research, 141980, Dubna, Moscow Region, Russia; 2 Yerevan State University, 375049, Yferevan, Armenia.

Abstract Wc show that the first principles of quantization and the experience of rclativistic quantum mechanics can lead to the definition of the observable time in quantum cosmology as a global quantity which coincides with the constrained action of the reduced phase space theory up to the energy factor. The quantum time coincides with the time of Friedman evolution for the "dust", with the confnrmal time for the radiation and describes the inflation in the case of the Jordan-Brans-Dicke theory without matter and radiation.

Introduction The cosmological evolution of the Universe has been established hard and fast in the classical theory of gravity , as well as by experimental data. A lot of papers have been devoted to the consideration of the same problem in quantum gravity (see refs.[1,2,3]). Among the main difficulties preventing a final solution to this problem are the ambiguities that appear in the quantization proccess [4] and in the interpretation of the quantum theory. In the present paper, where one more attempt to describe the quantum cosmological evolution is considered, we try to remove these ambiguities with the help of fundamental quantum principles and the experience gained in relativistic quantum mechanics [or the description of the time reparametrization invariant systems. To clear up the statement of the problem it is worth recalling two crucial peculiarities of the classical Friedmann evolution of the Universe [5]: i) the observable time and the evolution of the Universe arc defined by the matter content of the Universe, and ii) the proper time gauge (\/ffoo = 1) is physically distinguished. Wc emphasize that only in the classical cosmology •Jgoodt = dTp, this time is identified with that defined by the experimentally observed evolution of the Universe. Any other time (like ,for instance, the conformal time) is considered rather as a mathematical tool for solving equations. Under this statement of the problem we can see that the choice of a gauge is also an ambiguity of the quantization procedure. One of the reasons for the gauge ambiguity is the violation of the uncertainty principle for the nonphysical gauge field components in the conventional approach [1,6] which includes in the canonical scheme all physical and nonphysical components (see the discussion by Dirac of Eq. (2.28) in his Lectures [6], and the criticism by Schwinger [7]). Following Refs. [6,7], we can eliminate the nonphysical components from the canonical scheme yet on the classical level. This is just the idea of the reduced phase space quantization (RPSQ) [8,9] (which reproduces for QED the first Heisenberg- Pauti quantization [10]). We show that RPSQ leads to a definite gauge and a definite ordering of operators. The wave function of the Universe is completely defined by the initial Einstein action or the Jordan- Brans-Dicke theory [11] action in its reduced version. The experience of relativistic quantum mechanics and the correspondence principle help us to separate this action into two factors : the quantum cosmological time and the conserved energy. '397

The reduced action looks like a global quantity and cannot be reproduced by the differential equations without additional suppositions.

1. The Hamiltonian model of the homogeneous Universe In the first plase we shall consider the Einstein theory of gravity

a 1 w. w = J

м ±1 6/c3J *"*) --ГО' * = °' ' (2) for closed (+1), flat (0), and open (-1) spaces , where a is a physical dimensionless mctric variable (the spatial scale, or scale factor) and ro is the parameter of curvature. The matter dynamics is modelled by the energy-momentum tensor of dust (Мл) and radiation (er):

! f-ii- TS = 3 + Md 7J = 0, (3) V3(r0)a l2roa

where V(r0) is the spatial volume for a = 1. Then, in terms of the logarithmic spatial scale of the space fi = 'n(a) and its momentum P the action (1) has the form

W = J dt\(iP-aH-j , (t = x (4)

where

3 Я П = n V3(r0) + TS-: (5) 2 a»Vj(r0)

is Einstein's density of energy and a = y/goo is the time scale (or lapse function ), whose associated the equation of motion is the time-time component of Einstein's equations

H = 0. (6)

This equation is a first-class constraint [6]. The time-space components of Einstein's equations represent other first-class constraints WS)-* (7) The general solution to this last equation is the sum of the general solution to the homogeneous equation (7J = 0) and any particular solution to the inhomogeneous one (7j Ф 0). The former is simply the zero mode (dkP = 0j that describes the collective excitation of the metric fields. The Friedmann approximation corresponds just to the zero mode sector of the constraint (7), sector in which the theory is invariant with respect to the group of time reparametrization

t t'(t). (8) '398

2. The extended phase quantization Let us consider, first, the Dirac approach to quantization [6] in which both components of the metric a, a are included on an equal footing in the canonical scheme. The first-class constraints are

Pa = 0; {H, P„] = % = 0; (9) where H is defined by Eq. (5). These constraints, strictly speaking, should be supplemented by the secondary ones [6]. In the quantum theory, these constraints are imposed on the wave function

M>,VD= 0; (10)

This is the Whcclcr-DeWitt equation for the wave function of the Universe which is usually interpreted as a stationary state [1, 2]. According to this interpretation, the Friedmann time is identified with the local parameter of the phase of the wave function in the semiclassical approximation. Equation (10) presents a factor ordering ambiguity. To remove this ambiguity, it is useful to exclude the nonphysical component a from the canonical scheme of quantization (see refs. [6, 7]).

3. The reduced phase quantization In the reduced phase space quantization [8, 9] we should construct the reduced classical theory quantization by explicitly solving the first-class constraint (6). For the time scale Eq. (6) results in two solutions

P(±) = тП 1 K3(r0)

Those solutions determine two reduced actions (see Eq. (4) )

W1 W$(a(T)). <±> 2

Both reduced classical theories include only one component of the metric (a) and are invariant with respect to the initial group of time reparametrizations (8). Due to this in variance, the classical equations for the actions (11) become trivial identities, and the pure classical interpretation of the theory is contradictory. However, the variable ц in the action (11) has nontrivial conjugate momentum

SW"d pred — • -111 1 = d (12) - = T T F" (a). 'Ю fip 2dh\

The quantization of the reduced theories (11,12) leads to the quantum equations

d Ф± = ^iF"* (a) *(±). (13) dp

The total reduced phase space wave function can be represented as a sum of two wave functions with the coefficients

V«>{a) = /iWeW<"?<»> + ^t-)eilV(->(°) (14) '399

This result can be interpreted as a solution of the Wheeler-DeWitt equation (10) with a definite ordering of operators.

4. Global time To clear up the interpretation of the quantum theory (13), let us consider the simplest version of the initial theory (4,5) with the variable io instead of the spatial scale (a).

J 2 W = £dt [io/' - Qi(m - P ) which coincides with the action of a relativistic particle in the rest frame. In this case the expressions for the reduced actions [T W£= dlx0P = ^xDm (15) Jo lead to the wave function

0Г«» = Л(+)е-(10т + Л(-)е(гоп. which is the secondary quantized scalar field in the rest frame. Here Л'4' are the creation (+) and annihilation ( —) operators of a particle, and the role of the physical Lime is played by the invariant reduced action up to the energy factor (m). So we have two definitions of an invariant physical time: i) as an invariant interval (the Friedmann time)

JV A, - ^ца^УзЫ dlF = a(±)dt = ± Klf{a) (П) and ii) as an invariant reduced action (18) (global time). In the general case of a moving particle with velocity V, the physical time of the spectral

representation of the wave function, X0 = Tq (time of the "observer"), differs from the proper (Kriedmann) time of the particle by the Lorentz transformation factor

2 TQ = 'I'fIVT^V .

This example points out that the coefficients AW in Eq. (14) can be treated as creation and annihilation operators of the Universe in the homogeneous sector of quantum gravity, and the reduced action (11) coincides with the physical time of the spectral representation up to the conserved energy of the Universe [13, 14]

We-, = (18)

The reduced action (11) in the considered cases has the forms:

l k = +1: W(±) = T{*7 - cosr)o[sin rj0 + sinfo - no)]}(roMMd + eR)2~ ] (19) х fc = -l = + cosh 7/0[sinh t/0 + sinh(7j - 4a)]){r0MMd + ен)2~ ; (20) k = 0

l 2 —Щ^(2Ма + e) ' J (r0Mi + £я)- (21)

Where 2 1M г K £R о 2 а к '100

VVe сап see that in the limit of dust dominance, e = 0, the classical evolutions (cf. e.q. [12]) coincide with the quantum ones (19-21). This coincidence has an exact and global character and can be used (as the correspondence principle) for the separation of the energy factor in the more complicated case of radiation dominance. In this case, there is an "«inexplicit motion" оГ the radiation which is the reason for the difference between the global "quantum" time and the Friedmann one. These two times are connected by the nonlinear analogy of the Lorentz transformation. For pure radiation, the "quantum" time coincides with the conformal one, and in the closed space case the nonlinear transformation lias the form

Tp(TQ) ---- r0\fl [cos(7'

The quantum time (19-21) interpolates between the proper lime for the dust (cr = 0) and the conformal time for radiation dominance (с >> Л7), and docs not contradict the causality principle (in contrast with the Fricdmann lime). At the beginning of the expansion of the Universe, the quantum time strongly differs from the Friedmann one and a "quantum" observer sees the quantum Hubble constant and the corresponding "critical" density: (r4r)' (22)

III the second place wc shall consider the quantization of the homogeneous cosmological model in the Jordan-Brans-Dicke tensor-scalar theory of gravity [11]. Willi respect to the Einstein theory the JBD theory has an additional degree of freedom in the form of a scalar field IV = J d\dT° J-|L (<"rt - C<7„<7") + £„] , (23)

where or is the logarithm of the scalar field, £ is a dimensionless coupling constant. After a conform transformation a = Re-'!* = =

for the simplest fiat space case к — 0 and the radiation dominant epoch we receive

W = J dr + ) _ «« _ Ш _ , (24)

where

e + (25) •H - Я*ЫrD) j-j—+ 2 3T2C R R*\ '

P(/t) and P(ct) are canonical conjugate moments of the ft and a fields. Let us consider the action (23) on the explicil solutions of the constraint equation

, П = 0 =Ф- P(/i) = TF(K,l W)

The substitution of this solution in Eq. (23) leads to the reduced action

= Щг0)Р(г)<т т W"(P[a), R), (26) '401

W»{P{a),R) a EQTQ = K3(r0)^ (27) 0 +ЯМ/В + 1

The density of the massless scalar field dominates in the R —» 0 limiting case

EQ(P(,t), e = 0) = V3(r0)/>(

In another case, P(cr) —» 0, the density of the radiation dominates

£д(Р(<г)^0,£) = ад£г (29) and the time coincides with the conform time

TQ[P{„) = 0) = (30)

Therefore, we can assume for spectrum

EQ = K3(R0) >//«(<7) + E*, (31) then the quantum time will be determined from Eq. (27) as

2 + ln-T (32) Q y/Fw+12 «(3 + 2Q := у 1 + Я GK2 P2(ct)'

In с —» 0 limiting case, the inflational expansion of the space arises with respect to Tq time

Provided that R increases, the inflation stage gets over the expansion stage of the radiation dominant Universe (30).

5. Interpretation and conclusion We have got the reduced phase space wave function of the Universe (13)~(21) and have discovered that two different integrals for the Friedmann time (17) and for the phase of this wave function (13) coincide in the case of pure dust. This coincidence allows us to use the analogy of the "dust filled" Universe with a relativistic particle at rest (15) (the Ilamiltonian of which is the simplest version of the homogeneous gravity Hamiltonian (5)) and the experience of the interpretation of relativistic quantum mechanics. This interpretation consists in the comparison of measurements by two detectors i) of an external observer (E0) and ii) of an internal one (10). 1) For the particle at rest they are identified. The evolution of the wave function of the rest particle is determined by the proper time of the IO: Tf(X0) = X0 and by the physical mass (i.e. energy). The latter is defined from the reduced action and does not coincide with the Einstein Hamiltonian as a constraint. 2) For a moving particle in relativistic classical mechanics there are two equivalent times, those of the EO and the 10. They are connected by the relativistic transformation. In relativistic quantum mechanics (RQM), a wave function of a spectral representation depends only on the EO-time, so that the latter is distinguished and has the absolute character. '102

Let us apply these two points of the RQM interpretation to the quantum Universe, where we have two definitions of the time of evolution: the time as an invariant interval (or the Friedmann proper one), which we identify with the IO-tinie, and the time as an invariant reduced action which can be identified with the KO tiiric. 1) For the dust Universe (as for the rest particle in RQM) these two times coincide. 2) An external motion of the Universe can be caused by additional degrees of freedom of field excitations introduced in the explicit or implicit form (for example, by changing the state equation for the matter density). One of the examples of this external "motion" is the radiation . The wave function of the radiation Universe has the form of the spectral representation with respect to the conformal lime which plays here the role of the EO-time, according to the RQM interpretation. This КО-lime is distinguished at the quantum level. When we consider the beginning of the Universe in the quantum theory , a problem arises : what is the time describing physical quantum evolution of the Universe? Due to the acceleration, the 10-detector can measure different particle density and temperature than the EO-detector. One of the subtle points of the considered quantum theory is a consistent interpretation of the reduced gauge classical theory (11). It is worth recalling that the gauge theory has been formulated by Weyl [i5] as one of the types of relativistic quantum mechanics which does not have pure classical interpretation, and for its construction one can use the quantum principle»: i) observability, ii) uncertainty, iii) correspondence. The first principle allows one to use only invariant quantities (interval, or reduced action)for the definition of an observable time. The second principle forbids quantizing the gauge field components for which the constraints fix simultaneously the "momentum" and "coordinate" (of the type of the time scale) and leads just to the reduced phase space quantization. The third principle allows one to separate the quantum time from the quantum energy in the reduced action (11). Thus, the quantum principles completely determine the quantum evolution of the Universe.

Acknowledgment The authors thank Profs. A.Ashtekar, I.A. Hatalin, В M. Darbashov, A.T. Filippov, V.G. Kadyshevsky, D.A.Kirzhnitz and L.N. Lipatov, for useful discussions.

References

[1] U.DcWitt, Phys. Rev. 160,1113 (1967).

[2] A. D. Linde, "Elementary particle physics and inflation cosmology", Moscow, Nauka, 1990.

[3] Wilczek F. Erice Lecture on Cosmology, NSF-1TP-81-91. Ettorc Majorana Summer School, July, 1981.

[4] A. Ashtekar, G. T. Horowitz, Phys.Rev.D26, 3342 (1982). M.J. Gotay, Class. Quantum Grav. 3,487 (1986).

K. Kuchar, Contemp. Math. 71, 285 (1988).

[5] Л. Fried man n, Z.Phys.10, 376 (1922).

[6] P.A.M.Dirac. Lectures on Quantum Mechanics, Yeshiva University New-York, 1964.

[7j J.Schwinger, Phys.Rev. 127 324 (1962). '403

[8] V.N.Pervushin, Teor.Mat.Fiz. 45,1100 (1981); V.N. Pervushin, Riv.Nuovo Cimento 8 N10, 1 (1985); Nguen Suan Han, V.N.Pervushin, Fortsch.Phys. 37 611 (1989); N.P.IIieva, V.N.Pervushin, Sov. J.Par.Nucl. 22, 573 (1991).

[9] R.Loll, Phys.Rev. D 41, 3785 (1990).

[10] W Heisenberg, W.Pauli. Z.Phys.56, 1 (1929); ibid., 59, 168 (1930) .

[11] P.Jordan, "Schwerkraft und Weltall", Vieweg und Sohn, Braunschweig, 1955. C.Brans, R.Dicke, Phys.Rev. 124, 925, 1961.

[12] D. Kramer et.al. in "Exact Solutions of the Einstein Field Equations" Ed. E. Schmutzcr, Deutscher Verlag der Wissenschaften, Berlin, 1980.

[13] V.N.Pervushin, JINR Rapid Commun. 6/57/-Э2, 46 (1992); V.N.Pervushin, T.Towmasian. J.Moscow Phys.Soc. 3, 1 (1993).

[14] V.V.Papoyan, V.N.Pervushin, "Quantization of the Homogeneous Cosmological Model in the Jordan-Brans-Dicke theory and the Problem of Observable time", preprint JINR P2- 93-330 Dubna 1993.

[15] H.Weyl Z.Phys. 58, 330 (1929). '404

SYMPLECTIC MANIFOLDS WITH SYMMETRY AND WEAKLY G-INVARIANT COMPLEX STRUCTURES A. D. Popov

Laboratory of Theoretical Physics, JINR (Dubna), P.O.Box 79, 101 100 Moscow, Russia

Abstract The finite-dimensional phase manifolds M with the Hamiltonian action of the connected Lie group G and the complex structure J arc considered. We describe the homomorphism r: G -> AutV of the group G into the group of the automorphisms of the bundle V = V(M,Sp(2n, R)) over M and impose the condition of invariancc of the complex structure J under the group T(G) (weak G-invariance). The Kahler manifold (M,LJ, J) with weakly G-invariant complex structure J may be quantized.

1. Constrained systems arc often considered in physics and therefore their quantization de- serves particular attention. To quantize such systems one usually uses the method of canonical quantization or the method of path integral quantization. The method of geometric quantization of Kostant and Souriau [1-4] is a generalization of the standard canonical quantization on the curved phase manifolds M. Geometric quantization of the constrained systems have been considered in Refs.[5-ll]. In these papers it has been supposed that the (7-invariant polarization P exists on the symplcctic manifold (М,ш). We consider a positive Kahler polarization corresponding to the complex structure J on M and impose the weakened form of the symmetry requirement. Namely, wc describe the homomorphism г: С —» AutV of the group С into the group of the automorphisms of the bundle V = V[M,Sp(2n, R)) over M and impose the condition of invariancc of the complex structure J under the group r(G) (weak G-invariance). 2. Suppose wc choose a positive Kahler polarization of the symplectic manifold M. This polarization is in the one-to-one correspondence with the complex structure J on M. It consists of a linear operator J from TM to itself such that J2 = — 1 and

[JX.JK] = [X,Y] + J[JX,Y] + J[X,JY], X,Y&T(M,TM) (1) Condition (1) means that the almost complex structure J is integrable and one may introduce the Kahler mctric

g(X,Y) = u>(X,JY) (2) with Kahler form ui. 3. Let / be a dilfeomorphism of the manifold M and /. an isomorphism of the tangent

space Tj-\[Z)M onto the tangent space TXM. This isomorphism may be extended up to the isomorphism of the tensor algebra in Т]-цх)М and the tensor algebra in TXM [12]. This isomorphism we shall denote by f. For any tensor field В we shall define the tensor field /В in the following way:

(jB)x = I{Bj-Ht)), 16И. (3) 4. Let X be a vector field on M and /, = etX a one-parameter group of transformations, generated by X. For each t we have an automorphism /< of the tensor algebra on M. For any tensor field В on M a Lie derivative f-xB is defined by the formula [12]:

CxB = -\im\[ftB-B]. (4) '405

If the ^-parameter subgroup a(C) of the symplectomorphisms group acts oil M then we have к Hamiltonian vector fields X( (£ e 6), to which the one-parameter group of transformations

/, (g = exp[t(), a(g) = f3 = exp(tX()) corresponds. Thus, we may define the derivatives CxtB for any tensor field B. 5. Consider the principal Sp(2n, /i)-bundle

p-.V—>M (5) of symplectic frames on M and the group AutV of all automorphisms of V (which are bundle maps). A map A of "P onto V will be called an automorphism of principal fibration if A[qb) = A(q)b for every q € "Pi b £ 5p(2n, R). Bach automorphism A determines a transformation of the base M = P/Sp(2n, R); we shall denote this transformation by p(A). The group of automorphisms of the principal fibration V determining an identity transformation of the base will be denoted by Gauge Sp(2n, R). Gauge transformations of the space V are defined by smooth functions т(х) on M with values in the group Sp(2ra, R) and a set Gauge Sp(2n, R) of all т(х) may be identified with the space of sections of the associated bundle V XsP(2n.«) Sp{2n, R) -» M. We have a homomorphism о of the connected Lie group G into Symp(M, u). The group Symp(M, ui) of the canonical transformations of the manifold M is a subgroup in the group p[AufP) which preserves the symplectic form ш. Thus we defined the action of two groups on P: the action of the group a(G) С Symp(M, и) С AutV and the action of the group

Gauge Sp(2n, R) С AutV. Let us also consider a group Autc = a(G) x GaugeG, where GaugeG С GaugeSp(2n, R). The group Aulc. is the group of pairs (o(ff), T(I)), where a(g) 6

Q(G), T(X) € GaugeG and the product of pairs (a(g,), т,(х)) £ Auto, (a(g2), R2(I)) € Autc is a pair (o(ff), r(x)) given by formula

o(g) = a(g, )а(дг), т(х) = г, (х)т2(я). (6) 6. The action of a(G) on M induces an action on J. This action on the tensor of the complex structure is given by

c{g)J := fgJ (7) Analogously, the action of Gauge G on V induces the following action on J € Gauge Sp[2n, R):

1 I.J := TJT- <=> (f J)X = т(i)4r"'(i) (8) Finally, the action of the group Aula on J has the form

y(9,r)J ~ f(a{g)J). (9)

Suppose that there exists a homomorphism g —• {rs(x)} of the group G into the group Gauge G С Gauge Sp(2n, R). Then we may define a homomorphism r of the group G into the group Aula by the correspondence of an element

T(ff) = (otfo),Tt(x))6A«ie . (10) to the element G € G (cf. [13]). The action of T(G) on V induces the following action of the group t(G) С Aula on J-

r(g)J := t,(c(S)J) <=> (r(g)J)x = r3(x)(jsJ)xT-\x). (11) 7. The usual G-invariance of the complex structure means (see [6, 14]) that J is invariant under "'citomorphisms a(g) 6 <*(G), i.e.

<*(g)J = J, VJ E G. (12) '406

Locally the condition (12) is equivalent to the following condition:

CX(J = o, e g, (i3) where Cx( is a Lie derivative along the Hamiltonian vector field X( 6 <*.(S) on M. From (12) it follows that G С U(n) and therefore

1 r(g)J = Ts(a(g)J) = f,J = r,Jr," = 7, VS 6 G, because Ts(I) С Gt С (U(n))x. Thus, from the invariance of J under a(G) follows the invariant under the group T(G). 8. We would like to impose the weakened form of the symmetry requirement. We shall weaken the conditions (12) demanding the invariance of the complex structure J only under the 7--automorphisms:

r(g)J = J, Vs € G. (14) We shall call condition (14) the weak G-invariance condition of the complex structure.

To define the r-automorphisms one must require the existence of the fields тд(х) on M. Let us suppose that a symplectic connection V is defined on M. Let us also suppose that ther are

к = dirnG covariantly constant tensors W{ = х)"ц), Ws = 0, £ G Q, which constitute a basis of the Lie algebra Qr С sp*(2n, R) for every x € M. Then for g = exp(£) the function rs(x) may be expressed in the form

т,(а) = ex p(Wt(i)). (15)

It is clear that in virtue of covarianl constancy of the tensors Wj, all such functions Ts are completely determined by their value at the point x — 0 and parametrized by the group manifold G. 9. Consider the transformation (7). Let us denote by a(K) the subgroup in a(G) under which the complex structure J is invariant. The group и(А') is the image under the homo- inorphism a of the subgroup К of the group G. The Lie algebra Q may be decomposed in the following way e = £®<3, (16) where К is a Lie algebra of the Lie group A', and Q is a tangent space in the origin of the homogeneous space Q — G/K. We shall number the subspacc AC in Q by the indices i, j,... = 1 ,...,k — l, and the subspace Q in G — by the indices a, /3,... = 1,..., I. Then locally the condition (14) of the weak G-invariance may be written in the form:

Cx.J; = 0, Cx,ru = J^Wa)l-(Wa)lJ"x, (17) where (.Y;, Xa) are the Hamiltonian vector fields on M constituting the basis of the subspaces a.(K) and a.(Q) in the Lie algebra a.(G)- 2 10. Taking the Lie derivative Lxa of the identity J = — 1, we have

Jx + tf (Cx.Jx) = 0. (18)

0,1 Condition (18) means that CXAJ transforms the (—t) eigenspace Г' 'Л/ of J to the (+i) eigenspacc Г'1,0'Л/ and hence

CXAJ where T'I,0'M denotes the holomorphic vector bundle of the (1,0) tangent vectors. Linearizing the condition ui(X, JY)+ w(JX, Y) = 0, we find that {15, 16]:

CxJ = (19) '407 where ш = Л dzl is the Kalller form, a,b,... = 1, ...,n, and

are the C°° sections of the bundle Т<~1'°Ш ® T(1'°)M over M. Tensors B„ are symmetric (i.e. B°b = £'") and may also be decomposed with respect to the basis ® gfr with components

ЯГ = {•№)* - (21)

11. As already noted, the conditions (14) and (17) mean that there are к tensor fields W( on the manifold M. In particular there are I = dimG/K tensor fields Wa = {(Wo)!!} or, equivalently, I fields Ba — {Bj"}, which are the global sections of the bundle

П xG GjK —• M (22)

over M, associated with the principal G-bundle Ti[M,G) over M. Let us denote the total space of the bundle (22) by From the purely differential geometric point of view, this space is the product

Z = M x G/K, (23)

i.e. the trivial bundle. The conditions (14) (and (17)) Inean that the complex structure J depends on the / = dimG/K parameters ta (coordinatesion G/K) and may be represented in the form

J = г'Ч r„ (24)

where J0 is a fixed (canonical) complex structure and the element тд € GaugeG has been written out in (15). The complex structure J given by this formula coincides with Jo if and only if та[х) belongs to the subgroup Kx С Gx. Hence, on M we have the /-parametric family of covariantly constant complex structures which are weakly G-invariant by construction. Notice that the weak G-invariance of the complex structure introduced by us is analogous to tin generalized G-invariance of the connection in the principal fibre bundles studied in the papers[13, 17].

References

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SUSY LAX PAIR IN THE GELFAND-DICKEY HIERARCHY Ziemowit Popowicz1

Institute of Theoretical Physics, University of Wroclaw, pi. M. Borna 9, 50-204 Wroclaw, Poland

The idea of the supersymmetry appeared first time in the physics on the beginning of the 70-th years [1]. The main merit of it was to unify the bosons and fermions in order to treat on equal footing these objects. This could be achieved introducing the supcrmultiplet

(I,0) = C(I) + O-TI(X) (1)

where f(x) and u(x) are assumed to be the fermions (bosons) and bosons (fermions) operators respectively. Assuming that 0 is a Majorana spinor we can proved that our supermultiplet has good behaviour under the Lorentz transformations. However if we would like to consider the classical limit of such multiplet we encounter with the difficulties in the interpretation of the fermions field ((x). Usually such object are considered as the Grasstnan valued function (an- ticommuting object) which does not appeared on the classical level. This situation drastically has change if we consider more complicated supcrmultiplct

Ф (x,0,, 02) = ui(x) + 0iCi(i) + 0iC2(x) + в20,и(х) (2)

where now w and u are bosons operators while f, and (2 are fermions operators, 0, and 02 are two Majorana spinors and from that reasons such multiplet is called the extended (N = 2) supcrmultiplet. Several years ago such extended supersymmetry has been used to the construction [2,3] of the new soliton like equations. This idea based on the observation that if we

construct a some evolution for Ф (x, в,, 92) and consider the so called bosonic limit in which Q and (2 a" fermions fields vanishes we obtain the interacting system of evolutions equations on the functions u and w. Interestingly in this limit the function

Ф = и)+ад» (3)

could be considered on the abstract level on which 02 0X is the symbol with the properties J (02вi) = 0. Such functions have been considered by Russian mathematicians 011 the beginning of this century [4] as the generalization of the usual classical mechanics in the same pattern as the extension of the usual numbers to the complcx numbers. In order to investigate such bosonic limit it appears that it is much easier to work with the fully supersymmetric evolution equations and oil the end of our considerations to put

= (2 = 0. Hence the knowledge of the methods of the supcrsymmetrizations is very useful. However at the moment we do not have the unique prescription how the given classical theory can be supersymmetrized. An undoubtedly affirmative answer must be left for future work, but at the very least one should notice that there are three different frameworks of the supersymmetrization of the given classical theory: the geometric, algebraic and straightforward method. In the geometrical framework [5-6] the equation of motion (especially the solitons equations) are considered in the form of the Cartan-Maurer equations 011 the matrix 1-forms belonging to some Lie algebra of a Lie group. Then the supersymmetrization is performed by generalization of the Cartan-Maurer equation to the graded supersymmetric Lie algebras. In that way several interesting models have been Supersymmetrized including the extended supersymmetry [7-8].

'talk given on the International Workshop on "Symmetry Methods in Physics" in Memory of Y.A. Smorodin- ski, Dubna 06-10.08. 1993 '410

The second approach is based on the hidden symmetries of the integrable systems. For example Bogoyavlensky [9] discovered that the classical Toda lattice is connected with the simple Lie algebras and Leznov and Saveliev [10-11] showed that the periodic Toda lattice corresponds to the contragradient Lie algebras. In the Leznov-Saveliev approach the vanishing of the strength tensor of the non-abelian gauge group is utilized as the definition of the Toda lattice. This connection enable us to perform the generalization of the Toda lattice to the nonextended [12] and extended [13-14] supersymmetric case. The method presented in [15] allows us to obtain the extended superconformal symmetries which generalize the bosonic W- algebras. Other possibilities in this framework is to use the Lie algebraic interpretation of the Lax operator [15]. Indeed in this case one can associate with the afflne Lie algebra a scalar Lax operator of the n-th order and investigate the time evolution of the system via the Lax pair. This can be extended to the supersymmetric case by the use of the Lie superalgebras. This method has been adopted to the N = 2 supersymmetric Korteveg-de-Vries equation [16- 17] and recently Inami and Kanno [18] constructed the Lax pair for the supersymmetric N = 1 Boussinesq equation. Notice that this method is restricted to that case where we know explicitly the hidden symmetry. Interestingly in the case of the N = 2 supersymmetric KdV equation this method produces only one supersymmetric extension but we know that there are three different extension of these equation. How to fit remaining two cases [19-20] in this method is still an open problem. From that reasons we prefer to use the straightforward method in which we simply to rewrite the given equation or Lax pair in terms of the superfields and the covariant derivatives. However such a superfield generalization is not always unique but this freedom can be restricted assuming an additional condition or conditions ( as hamiltonian structure, completely integrability etc.) on the system. Such method uae the tremendous computations and hence in order to simplify it the use of computer symbolic algebra is very useful. In this talk I will explain how it is possible to obtain the supersymmetric extension of the so called Gelfand-Dickey hierarchy which describes the huge class of the completely integrable Hamiltonian systems. This direct method based on the use of the "golden rule" of the supersymmetrization which could be explained as follows: 1: We use the following supersymmetric derivatives

D\ = дв, + в,дх, Dj = да, + в2д, (4) D) = D\ = dx, DY D2 = -DIDI or

1 1 0i = dh - -в2э, Di = д0г - jjM* (5)

DJ = D\ = 0, DiDj + D3Di = -dx 2: The space of the supersymmetric functions is defined by the following vectors

K(Ф) = Кш + + в2К0 + в2 вгК„ (6)

and by the following covectors

7(Ф) = 7« + 7c. 01 ~ К,в2 + LJ2 01 (7)

for which the scalar product

(K,7) := j (Ku7ui-Kw7„+K

is well defined. We use the Berezin notion on the

Bi d = Stj (9)

3: The conformal dimension could be associated with the scaling properties of the given variables, function and equation. Having in mind these properties let us formulate the "golden rule" as the following rule: Everywhere in the classical equation where appears the function u alone let us replace it by the superfunction in the form of (2). Everywhere where appears the product of two function for example f and g let us replace it by all combination of the corresponding superfunctions, its (super)derivatives in such a way that to obtain the conformal dimension one less then the classical product of / and g. For example applying this to the famous KdV equation we obtain the following generalization

2 2 Ф< = - Фттт + оаф* + а2ф фх + а3 (А й2ф) ф +

+ сцф(01ф)(01ф) + ав(01ф1)(02ф) + (10)

А + ОВ (АФ) (П2ФХ) + ГФХ (А А Ф)

+ а8 ^(Di А where а,- are an arbitrary constants. From this formula we see that such way is indeed not unique. How it can be restricted ? Does other properties of the KdV equation survive in this procedure? Hence we can ask on 1. In the bosonic limit we should obtain the usual KdV equation. 2. Is it possible to find the bihamiltonian formulation of super KdV equation. 3. How one can construct the Lax operator. 4. What about the integrability. 5. What about the completely integrability. In the case of the KdV equation these problems have found the solutions in the following papers [2,20,19]. What we can say if we would like to apply this strategy to the whole Gelfand- Dickey hierarchy [21] ? The present author considered in [22,23] the extended supersymmetrization of the "classical" Lax operator

L = dxxx + vdx + u (11) where v and u are two functions, which produce the "classical" Boussinesq equation. The most general supersymmetric partner of (11) could be assumed as

L = adxxx + pDxD2dIX + zldx + zO , (12) where a and /9 are arbitrary constants and zl, zO are the elements constructed from all possible

combinations of the elements Di,D2,dz and two superboson fields of the conformal dimensions 1 and 2 in such a way that the object zl and zO have the conformal dimension 2 and 3 respectively. By using the symbolic manipulation package REDUCE I were able to compute the Z>''3 and compute the commutator

Li = [LT'I] (13)

which give us the equation of motion. As the result I found that the following Lax operator

I = A[3« + Jdx + T]D2 (14) '112 reproduces the N — 2 supersymmctric Boussinesq equation

- J.г - Jv72 + 2Г (15)

' = 11 - 2J „ - 2J J + -ЗГ„ + 2 (T J - TJ ) Л z It X r (16)

-2(D2Jx) (Л, J) + 2 (D,T) (D2 J) + 2 (D2T) (A J) j •

These equation can be easily obtained if wc use the obvious relation

DiAD2DlBD2 - -DiAdBDi I A (DlA)(D2B) D2, (17) where A and В are an arbitrary superbosons functions. Notice that the equations (15-16) can be reduced to the simpler form by shifting the superboson T, as follows

(18)

After such transformation, our equations take the simpler form

d (19)

J„ + 2 (£>2 J) i- dt и,+ (20) TJr - (D,D (2), J) - OhT) (B, J-)] .

This equation coincides with the equation obtained by Ivanow and Krivonos [24] for the special value of the free parameter (in their notation ct =- —4/c) if we transform this to the same form as the equation (19) and scale our ,/ superfunction. The equation (13) could be generalized to the whole hierarchy of the integrable equations if ive generalize this as

Llq = [L"+,L], (21) where q may assume suitable integer or rational values. Tim choice

A = 9 H-tuS"1 + ..., (22)

give us the Kadomtsev-Pietviashvilli hierarchy in the Sato approach, while the choice

N N 2 L = d + UN-2d - + ...Uo, (23)

with the fractional power of q leads to the Gelfand-Dikii construction. Now using the relation (14) it is easy to compute the fractional power of the Lax operator and then the equation of motion. Let us present these computation for the second member in the hierarchy for which q = jj and N ф 2 case. Then we have

N l £ = £>, [a + g w^j D2, (24)

/Л = я, (a + luv-,) D2 , (15) '413 with the following equation of motion

jtWN_2 = (2-1V) UV-2,2 + 2WW, + ^(2-N) WN-2WN-V, (26)

N JTW, = - ~ ( A ) WN.2,N-, + 2W..,., + ^Wn-JW,., +

tee 4 '

(27) +1 ё (0 (o.w.x^^,.,) +

+ W„2 + -l(D,Vi/N.2) (D2W,),

for s Ф 0, N - 2,

d 2 2 -lib = - 2.Л-- 4- t=0

9 w-j

»«0

+ WO,2 + JJ WN-2 ^0,1 - |R (DIWN-2) (D2WO) ,

where Wnim denotes the usual m-th order derivative of the WN superfunction. These formulas give us the extended supersymmetrization of the second member in the Gelfand-Dikii hierarchies. Using the similar procedure one can generalize it to the arbitrary member in this hierarchy. The case N = 2 in the formula (24) corresponds to the KdV equation and it should be considered independently because then we have to put q = 3. For this case we obtained one of the extensions of the N = 2 supersymmetric versions of the KdV equation which corresponds to the a = 2 case.

This paper has been supported by KBN grant 106/p3/92/03.

References

[1] J. Wess, J. Bagger, "Supersymmetry and Supergravity" Princeton Series in Physics (1983).

[2] C.A. Laberge, P. Mathieu, Phys. Lett. В 215 (1988) 718.

[3] M. Chaichian, J. Lukierski, Phys. Lett B212 (1989) 461.

[4] F. Dimentberg, " Vintovoe ischislenie i jego prilojenie v mehanike" Nauka 1965. (in Russian) '414

[5] R. D'Auria, Т. Rcggc, S. Sciuto Nucl. Phys. E 171 (1980) 167.

[6] R.'D'Autia, S. Sciuto, Nucl. Phys. В 171 (1980) 189.

[7] E. Napolitano, S. Sciuto, Phys. Lett. ВИЗ (1982) 43.

[8] E. Ivauov S. Krivorios, Lett. Math. Pliys. 7 (1983 ) 523.

[9] 0. Bogoyavlensky, Comm. Math. Phys. 51 (1976) 201.

[10] A. Lcznov, M. Savclicv, Lett. Math. Phys. 3 (1979) 489.

[11] A. Lcznov, M. Savclicv, Physics of Elementary Particles and Atomic Nuclei 11 (1980) 40, 12 (19H0) 123.

[12] M. Olshanetsky, Comm. Math. Phys. 88 (1983) 63.

[13] '/. Popowicz, .1. Phys. A: Math. Gen. 19 (1986) 1135.

[14] J. Evans, T. Hollowood, Nucl. Phys. В 352 (1991) 723.

[15] V.G. Drinfcld, V. Sokolov, Sov. J. Math. 30 (1975) 1975

[16] T. tiiami, H. Kanno, Comm. Math. Phys. 136 (1991) 519.

[17] C. Morosi, L. Pizzocchero, "On the Bihamiltonian Structure of the Supersymmetric KdV Hierarchies. A Lie Superalgebraic Approach" preprint of Dipartimento di Mathematica, Politecnico di Milano 1992 (to appear in Comm. Math. Phys. (1993)).

[18] T. Inami, H. Kanno, J. Phys. A: Math. Gen 25 (1992) 3729.

[19] Z. Popowicz, Phys. Lett A 174 (1993) 411.

[20] W. Oevel, Z. Popowicz, Commun. Math. Phys. 139 (1991) 441.

[21] L.A. Dickey, "Solitons Equations and Hamiltonian Systems" World Scientific 1991.

[22] Z. Popowicz, "The super W3 Conformal Algebra and the Boussinesq Hierarchy " preprint University of Helsinki HU-TFT-93-27, (hep-th/ 9305059).

[23] Z. Popowicz, " The Lax Formulation of the N = 2 SUSY Boussinesq equation" to appear in Phys. Lett B.

[24] E. Ivanov, S. Krivonos, Phys. Lett В 291 (1992) 63. 415

TOWARDS THE CLASSIFICATION

OF THE DIFFERENTIAL CALCULI ON GLq(N) P. N. Pyatov and A. P. Isaev Bogolubov Theoretical Laboratory, JINR, Dubna, SU-101 000 Moscow, Russia

Abstract

All the possible external algebra structures for Cartan's 1-forms (fl) on GLq(N) are classified. The external algebras obtained are supplied with the differential mapping d , obeying the usual nilpotence condition (d2 = 0), and the possibly deformed version of Leibniz rules. The status of the known examples of G£,(iV}-differential calculus in this classification scheme, and the problems of SZ/,(7V)-re9triction are discussed.

The general framework for construction of differential calculus on matrix quantum groups was formulated by S.L.Woronowicz in his remarkable work [1]. This paper clarified considerably the basic algebraic principles of quantum group differential geometry, but when tempting to perform an explicit construction for each quantum group we still meet the serious problems. Namely, Woronowicz's scheme possesses variety of differential calculi for each quantum group, but doesn't contain neither the regular procedure to find all the possible differential calculi, nor the criteria to choose the most appropriate one. An attempts to find the most natural differential calculus on matrix quantum groups starting from the differential calculus on quantum hyperplane were initiated by the works [7, 5, 8]. Not surprisingly the Я-matrix formalism [2] appears to be an adequate technical tool to approach this problem. In Refs. [9, 10, 11] this formalism was applied in finding the (possibly) most patural differential calculus on G£,(JV). But, unfortunately, when restricting to 5i,(iV) the calculus obtained reveals some unfavorable properties (see discussion below). So we forced up to search for an other possibilities, and the classification of differential calculi on quantum groups becomes an actual problem once again. In the present paper we make an attempt to use the natural algebraic conditions (most of them coincide with Woronowicz's basic principles) in finding and classifying all the possible differential calculi on GLq(N) and SL,(N). Namely, we present the classification for the differential complexes of invariant forms on GL,(N) and on SL,[N).

We consider Hopf algebra Fun(GL4(N)) generated by the elements of N x N matrix T = ||T;,||, i, j = 1,..., Ar obeying the following relations: RTT' = TT'R. (1)

Here T s Tj = T ® 1, T's T2 = 1 ® T, 1 is N ® N identity matrix, Rs Ru = Р12Я12, P12 is 1 permutation matrix and R12 is GL4{N) Я-matrix satisfying quantum Yang-Baxter equations and Hecke condition respectively

RR'R = R'RR', (2) l (R — ?1)(R + 9~ l) = 0, R*-AR + 1 = 0, (3) where A = q — q~l, R' = RM = -PMRM and 1 is N2 x N2 identity matrix. In accordance with (3), for q2 Ф —1 the matrix R decomposes as R =qP+- q~lP~ , (21) P* =(9 + ?-,)-1{9t,1±R}

'For the explicit form of GL,(N) Д-matrix see Refs.[3, 2]. '416 where the projectors P+ and P~ are quantum analogues of antisymmetrizer and symmetrizer respectively. The comultiplication is defined on the generating elements as ДЗу = Тц. ® Ту , and the antipode S(.) obeys the conditions 2 = T^SCty) = 8ц\ , so in what follows we use the notation Г"1 instead of S(T).

Consider some jV'-dime.isional adjoint Fj«i(G^,(N))-comodule A. We arrange it's basic elements into N к N matrix А = ||Л;>||, i,j = 1, -,N- The adjoint соaction is

A) - T',S(T)> ® A). = (TAT'*)) , (5) where in the last part of the formula (5)) the standard notation is introduced to be used below. The comodule A is reducible, and the irreducible subspaces in A can be extracted by use of the so called quantum trace ((/-trace) [2] (see also [6, 11, 13}). In the case of Fun(GLq(N)) it has the form:

-w-1+2 N TrqA = Tr(DA) s ^

Tr,(TAT~x) = 1 ® Tr,A = Tr„A , i.e. TrqA is the scalar part of the comodule .4, while the ?-traceless part of A forme the basis 2 of (N — l)-dimensional irreducible Fun(GLq(/V))-adjoint comodule. Consider now the associative unital C-algebra С < Aij > freely generated by the basic elements of A. As a vector space С < Лу > naturally carries the Fun(GL, (iV))-comodule structure. Now we introduce GL4(N)- covariant quantum algebra (CA) as the factoralgebra of С < Aij >, possessing the following properties [12]: (A) The multiplication in this algebra is defined by a set {a) of quadratic in Aij polynomial identities:

C°jkiAi]Ak, = CfjAij + C". (7)

In other words, CA is the factor algebra of С < A(j > by the biideal generated by (7). (B) Considered as a vector space CA is a GL,(N)-adjoint comodule, so the coefficients C°jkl in (7) are q-analogues of the Clebsh-Gordon coefficients coupling two adjoint representations, and the set of the relations (7) is divided into several subsets corresponding to different irreducible Fun(Gi,(A'))-comodules in A?) A. Parameters Cg are not equal to zero when couple A® A into the adjoint G^fA'J-comodule again, while С" ф 0 only if С^А^Аы are the scalars. (C) All the monomials in CA can be ordered lexicographically due to (7). (D) All the nonvanishing ordered monomials in CA are linearly independent and form the basis in CA. Now we recall that for the classical case (q = 1) the dimensions of the irreducible Fun(GL{N))-subcomodules in «4® «4 are given by Weyl formula [14]:

dim Л ® Д = [(ЛГ2 - 1) + 1]J = 2 • [1] © (3 + вц,г) • [A/3 — 1] ® (8)

2 2*„,2 . [i^LlMlzil] e + ф^. |iV (iV + l)(yV-3)j _

'Strictly speaking in order to define the antipodal mapping on Fun(GC,(N)) we must add one more generator (dct,(T))~l to the initial set {Jy} (see [2]). '417

where BNJA = {1 for N > M; 0 for N < M}. Thus, A® A divides into 2 scalar subcomodules, 4 (3 for N = 2) adjoint (traceless) subcomodules and 4 (1 for Л' = 2, 3 for /V = 3) higher-dimensional mutually inequivalent subcomodules. In quantum case according to the results of Refs. [15] the situation generally does not change (the exception is for q being root of unity). Below we employ the g-(anti)symmetrization projectors P± and 9-trace to extract the (irreducible) subcomodules in Д® A, therefore supposing from the initial that q ф -1 and ZY,1 = A"1 (qN - q~N) = [N\, ф 0. First, we shall obtain the sets of quadratic in combinations, that correspond to the left hand side of (7) and contain a four higher dimensional fun(GZ,,(Ar))-subcomodules (see (8)). Let us start with N2 x N2 matrix ARA (A = Л, = Ag 1) containing all the N4 independent quadratic in Ay combinations and having convenient comodule transformation properties:

ARA (TT')ARA(TT')-1. (9)

FVom (1), (4) it follows that P*TT' = TT'P*, hence we can split ARA into four independently traasforming (for N > 3) parts:

X±iz = P±ARAP± .X** = ARAP=f. (10)

Namely the q-traceless (in both 1st- ai.-4 2nd- spaces) parts of X and Х±т are the four higher dimensional subcomodules in A ® A with dimensions: -

1 1 1 A ; (Tr,A)A, A(TrsA), A*A = Tr,l2)(R- ARAR- ). (11)

Here we denote the 7Y,-operation acting in t-th space. The g-traceless parts of these combinations correspond to the irreducible adjoint subcomodules in A ® A. Applying Trq to the Eqs.(ll) once again we result in two independent combinations

(7V,A)2, 7Y,(A>), (12) corresponding to the scalar subcomodules. We refer to the expressions (10), (11) and (12) as higher-dimensional (h-dimensional), adjoint and scalar terms respectively. As >t was argued in [12], in order to satisfy the condition (C) for CA, the left hand side of the relations (7) must contain independently either X++ with X—, or X+~ with X~+. One can combine these purs into concise single expressions:

(9 + 9-1)(X++ - X~) = RARA + ARAR'1 , (13) (q + q-1 )(X"+ - X+") = RARA - ARAR . (14) The way of the combining the quantities (10) is not important. We choose the concise forms

(13), (14) because in the classical limit they are nothing but the anticommutator [A2, Aj]+ and commutator [A2, A^,. So it is natural to call (13) and (14) 9-anticommutator and q- commutator respectively. In view of this all the OA's with the defining relation (7) are classified into two types depending on whether their defining relations contain g-anticommutator or q- commutator. The first will be called further antisymmetric CA (АСА) and the last - symmetric CA (SCA). At the moment we still fix the h-dimensional terms in a quadratic part of the relations (7), but there remains an uncertainty in the choice of the adjoint and the scalar terms. Let 3For JV = 2 or q2N±i s 1 the last combination in (11) appears to be expressed as a combination of the first ones. 418 us show it explicitly. First of all we employ the simple dimensional arguments. In order to satisfy the ordering condition (C) at a quadratic level we must include at least independent relations in (7) (e.g. for the classical case of gl(N) it corresponds to the number of the commutators (A 3 case (5 combinations for N = 2) and are forced to include in (7) 2(1 for N ~ 2) independent q-traceless adjoint terms and two independent scalar termB. With this inclusion ACA's are defined by the set of ^С^+Ч relations. Thus, we have determined the number of independent adjoint and scalar terms in symmetric and antisymmetric CA's, Note that ^-commutator and g-anticommutator itselve contain the true number of adjoints and scalars, which is demonstrated by the following symmetry properties:

P±{RARA - ARAR}P± = 0 , (15) /^{RARA + ARAR"1}^ = 0 , (16)

But there is an opportunity to change the form of the adjoint terms in the left hand side of Eq.(7) (represented by the combinations (13,14)) without changing of their number. Indeed, consider the quantities ЫЩЛ)) = RU(/1)R±1 ± U(A), (17)

U{A) = o(R)AJ + (&(R) — e(R))(7V,/l)A+ +(c(R) - e(R))A(7Y,A) + d(R)(A * A), (18) where /(R) = /, + ftR and e(R) = j^(a(R) + q~Nd(K) - 1). We make the e(R)-shift of the parameters 6(R) and c(R) for future convenience. Expressions Д± are the most general covariant combinations which contain only adjoint and scalar (for A+) terms and also satisfy symmetry properties ± ± Р А^Р = P±A+P* = 0 . (19) Hence, we can use Д+ and A_ (17) in the variation of the adjoint part of antisymmetric and symmetric CA's respectively:

RARA ± ARAR*1 = Д± . (20)

FVom until now we will concentrate on studying the homogeneous (pure quadratic) ACA's, which possess the natural Zrgrading and may be interpreted as an external algebras of the

invariant forms on GL4(N). To emphasize this step we change notations from A to П. All the other cases can be considered following the same way. As it was shown, the general form of defining relations for homogeneous ACA's is (20).

These relations contain 8 random parameters a,, 6(, с,-, d,-, (t = 1,2), but actually this parametrization of the whole variety of homogeneous ACA's is excessive. To minimize the number of parameters in (20), let us pass to the new set of generators:

f« = Tr,Sl (21) П = П-lptjj, Гг,П = 0, '419

Using these new variables one can extract the first scalar relation ш3 = 0 and (20) is changed slightly to ' RHRO + nRflR-1 = Д+(1/(П)), (:г*w* := 0г , — (22) where Д+({/(П)) = Rl/R + U and

U(Q) = a(R)n5 + 4(R)u»n + c(R)Ou + d{R)(Q * П) . (23)

Here as usual we imply that fi = П ® 1. Applying the operations TV^.^R"..^], i = 1,2, a,b = 0, ±1, after some straightforward but rather tedious algebraic manipulations one can rewrite (22) in the form

w J J 2 RftRn + OROR" =i(R)(5 + 9 R )((Rn R + n ) + rw(RnR + n)) , Пш = -ршП + trft , (24) ( 2 ш =0, where

x(R) = (qN + q~N) + ([AT], + Aq")R , (25)

2 N *(R) = (l(R))-' = [N _ + ^ (-/V + q- ) + ([A'), + \q )R) . (26)

Here we suppose that (JV + 2],[JV —2], ф 0. Note, that the case N = 2 can be treated similarly, if we exclude d(R)-term from the beginning (see (23)), since corresponding adjoint term П * П is not independable here. The system of relations (24) depends on the minimal set of four parameters: S, r, p and

1 N N S = ф- ((an - Aq )c - (a, - q )c„) , Р = Ф~* ((<*/ - 1)6/, - , с = ф-1 ((о, - l)d„ - (an - \qN)(d, - 1)) , г = (b„c, - Ь,с„) , ф = (di — 1)с// — due, ,

where (// + ///R) = x(R)(/1 + /3R) for / = {a,b,c,d) . Finally we should mention that relations (24) present explicitly one scalar term and one adjoint term. The resting adjoint and scalar terms (together with the pair of h-dimensional terms) are contained in first of relations (24). They can be extracted from this relation by the use of TV,-operation. The result is:

П * П = + тшй , Гг,(П2) = 0 . (27)

The next step to restrict the possible form of homogeneous АСА is to consider the reordering of cubic in П monomials. Let us consider two basjc cubic monomials: (RO)3u> and (R'RI2)3. In the classical limit these combinations become ilijtliu and П3П2П1 respectively, and for the ordinary external algebra of invariant forms on GL(N) the procedure of their ordering looks like Qjftiw —• ыП|П] and П3П2П1 —> П1П2П3. The quantum analog of this procedure is

f (RO)3w -wfiRnR-1 + ... 1 (R'RO)3 -» -nRR'flRR'-'nR-'R-' + ..., ( J

where by dots we denote some additional terms. The point is that such an ordering can be performed by two different ways, depending on whether we first permute the left pair of the generators or the right one. According to the condition D both results must be identical, i.e. the additional cubic terms in (28) calculated in two ways must coincide modulo the relations (24), otherwise the ordered cubic monomials would not be linearly independent. Omitting the '420 lengthy and rather cumbersome calculations we present the result in the following Theorem 1: For n general values of deformational parameter q there exist precisely four one- parametric families of homogeneous ACA's. The defining relations for the first pair of them looks like , ... . -J x I RTTIM + SIRS2R-' = /С, [SI + RFT RJ , \ ы2 = 0 ;

where «, = q ф -1, [Л/]„ ф 0, [N ± 1], ф ±qN[N ± 1], ф ±qN*\ [N ± 21, * 0, and I type: tlw = —ршй, рф 0; (30) II type: [fi,w]+ = crfi2, о ф 0, (31)

For both cases the following remarkable relation is satisfied:

RnJR« - ORf^R = 0 . (32)

The resting pair of families is connected with the first one by the -symmetry transformation and the defining relations for them are obtained from (29), (30), (31), (38) by the following substitution: R м R"1 , q <-> q~l (33) Finally there exists one more exceptional classical (q = 1) homogeneous АСА, defined by

< [ft,ui]+ = 0, (34) [ u,1 = 0 .

Let us make some comments:

1. In principle to convince ourselve that the algebras presented satisfy condition D we have to check the ordering of all the monomials of the type (Rn-j,n-i • ... • Ris^i)"-1" and (R-n-i.n •..- • Ri2Hi)n for n > 3. We carry out this procedure only for n — .'i case, but allowing to the Manin's general remark (4] for an associative algebras it is actually sufficient to check the consistency of the ordering procedure with condition D at a cubic level.

2. The parameters а ф 0 ( г ф 0 ) for type JI (exceptional classical) algebras are inessential. They can be removed from the defining relations by simple rescalings of generators u> (П).

3. Note, that when trying to find the explicit ordering prescriptions from the covariant formulas (29), (30), (31) and (34) we may obtain some additional restrictions on the possible values of parameters p, IT, T.

4. Finally, let us present the defining relations for homogeneous ACA's in terms of fi's (see (21)) I type : (RIZR + П) 1 RfiRfi + S2ROR-1 = к, (RJ^R + n1) + "type: R«2R + n2) \ ' (35) Note, that with the choice

Rimn + mmR-1 = о. (36) '421

As we argued before among the algebras presented in Theorem 1 there exists the true algebra (or maybe the set of such algebras) of invariant differential form on GLq(N). To make this connection more clear we shall supply the homogeneous ACA's listed in Theorem 1 with grade-1 nilpotent operator d of external derivation. The definition of d must respects the covariauce properties (5) of Cartan 1-forms, i.e. d must commute with the adjoint GL,(N)-coaction on CI. Hence, the following general anzats is allowed: 1 Г dofi = xti + ушй - zCl о d (3?. \ dou — —tuiod. 1 '

Note, that the last term in the right hand side of (37) defines the deformed version of Leibniz rules for differential forms. The ordinary Leibniz rules are restc'.ed under the limit z = t = 1. We stress here that the idea of deformations of Leibniz rules in the noncommutative differential calculi is not new (see e.g. [16]). Now it is straightforward to obtain Theorem 2: Under the restrictions of Theorem 1 there exists two distinct envariant differential complexes for type / algebras, defined by

d typeIA:{ .°Й = v(38) l ЙОЫ = -/Xii о d; '

, ia Г do& = шй-zClod, ,„„, type!В : < , , ' (39v ) [ aou = —ш о d. ' The differential complexes for type II and exceptional classical algebras are defined uniquely:

. ,, / doQ = П5-Hod, ,„„. typell: < ' (40) ( flow = -ш о d\ v '

Exceptional (don = w&-&od, , . classical case : [ (iou = —ш о d. Here all the inessential parameters are removed by ui- and Ct-rescalings. Proof: These restrictions are easily obtained by demanding

In conclusion, let us discuss the present status of the problem of constructing GL4(N) (5£,(A^))-bicovariant differential calculus (calculi). For the general values of N the only two examples (connected with each other by the symmetry transformation (33)) of such calculus are still constructed [8, 9, 10, 11, 12] (for the detailed analysis of N = 2 case see [17, 18]). The external algebras of invariant forms presented by these examples are of the second type (see remark 4 to Theorem 1). An attractive feature of this type calculi is the possibility to construct the local coordinate representation П = dT • TBut. there is also the serious shortcoming: unlikely to the classical case here the adjoint combinations 2 [П,ш]+ and П are not independent, so the simple classical recipe of passing from GL,(JV)- to 5i,(A')-case, i.e. to put ш = 0, fails at quantum level. As a result, in attempts to construct differential calculus on SL4(N) starting from external algebra of second type ( see [11, 18] ) the number of invariant 1-forms exceeds that of invariant vector fields and of local coordinate functions by 1. So, strictly speaking, we can't treat these quantum differential complexes as true deformations of differential calculus on SL(N).

A possible way to overcome this difficulty is to use type I external algebras instead type II ones. Since type IB differential complex have no conventional classical limit, the only '122

candidate here is type IA complex with an additional condition lim,_i p = 1. In this case the constraint w = 0 is noncontradictive and transition from GLq(N)- to SL,[N)- invariant forms is straightforward. But, due to the deformation of I.eibniz rules, for p ^ 1 the problems of construction of local coordinate representation, and of finding the suitable crossproduct for fl's and T's seems to be unsolvable. So type IA differential complex with p = 1 appears to be the only candidate for construction of consistent differential calculus on GL4(N) with it's possible restriction to SLq(N). We hope to revert to these problems in further publications. Acknowledgments This work was supported in part by the Russian Foundation of Fundamental Research (grant 93-02-382").

References

[1] S.L.Woronowicz, Comm.Math.Phys. 122 (1989)125. [2] L. D. Faddeev, N. Reshetikhin and L. Takhtajan: Alg. i Anal. 1 (1989)178. [3] M. Jimbo: Lr.lt.Math.Phys. 10(1985)63, ibid. 11 (1986)247. [4] Yu.Manin, Comm.Math.Phys. 122 (1989)163. [5] Yu.Manin, Notes on Quantum Groups and Quantum de Rahm complexes, Bonn Prep, MP1/91-60 (1991); Tear.Mat. Fiz. 92 No.3 (1992)425. [6] B.Zumino, Introduction to the Differential Geometry of Quantum Group, Preprint University of California UCB-FTH-62/91 (1991) and in Proc. of X-th 1AMP Conf., Leipzig 1991, Springer-Verlag (1992) p.20. [7] J.Wess and B.Zumino, Nucl.Phys. (Proc. Suppl.) 18B (1990)302. |8] G.Maltsiniotis, C.R.Acad.Sci. Pans, 331 (1990)831-834; Calcul diffe'rentiel sur le groupe line'arie quantique, Prep. ENS(1990). [9] A.Sudbery, Phys.Lett. B284 (1992)61. [10] A.Schirrmacher, in: Groups and Related Topics (R.Gielerak et al. Eds.), Kluwor Academic Publishers (1992) p.55. [11] P.Schupp, P.Watts and B.Zumino, Lett.Math.Phys. 25 (1992)139. [12] A.P.Isaev and P.N.Pyatov, Phys.Lett., A179 (1993)81. [13] A.P.Isaev and Z.Popowicz, Phys.Lett. B281 (1992)271; A.P.Isaev and R.P.Malik, Phys.Lett. B280 (1992)219. [14] H.Weyl, Theory of Groups and Quantum Mechanics, Dover Publications, Inc. (1931). [15] G.Lusztig, Adv. in Math. 70 (1988)237; M.Rosso, C.R.Acad.Sci. Paris, Ser. I, 305 (1987)587. [16] P.P.Kulish, Zap. Nauch. Sem. POMI, 205 (1993)65. [17] F.Muller-Hoissen, J. Phys. A 25 (1992)1703.

[18] F.Miiller-Hoissen and C.Reuten, Bicovariant Differential Calculi on GLn(2) and Quantum Subgroups, Prep. GOET-TP 66/92(1992). '423

MAXIMALLY INVARIANT CONFIGURATIONS IN THE SUfo) SKYRME MODEL Yurii P. Rybakov

Theoretical Physics Department People's Friendship University of Russia 6, Miklukho-Maklai Str., 117198 Moscow, Russia

Abstract Some classes of G-invariant fields for chiral models invariant under the action of a compact group G are considered. The conditions, under which the G-invariant critical point of the Hamiltonian proves to be its minimum, are formulated. For the SU(3) Skyrme model two classes of maximally invariant soliton configurations are specified. The existence of regular soliton configurations, parametrized by two radial functions and endowed with even topological charges, is proved.

1. Introduction Over the last decade the ingenious idea of T. Skyrme1'2 on representing the nucleon as a chiral soliton became popular with physicists. The main point of Skyrme's approach was the choice of nonlinear field manifold originated in spotaneous breaking of the chiral symmetry. In chiral models the principal chiral field U(x) takes values in a compact group G or in a homogenious 3 space G/G0, G0 С G. Thus for the case of G = SU(3) the mapping U(x) : Я -> SU(3) is realized. The finiteness of energy implies the natural boundary conditions

Q = deg [S3 -» 50(3) С St/(3)] 6 Z. (1)

Using the left invariant vector fields (left chiral currents) = U*dhU, ft = 0,1,2,3; on SU(3) one can construct the topological charge Q = 3 ~tb Jd xc'*Tr(h-lJtlk), (2) and the Lagrangian density in the Skyrme model

3 £ = -^Tr/„ + ^Tr[fM,/,,]', (3)

with e,A being physical constants of the model. The expressions (2) and (3) amount to the lower estimate for the Hamiltonian H through the topological charge Q:

J H > 6V2*r (£/A)|C?|, (4)

In view of (4) one concludes that the minimum of H can be realized in the given homotopy class. It turns out that for Q = 1 the energy minimum is attainable on the critical field configuration, known as the hedgehog ansatz3'4:

U = exp{i(n • Ai).F(r)}, (5)

where n = x/r, r = |z|, and Aj stands for the triple Ai = (Ai, А2, Аз)/2 of the su(2) algebra generators, realized by Gell-Mann matrices A,. However, there exists an inequivalent realization '424

of the su(2) algebra, given by the generators K2 = (A2, At, yielding another imbeding 50(3) С SU(3), used in (1) and (2). It appeares that the corresponding field configurations are endowed with the even topological charge Q = 2n. 2. The Structure of Maximally Invariant Configurations As was revealed in searching for the minimum of G-invariant functionals3'3, there appear special classes of equivariant field configurations фо(х), defined by the condition

Фо(х) = Тефа(д~'х), g € G' С G, (6) with Ts being the representation operator of the subgroup G' С G. In our case of the group G = SU(3) ® S0(3)s, where S0(3)s is the group of spatial rotations, two subgroups G' play the important role, namely

Gi =diag[50(3),®50(3)s],

G2 =diag[50(2)/®50(2)s],

with 50(3); being the group of isospin rotations. Following the recipe (6), one obtains two kinds of field configurations known as spherically symmetric or axially symmetric onis, for the choice of the group Gj or G?, respectively. Then the following theorem is valid3,6:

Theorem. Let a Gk-invariant field фв, к = 1,2, realizes minimum of the Hamiltonian H, restricted to the Gk-invarianct class. Let a bo the functional Н[ф] be convex with respect to the

derivatives д;ф at the point ф— Фа- Then the field ф0 realizes the true minimum of Н[ф]. Proof. Write down the second variation of the Hamiltonian H in the vicinity of the field фа in the form of the scalar product:

РН={у,к(х)у), у = 6ф,

where K(x) is the Jacobi operator for Н[ф]. Due to the G.-invariance of Я the operator K(x) should be also Gi-invariant, that is

l 1 TgK{g- x)T- = K(x), g e G,-.

This fact implies that one can express К in terms of the Casimir operators Gi = (У + I)2 or 7 C2 = (J3 + I3) of the correspondent groups, with Д- and /, being generators of S0[3)s and 50(3)/, respectively. Taking into account the convexity of Я in derivatives we infer that the ojperator к is elliptic, and so its spectrum A monotoniously increases with the spectrum a of Ci, that is _ о On the other hand, for an invariant eigenfunction yo of к one gets C;t/o = 0 and kya = •Wo, Ao > 0, as фо realizes minimum of H in the invariant class. If now one consideres a

noninvariant eigenfunction y„ of К with the eigenvalue An, then one derives

С(У„ = ot„y„, o„ > 0.

Therefore, in view of (7) one concludes that A„ > A0 > 0 i.e. S*H > 0. The theorem is proved. Now consider Gi-invariant configurations corresponding to A] realization of the su(2) algebra. Then the equivariance condition (6) reads

-i(xx V)U+ [Л2, V] = 0. (8) '425

General solution to the Eq. 8 can be written in the form8,7:

U = e'"(I — A2) + c~iv/2 • (Л2 cos и - iA sin v), (9) with u, v real radial (unctions and Л = (Кг • n). Notice that the boundary conditions at space infinity emerge from the finiteness of energy:

u(oo) = u(oo) = 0.

As for the conditions at the origin r = 0, they can be derived from the regularity of the mapping U : Я3 -» 51/(3). In fact, detU = 1 implies the relation

U(0) = lexp{i2Trm/3}, ,n g Z which is consistent with (9) if

v(0)=2;r^, и(0) = шг, (-l)m = (-1)"; «e.Z. (10) u Inserting (9) in (2), one obtains the density of the topological charge in the form

P = - [u ~ sin u C0S(3W2)] ,

that permits, with (10) in mind, to accomplish the integration in (2) and to find

Q =; 2u(0)/x = 2n.

Thus, the topological charge of the configuration turns out to be even, the minimal one Q = 2, corresponding to the choice m = n = 1. 3. Existence of Invariant Configurations The existence of weak solutions with the structure (9) was first established in Ref. 5. Our aim is to prove the existence of regular (classical) solutions (9) in the Skyrme model (3). To this end it is convenient to use the substitution 2 " = / + 9, v = з (/ - 9),

imposing the boundary conditions

/(0) = x, s(0) = /(oo) = 9(00) = 0. (11)

Performing the scale deformation 1—> reA(3/2)I/'', one reduces the Hamiltonian to the form

where the functional /[/,3] reads 00 00 1 = JdrH = J dr |r2(/'2 + /V + /J) + 3(sin2 / + sin2 g) + о 0 +/'2 sin2 / + s'2 sin2 g + ^ [(sin2 / - sin2 gf + sin2 / sin2 g)}. (12)

Prove that the lower bound of H is attainable in the class of smooth functions, satisfying the

boundary conditions (11). Consider first the modified functional 1аь = Ja' drH, defined on the '426 compact interval [a, 6], a > 0, b < oo, with the conditions (11) being preserved at the end points. Let {/mSn} be a minimizing sequence of correspondent smooth functions. Since the functional /'j* being bounded from above, one concludes that for y„ = /„ or gn ь JdrrYl

Using the Schwarz inequality, one derives from (13) that for r0 G (a, b) ь ь ь

3 l |Уп(г0)| < Jdr\y'„\ <(/$•/Jrr • < C,r-0 '\ (14) Го ГО ГО

In view of (13) and (14) f„,gn belong to the Sobolev space #'(a, 6), that implies the weak convergence {/„,jn} ^ {f,g}- Now represent as the sum of squared norms in £2(0,6):

*=i

where Aj*' denotes the following vectors of Za(a, A):

l/5 n n) A$ = 2- rj,n', hp = 6< > + *<">, Ai = 4^sinj,„,

n) 4 3 = »»' sin У„, Ai = 2(sin /„ - sin gn)/r, h^ = 2 sin /„ sin tf„/r.

Due to the compact imbedding Я'(а,Ь) С С (а, 6) one concludes that sin yn Д siny, Л J."' & Afc(y). Taking into account that the squared norm is weakly lower semicontinuous in Ьг(а, b), one finds that inf /аь is weakly attainable. Now, to prove the classical origin of this weak solution, write down the equations of motion in the form y" = F(r,y,y'). (15)

The functional 1аь being upper bounded, one finds that F £ Lx(a,b), whence y' 6 C(a,b), and in view of (15) у" € C(a, b). Repeating of this procedure implies у S С°°((^6). Now, to make the fined step, one must put о —> 0, b —* 00. The limiting process а -• 0 proves to be regular one due to the a priori estimate |y'(0)| < 00. In fact, if this estimate fails, then |j/'| —» 00 as r —» 0, and the functions x = (f — ir)J, z = g1 satisfy the equations:

x" = 2" = (1 + z)/rJ,

with the regular solution x = z = CJr3. This contradiction proves the validity of estimate. As for the limit b —• 00, one can deduce from the integral form of the equation of motion, that y(b) = 0(6-2), y' — 0(i-3), thus proving the regular character of the limiting process U /• '427

References

[1] Skyrme TM.K.J / Proc.Roy.Soc. (Loni.), (1961), 260 A p. 127.

[2] Makhankov V.G., Rybakov Yu.P., Sanyuk V.I., The Skyrme Model: Fundamentals, Meth- ods, Applications (Springer-Verlag, Heidelberg-Berlin-New York, 1993) 265 pp.

[3] Rybakov Yu.P,,//Problems of the Theory of Gravitation and Elementary Particles, (Moscow, Energoatomizdat, 1982) Vol. 13, p. 187.

[4] Kozhevnikov I.R., Rybakov Yu.P., Fomin M.B.//ТНеог.в Math. Phys., (1988), 75 p. 353.

[5] Rybakov Yu. P. // Problems of High Energy Physics and Field Theory. Proc. of X Work- shop (Nauka, Moscow, 1988), p. 349.

[6] Romanov V.N., Frolov I.V., Schwartz A.S.//Thtor.& Math. Phys., (1978), 37 p. 305.

[7] Balachandran A.P., Lizzi F., Rogers V.G.J., Stern k.//Nucl. Phys., (1985), 256B p. 525. '428

SYMMETRY AND STATISTICS OP THE ELEMENTARY COLLECTIVE EXCITATIONS IN SOLIDS Vladimir L. Safonov Institute of Molecular Physics, Russian Research Center "Kurchatov Institute", 123Ш Moscow, Russia

Abstract New phenomenological approach for the description of quasiparticles in many-body interacting systems is proposed. The equations of motion for the elementary collcctive modes are supposed to have the same structure as the equations for real particles in vacuum. The cristalline anisotropy is introduced by means of corresponding spaco-time geometry.

1. Introduction Concept of quasiparticle as the quantum of collective excitation in many-body systems is a very useful tool in solid state physics (see, e.g., [1,2]). In many respects quasiparticle behaves as usual particle, however, the specificity of a crystal (which plays a role of "vacuum") imposes peculiarities on their properties. As far as quasiparticle is a product of excitation of all atoms ir. crystal, it is natural to study the elementary excitations in solids on the base of general symmetry considerations. In the present paper 1 propose to study quasiparticles on the base of relativistic wave equations. This idea has the following motivation. The equations of motion for real particles are (to some extent) the phenomenological ones. Vacuum can be considered as a kind of medium with a spherical symmetry. Crystal structure can be considered as a kind of anisotropic space or anisotropic vacuum. We can assume that the equations of motion for quasiparticles have the same structure as the equations for real particles [3,4]. The cristalline anisotropy in this case is introduced by means of corresponding space-time geometry. Unknown internal forces of the medium are included into a metric tensor. In order to illustrate the above idea below the generalized Dirac equation will be constructed and the non-relativistic Hamiltonian containing electromagnetic fields will be calculated in the longwavelength approximation. This Hamiltonian is expected to be true for the description of conduction electron (and hole) in highly anisotropic crystalline systems.

2. Particle in the anisotropic space Consider the most simple model of a crystal as the uniform anisotropic medium. The space-time interval for it can be written in the form з (1) ;=i where с is the light velocity, G/ are the diagonal components of metric tensor. The action for a free particle with a rest mass mo and a charge e moving through electromagnetic fields is defined by [5]

s (2) '429

where A = (Aj, <43), ф are the vector and scalar potentials, respectively, С is the Lagrangian. The generalized momentum and the energy are P = dCjd\ and £ = (vdC/dv)— С , respectively. For the energy one obtains the following expression

2 2 2 p 1 2 £ = [(m0c ) + c £ ( < ~ -Aif/Gi ] ' + еф. (3) •=i c

The corresponding non-relativistic energy is

з P ! 3 Sn„i = Y2 ( < - Д) /2т* + еф. (4) i=i 0

Thus iTi* = G,m0 can be considered as the tensor of effective masses (in a diagonal form).

3. Relatr/istic wave equation Taking into account that the operators of momentum and Hamiltonian are defined by P =

—t'fcV and H = ih§-t , from (3) one can obtaii? generalized Klein-Gordon-Fock equation for the spinless quasiparticle

1 2 2 (i&- еф) + с £ (Pi ~-Aif/Gi + (m„c ) Ф = 0. (5)

In order to construct a generalized Dirac equation we start £from a wave equation of the form з Е /3m0c' + CY2<*J(PJ ~ + Ф Ф (6) i=i ° in which P and a, are 4x4 matrices. Comparing the Hamiltonian of Eq.(6) with the relativistic energy (3) in the absence of electromagnetic fields, one obtains

2 a,a: +ajQi = 2«0/G„ аф + = 0, P = 1. (7)

These relations may be satisfied by ?)• -а л), -a ?)• » if all Gj > 0 and

аз = J), (9) when Gi <0, Gj < 0, G3 >. 0 . Here ay are the Pauli matrices. Note that the Eqs.(5),(6) can be obtained from corresponding Lagrangians. Second quantization of (6) (and (5)) is analogous to that of given in [6] for the isotropic case. '430

4. Non-relativistic Hamiltonian The non-relativistic Hamiltonian containing the relativistic terms of the order of c~2 can be received by a canonical transformation due to Foldy and Wouthuysen (see, e.g., [6]). The resulting Hamiltonian is

2 3 m0c + £(Pj - - А,?/2т; - (8m »c*)->(£ j=i J=I

+ " + a») where /2 72 l/2 Па = -(GsG3)-' FlCTl - (СзС,)"' ^ - (G, 0 and

1/J ,/s 1 = (I Ga | G3)- F,a^ + (G3 | G, |)- iW - (^G,)- /^ (12) if G, < 0, Gj < 0, G3 > 0 .

We use the following notations:

s eft + 4rn c 1 ( ° )" [iK"iE)J + 2(E x P).]},

H = rotA, E = -c"'3AIdi - дтаЛф. H, E are the magnetic and electric fields, respectively. It should be mentioned that the Maxwell equations in the anisotropic space have the same structure as the equations in the isotropic space. For the positive energy solutions in (10) 0=1. Note that if all G, = 1, then the Hamiltonian (10) coinsides with the classical result.

5. Discussion For simplicity let rewrite from (10), (11) only the Zeeman Hamiltonian

1 3 = HwoOjHj, (13) i=i

where po = eh/2m0c is the Bohr's magneton and

gj = 2т0/т-ф (14)

1 3 1 2 1 2 m-cl = (mjm;) / , m'c2 = (mjmj) / , m^ = (т^) ' . We can see that the ^-factors depend on corresponding cyclotron effective masses. Cohen and Blount [7] have obtained the analogous result by means of traditional approach in the limit of large spin-orbit coupling. This result gave a good description of experimental data (de Haas - van Alphen oscillations) in Bi. In other metals an interaction between quasiparticles seems should be taken into account. '431

In the two-dimensional case (mj ф 0, mj ф 0, mj —» oo), according to (14), we can observe only one, perpendicular to the 2D plane, spin component. This fact is in a good agreement with the magnetotransport measurements [8] on strained quantum wells and it can not be explained on the base of Pauli equation. In the one-dimensional case (mj ф 0, m\ —» oo, mj —> oo) the spin of quasiparticle falls out of Hamiltonian. This fact can be interpreted as the spin is a hidden coordinate in this case. Following such a hypothesis, tiie conduction electrons (or holes) with up and down spin components are indistinguishable. This means that at once two (instead of one) quasiparticles can occupy the same energy level. In other words, one can expect a para-Fermi statistics with the rank p = 2 for "one-dimensional" electrons instead of Fermi statistics. Several questions concerning systems with para-Fermi quasiparticles have been considered in [9]. The above analysis concerns mainly quasiparticle with the spin S = 5, however one can develop quite analogous approach for 5 > 1 quasiparticles. It is interesting that for 5 = 1 quasiparticles we can expect para-Bose statistics with a rank p = 3 for \D case and so forth. In the present paper we considered quasiparticle in crystal as a particle in the uniform anisotropic space. One can also develop more general models in which the metric tensor G„(г) has a spatial dependence (for example, periodic dependence which models a crystal lattice). In principle, it is possible to construct an additional equation for Gy(r, t) which can take into account a dynamics of many-body system.

Acknowledgements I would like to thank Prof. Yu. A. Danilov, Prof. I. Harada, Prof. M. I. Kaganov, Dr. A. K. Khitrin, Prof. D. A. Kirzhnits, Dr. V. V. Losyakov for helpful discussions. This work was supported, in part, by a Soros Foundation Grant awarded by the American Physical Society.

References

[1] M. I. Kaganov and I. M. Lifshits, Quasiparticles (in Russian), Nauka, Moscow (1989).

[2] H. J. Zeiger and G. W. Pratt, Magnetic Interactions in Solids, Clarendon Press, Oxford (1973).

[3] V. L. Safonov, Phys. stat. sol. (b) 171 (1992) K19.

[4] V. L. Safonov, Phys. stat. sol. (b) 176 (1993) K55; Int. J. Modn. Phys. B7 (1993) in press.

[5] L. D. Landau and E. M. Lifshits, Field Theory (in Russian), Nauka, Moscow (1973).

[6] J. D. Bjorken and S. D. Drell, Relativislic Quantum Mechanics, Relativistic Quantum Fields, McGraw-Hill Book Co., New York (1965).

[7] M. H. Cohen and E. I. Blount, Phil. Mag. 5 (1960) 115.

[8] R. W. Martin, R. J. Nicholas, G. J. Rees et al, Phys. Rev. B42 (1990) 9237.

[9] V. L. Safonov, Phys. stat. sol. (b) 167 (1991) 109; ibidem 174 (1992) 223. '432

APPLICATION OF ISOSYMMETRIES/ Q-OPERATOR-DEFORMATIONS TO THE COLD FUSION OF ELEMENTARY PARTICLES R. M. Santilli

The Institute for Basic Research, Box 1577, Palm Harbor, FL 34682 Fax: +1-813-934 9275, E-mail: [email protected]

In preceding papers, Kadeisvili [1] reviews the nonlinear-nonlocal-noncanonical i so topics of Lie's theory, and Lopez [2] reviews the axiom-preserving isotopies of quantum mechanics(QM), called hadronic mechanics (HM) (originally submitted in |3], see ref.s [4] for details), and their axiomatization of Q-operator-deformations, here called Q-isodeformations. In this note we shall apply the isosymmetries/$-isodeformations to a speculative, yet intriguing novel problem, the cold fusion (or chemical synthesis1 j of elementary particles, defined as the apparent tendency of massive particles to form a bound state at short distances (< 1 fm) in singlet state which is enhanced at low temperature (or very low energies). According to these novel views, we expect that unstable elementary particles can be artificially produced via the chemical synthesis of lighter massive particles suitably selected in their spontaneous decays. Such a chemical synthesis occurs for each individual particle in our space-time only, without any unitary interior space and, thus, without the possibility of even defining a quark. Nevertheless, compatibility with quark theories is apparently achieved by considering families of particles via the addition of unitary internal spaces. This aspect is studied elsewhere [5] via the isotopies SUq(3) и SU(3) characterizing isoquarks, which have all conventional quantum numbers, yet more general nonlinear-nonlocal-nonhamiltonian interactions, exact isoconfinement and convergent isoperturbative series. A central problem in the achievement of the above cold fusions/chemical syntheses is the need for new renormalizations of the intrinsic characteristics of particles. Their physical origin is seen in the nonlinear-nonlocal-nonhamiltonian interactions expected in the total mutual penetration of wavepackets-wavelengths-charge distributions of particles one inside the other, and represented with theisotopic operator Q = (3(х,х,х,ф,дф,ддф,...) [1]-[5]. Their novelty then follows from their nonlagrangian-nonhamiltonian character. Among all possible deformations, we select the isotopies Oq(3.1) and Aj(3.1) of the Lorentz 0(3.1) and Poincare Я(3.1) symmetries introduced by the author back in 1983 [6] 2, which are constructed with respect to the isounit / = Q'1, and imply a generalization of the very notion of particle into the covering notion of isoparticle possessing precisely the generalized characteristic needed for the cold fusion, while preserving the original symmetries (but evidently not the original transformations) as exact, Oq(3.1) ss 0(3.1), PQ(3.1) и P(3.1). In this note we shall denote all ordinary particles characterized by P(3.1) with the familiar symbols e/i*, p*, etc. and use the symbols e*, [i*, p*, etc. for their isotopic conditions characterized by Phadrons, called hadronic media [3]. A first understanding is that all Q-operator are selected in such a way to recover the trivial unit I = diag.(l, 1,1,1) for distances > 1 fm, when motion returns to be in vacuum, in which case HM recovers QM identically. Also, in the transition from interior motion within a hadronic medium to exterior motion in vacuum, isoparticles reacquire their conventional P(3.1)- invariance, plus possible secondary emissions (e.g., of 7 and u) due to the original excitations.

'The author would like to thank A. N. Sissakian of the JINR for suggesting that name. 2In all preceding works the isotopic element is indicated with the symbol T rather than Q. '433

Quantitative representations of the following cold fusions are now available [3, 4], here expressed in self-explanatory notations (see later for secondary emissions)

Electron pairs = (e , C-)QM => Cooper Pair = (ё~,ё~)нм, (la)

Positronium = (e~,e+)qM Jr° = (£" ,ё+)нм. (16)

+ + Muonium = (ft~,/J )qM => т/= , ft )HM, (lc)

+ + Pionium = (ir~,T )qM =Ф Кf = (тг~,* )нм> (Id)

+ + Hydrogen atom = (p ,e")QM " = (p ,ё")нм- (le) Numerous other chemical synthesis are then consequential, йлсЬ as A = (h,х°)цм, 2* = (Ajir^HM) etc. A primary objective of the isosymmetry Л?(3.1) is therefore the reduction of aII massive elementary particles to electrons and protons, for which purpose the construction of hadronic mechanic was suggested [3]. In this approach isoquarks emerge as being suitable HM bound states of electrons and/or protons in isotopic conditions, resulting in fractional charges which are notoriously extraneous to P(3.1), but rather natural for Pq(3.1)-invariance under generalized interactions. The first and most fundamental cold fusion (la) is fully established experimentally, and it is given by the Cooper Pair (CP) in superconductivity (see, e.g., [7] and quoted references). Its interpretation via conventional QM is manifestly problematic owing to the highly repulsive Coulomb interactions at short distances. However, the Pg(3.1)-isoinvariance does permit a consistent interpretation of the CP via the particular isounit I = exp[tN j сРх'ф+(х')р(х')} which merely represents the overlapping of the wavepackets of the two electrons [8]. Once the experimental evidence of the (e~,e") cold fusion is admitted, one has the inevitability of the (e",e+) cold fusion (also called compressed positronium). In fact, as shown since 1978 (see [3], Sect. 5), under the use of the same isounit of isotopy (la), the charge radius of 1 fm and the meanlifeof 0.83 • 10~16 sec, the state (ё~,ё+)нм represents all characteristics of the such as rest energy (134.96 Mev), spin, charge, meanlife, electric and magnetic moments, etc., as well as the decay with lowest mode ir° —» e+, e~(< 2 • 10"6) as a tunnel effect of the constituents. Once the mechanism of the cold fusion is understood at the level of electrons, its extension to the remaining mesons is straightforward. In fact, the compression of the muonic (lc) and of the pionic (Id) atoms follows the same rules as those of the electrons. The identification of the states with the г/ and Kg particles, respectively is rendered inevitable by the uniqueness of the total characteristics, as it was inevitable for identification (lb) (see below). The isopoincare symmetry PQ(3.1) then permits the interpretation of all remaining mesons as cold fusion of lighter (massive) particles suitably selected in their spontaneous decays [4]. In the transition to baryons, new fundamental problems expectedly emerge whose solution "required systematic studies on the isotopies SUQ(2) of the S£/(2)-spin symmetry [4, 9] (see also the review in [1] for the 0neutron as a compressed hydrogen atom. The existence of the neutron was subsequently confirmed by Chadwick [11], but' Ruther- ford's conception of the neutron was abandoned because contrary to QM on numerous counts (impossibility to represent the total rest energy of the neutron because of the need of a "positive binding energy", inability to represent the total spin, mean life, size and other characteristics of the neutron). As well known, these difficulties lead to the conception of the isospin SU[2) which subsequently lead to the SU(3) theories. The above QM difficulties were however inconclusive, because of the unplausibility of the underlying assumptions such as: the electron freely orbits within the densest medium measured in laboratory; the treatment deals with a tiny acorn inside the proton; etc. As a. result of '434

such inconclusive character, studies on Rutherford's historical conception of the neutron were continued by various authors. The most salient recent results are the following. The first representation of all characteristics of the neutron via HM, including spin via the use of the SUg(2) symmetry, was reached in rcf.s [12] of 1990. However, the problem of the total spin of the neutron (which requires a null total angular momentum for Rutherford's electron when compressed inside the proton) was first solved by Dirac (without his knowledge) in two of his last (and little known) papers of 1971- 1ЭТ2 [13] where he introduced a generalization of his equation, which subsequently emerged as possessing an essential isotopic structure. The isosymmetry Яд(3.1), its related isospinorial

covering PQ(3.1) = SLQ(2.C) X rq(3.1) and St/g-little group, where finalized in [14]. The first preliminary, yet direct and impressive experimental verifications of the cold fusion of photons and electrons into rtbutrons via the reaction at low energy p+ + e~ —» n 4- v has been achieved by an experimental team headed by the (late) don Borghi [15], reviewed later on with a number of indirect experimental confirmations. It is hoped the reader can see the intriguing implications of cold fusion (le).In fact, if confirmed, it will imply the possibility not only of producing unstable particles via chemical synthesis, but also their artificial disintegration. By recalling that currently available tech- nologies are based on mechanisms in the structure of molecules, atoms and nuclei, the studies of this note are motivated by the possibility of resulting in a new technology, called hadronic technology [12], which is based on mechanisms, this time, in the interior of hadrons. The nonrelativistic radial treatment of the cold fusion of particles has been known since 1978 [3] (see [4, 16] for recent accounts). The central hypothesis is the generalization of Planck's constant h = 1 into the isounit I = Q~r = 1(1,г,р,р,ф,рф,ддф,...) > 0 which represents nonlinear-nonlocal-nonhamiltonian interactions, although / = h for mutual distances г > 1 fm. The isotopy of the unit then implies corresponding compatible isotopies of the totality of the structure of QM into that of HM [4] (see the outline [2]), including: isotopy of field F(n,+,x) => /g(n,+,*), / = Q-1, * = x Eq(r,6, Я), 6 = Q6] isotopy of Hilbert spaces H :< Ф\Ф >€ С =S- "HQ :< Ф\Я\Ф > Ф> I 6 CQ\ isotopy of eigenvalue equations Н\ф >= Е'[ф >=*• H *\ф >= >~ Ё*\ф >= Е\ф >, Е ф Е°\ isotopy. of enveloping operator algebras, Lie algebras, Lie groups, representation theory, etc. Hereon we shall assume for simplicity that 8Q/dr = 0. The Q-isodeformation operator is then selected to yield the isotopy [3] [-=№"! * ?] * - « - Ыда* * ?] • * - * -

where Q" and R° a.re positive constants. By recalling that the Hulten potential behaves at small distances like the Coulomb one, isotopy (2) can then be reduced to

where V = V° —(±е2Я°). The radial structure equations of the cold fusion submitted in ref. [3], Eq.(5.1.14), p.836, are then given by (ft = 1)

ф(Г) = Еф(г), (4a) [ 2m r2 dr dr 1 — e~R '435

1Ы 2 bin 1 Е = cMl. - E(MeV), r"' = 2A|0(O)| aS (sec- ), Я°(ст), (4b) where EM, r~', and П° are the total energy, raeanlife and charge radius, respectively, of the isobound state. For applications to cold fusion (la) one may inspect [8]. When applied to cold fusion (lb), the above model permitted the representation of all characteristics of the 7r°, beginning with the suppression of the original spectrum of the positronium into one, single, admissible energy level: 134.896 MeV [3]. A comprehensive study is provided in [4, 16]. We then have the following isopostulates of the cold fusion of particles [3, 4, 16]: I) ISORENORMALIZATIONS. The intrinsic characteristics of the constituents of cold fusions (1) are isorenormalized ( in the language of[3]) because of the nonlinear— nonlocal-nonhamiltonian interactions expected in the total mutual penetration of their wmie- packets-wavelengths-charge distributions. II) ENERGY BALANCE: Conventional QM bound states have a "negative binding energy" because Ет < All cold fusions (1) instead, if QM treated, would require a because Ет > thus resulting in inconsistent indicial equations [3, 4, 16]. A necessary condition to resolve this problem is the isorenormalization of the rest energy of the constituents given in Eq.s (3) by mcj =>• mCgQunder which binding energies can return to be negative. While conventional fusion processes energy, cold fusions (I) energy. 3 1П) SUPPRESSION OF TRIPLET STATES. Only singlet isobound slates of spinning particles are stable, because triplet couplings under total mutual penetration imply highly repulsive forces due to the spinning of each particle inside and against that of the-other (this occurrence was represented in [3] via the so-called ). IV) CHARGE INDEPENDENCE: The mechanism of cold fusions (1) is the dominance of nonlinear-nonlocal-nonhamiltonian forces at distances < 1 fm which are attractive in singlet couplings and absorb the Coulomb interactions resulting in attractive total interactions irrespective of the attractive or repulsive character of the original Coulomb interactions. V) CONSTANCY OF SIZE: Another difference between bound states in QM and HM is that the size of the former increases with mass, as established in nuclear and atomic structures, while the size of the latter remains approximately constant with the increase of the mass, as established for hadrons. This occurrence is quantitatively interpreted by HM via the unification of the following disparate occurrences into 1 fm: I) the range of the nonlin ear-nonlocal— nonhamiltonian interactions due to total mutual penetration; 8) the range of the strong interactions and 3) the minimal hole needed to activate Hulten's potential. The addition to an isobound state of a further constituent does then increase its rest energy (and density, thus increasing the isorenormalizations), but leaves the size essentially unaffected. VI) SUPPRESSION OF ATOMIC ENERGY SPECTRA. Yet another difference between QM and HM bound states is that the former generally have a spectrum of energy, while the latter admit only one, single, energy level (this occurrence was called Kspectrum suppression> in [3]). Each given cold fusion (J) therefore has no excited isostates at distances < 1 fm, because all exited states imply greater distances, thus recovering conventional QM energy levels at distances > 1 fm. In fact, the Hulten potential has a finite spectrum of energy levels, as well known. When all conditions (4b) are imposed, this finite spectrum reduces to only one level (see [3, 4, 12, 16],). VII) NEARLY FREE CONSTITUENTS. The notion of potential energy has no mathematical or physical meaning for the contract nonhamiltonian interactions responsible of cold fusions (1). The binding energies are then generally small, an occurrence which is remi- niscent of "asymptotic freedom" in quark theories [5].

3Novel forms of energy from cold fusions (1) should not be rules out, because they are conceivable via mechanisms different than conventional ones. '436

We now study the relativistic treatment of cold fusions (1) which is presented at this Meet- ing for the first time. Amore detailed presentation is available in the JINR Communication [16]. The treatment requires a knowledge of the isotopies of the Poincare group -P(3.1) =>• Рд(ЗЛ) studied in details in [14] (see also [19]), which cannot be reviewed here for brevity. We only mention for notational convenience that flj(3.1) is the invarance of all infinitely possible isotopies of the Minkowski space [6].

MC}(X,T),R):V = Q(X,X,X,...)V, V = diag.(l, 1,1,-1), 1 = Q~', (5a)

(x-yf = [(x*-yu)fj^(x,x,x,...)(x,'-x'")]ie RQ, x = (x,x<), x4=c„t, (5b)

Q = diag.(62, bj, bl 63), b„ = 6„(x, x,£,...)> О, Mg « M. (5c)

The isocommutation rules of the connected component Р£(3.1) are given by [loc. cit]

— [Muv, 'Map] - iiflvaMp (i VnnMp i/ зМа№ + VtfMuv), (6a)

[M(l„,-P0] = i(^[>P„-7j„t(fB), [FM,"P„] = 0, (66) where [A,' B\ :=A*B — R*A = AQB — BQA is the product of the Lie-isotopic theory [1, 3]. The isocasimirs are then given by [6]

C(0) = /, C<" = = P*P = PrfVP,, (7a)

C<2> = H'5 = К = c^.J"11 * P". (76) The general isopoincarc transformations are then given by (Iol. cit]

x' = A * x isolorentz transforms, (8a)

x' = x + A(x,x,x,. •.), isotranslations, (86)

4 x' = jrr * x = (—x, r ), x' = irt * x = (—x,x*), isoinversions, (8c)

2 A„ = a,{bl + a°[6 ,"Pa)f\[ + aV[[6j, *Pa], 'Pp]/2\ + }. (8d) The general isolorentz transformations are given by the isorotations reviewed in [1], and the isoboosts first constructed in [6] x" = x1, xn = x2, (9a)

3 3 4 3 x' = x созЬИбзМ - Х (Ь4/ЬЗ)sinh[u(6364)] = 7(x - /5i"), (96) 4 3 4 4 3 x' = -x (63/6„) sinh[v(6364)] + x cashMfcfc*)] = 7(x - 0x , (9c) where /? = v/co, $ = v'blvb/obfa, (10«)

1/2 coshM6364)] = 7 = |l-/3T , sinh[«(6364)] = fa. (106) The special isolorentz and isopoincare transformations occur when the quantities called characteristic functions of the medium, are averaged into constants 6J = Aver.(6„) > 0. It is easy to prove the local isomorphisms Pq(3.1) m P(3.1) for all Q > 0 (but not otherwise). Yet, transformations (9) are nonlinear-nonlocal-noncanonical owing to the unrestricted

functional dependence of the 6M-functions. This illustrates that the Lorentz transformations are necessarily "inapplicable" (and not "violated") under isotopics, but the Poincare symmetry is preserved in an exact form, only realized in its most general possible (rather than simplest possible) form. The isopoincare invariance implies the isotopies of all basic postulates of the special relativity [6, 14]. Those important for this note are the following isopostulates for realization (5) with dbp/dx" = 0 and 61 = 62 = 63^ 64: '437

I: The invariant speed is the "maximal causal speed"

VM„, = |

II: The addition of speeds и and v is given by the "ifotopic addition law"

v' = (u + v)/(l+ukblvls/4b]y, (12)

Ш: Time intervals and lengths follow the isodilation and isocontraclion laws

т = ^Tq, A L = 7ДХ.0, (13)

IV: The frequency follows the "isodoppler shift law" (for aberration a = 0)

ш' — 7U>; (14)

V: The energy equivalence of mass follows the "isoequivalence principle"

В — тс2 = т<%Ь\. (15)

The above generalized postulates are implicit in the preceding formulations; they recover identically the conventional postulates in vacuum for which b^ = 1; and they coincide with the conventional postulate at the abstract, realization-free level, where we lose all distinctions between } and /, iJ and хг, в2 and /32, f and r, u> and u>, E and E, etc. A most visible departure from the conventional postulates is the abandonment of the speed of light as the invariant speed in favor of quantity (11) which is intrinsic of the isominkowski geometry and represents the maximal causal speed as characterized by an effect following a cause due to particles, fields or other means. Note that in vacuum VM«* = C0 by therefore recovering as a particular case the speed of light as the maximal causal speed (see [14] for verifications and details). A primary function of the isominkowski spaces (as well as a primary mean for their experimental verification) is the geometrization of inhomogeneous and anisotropic physical media at large, and the media in the interior of hadrons, in particular. By recalling [1,4, 19] that systems are now characterized by a conventional Lagrangian or a Hamiltonian plus the isotopic element Q, the desired novel isorenormalizations are expected to originate directly from the isominkowskian geometrization. The operator relativistic isokinematics on MQ(X,C), R) for constant characteristic quantities b° is characterized by the basic expressions [9, 19]

и = Я = (Р ) ("') = (m07ct;*,m07c), m = m07i с = соб;, (16)

isoeigenvalue form p„ * ф = = (17) and fundamental isoinvariant

2 3 р *ф = 7~fp„ *р„*ф = (b\pk *pk - c p4 *рл)*ф- (-т1с*)ф. (18)

It is then easy to see that the isorenormalization needed for the cold fusion is provided by isopostulates I-V themselves. As an example, in going from motion in vacuum to motion within a physical medium, a particle experiences the following isorenormalization of the rest energy E = mc2 =» E' - mc2 = mc^bf, (19) which is precisely the relativistic version of the nonrelativistic isonormalization of Eq.(3). '438

A similar occurrence holds for all remaining intrinsic characteristics. This is expected from the alteration of the conventional Casimir invariants into form (7). The isotopies of the Minkowski space and of the Рогасагё symmetry imply the lifting of Dirac equation into the isotopic form [9, 14, 16]

(% * P" + * = 0, rh = mi. (20a)

{% } = + %Q% = 2GUJ, 1 = Q~\ (206) 4 7 d 7* = ЬЧ ( Д ) , 7 = ib4 ( 0' J), /. = diag.(l, 1), l s = -/. (20c) here presented for simplicity also for the case дЬл/дх" = 0 or for constants = Aver.(bM) > 0. The orbital and intrinsic angular momentum of particles for the orbital hadronic weight I = 1 are then characterized by

OQ( 3): Lk = ekijriPj, [Lu Lj] = tijkbfLk, (21a)

3 J 2 I* * Ф = (B- IJ + + Ь£ Ь?)Ф, 13*Ф = Ь-Ч^Ф; (2ib)

X SU(2) : Sk = -tkify * 7j, [5,-, 5,] = cijkb\Sk, (21c)

£?*Ф = (1/4)(ЩЬ1 + Ь1Ь1 + %Ь1)Ф, SW = ib,62 ф, (21 d)

which confirm the existence of the desired nontrivial isorenormalizations. A simple isotopy of the conventional derivation, yields the magnetic and electric isodipole moments (assumed for simplicity along the third axis)

A=£m, rh=£m, (22)

first derived in [3], Eq.s (4.20.16), p.803, andthen isotopically reformulated in [15] (see [4, 16]). The isosymmetry of (20a)_, the isotope SL{2.C) of the spinorial symmetry SL(2.C) is char-

acterized by the generators Sk — \tkijii * 7j, Гк = %% * 7< and also verifies isocommutation rules (6a). By adding isotranslations, Eq.s (21) characterize the spinorial covering VQ(Z.\) of the isopoincare symmetry Pg(3.1). The proof that isodirac's equation transforms isocovariantly under PQ(3.1) is instructive (4, I4|. We now specialize Eq.s (20) for the characterization apparently for the first time, of cold fusion (le). Recall that Dirac's equation describes the ordinary electron e~ under the external field of the proton p+. Eq.s (20) are therefore ideally set to describe Rutherford's electron when immersed within the hadronic medium inside the proton considered as external. As a first step, we can therefore study the cold fusion n = (р+,ё~)нм where the proton is unperturbed owing to its much greater mass, and the isoelectron is represented by (20) with the A„-functions averaged to constants thus averaging all interactions, whether Lagrangians or not. 4 These assumptions then permit the following remarkably simple isorenormalizations: Isorenormalization of rest energy. The energies involved in cold fusion (le) are: E° = 939.57 MeV; E° = 938.28; and E° = 0.5 MeV. As we shall see, the binding energy is very small

4The understanding of this point requires the knowledge that conventional electromagnetic interactions can also be represented via the generalized Lie tensor (the (-functions), with the Lagrangian representing only the kinetic energy [20], pp. 98-101. Despite its misleading appearance, Eq.e (20) represent, as written (i.,e., without conventional interactions), an electron under the most general known linear and nonlinear, local and nonlocal, as well as Lagrangian and nohlagrangian interactions. The averaging of the ip-functions into constant 4J is possible because of the stability of the cold fusion considered. '439 and can be assumed to be null in first approximation. This requires the isorenormalizalion via (19) E\ = m°cl = 0.5 MeV El = m^bj »s 1.3 MeV, Ц « 1.62, (23) first predicted in [15], p. 527. 5 Isorenormalization of total angular momentum. The isoelectron c~ must have a null total angular momentum to permit a consistent cold fusion (le). This result was first reached by Dirac [13] via his generalization of his own equation, which results to have an essential isotopic structure with a nondiagonal Q-matrix (denoted f3 in [13]). The total angular momentum is then subjected to the isorenormalization L+ j =>• (n + n')/2, n, n' = 0,1,2,..., thus being null for the ground state (see [14, 16] for detailed review and isotopic reinterpretation). The isotopic Si/(2)-spin theory [9] permits a rigorous confirmation of this result because the only allowed addition of angular momentum and spin for a light particle immersed within the hyperdense medium inside the proton is that for which [12]: 1) the spin-spin coupling is a singlet; 2) the orbital angular momentum of e~ is- along the spin of the proton; and 3) at the limit of compression of the electron to the center of the proton, its orbital and intrinsic angular momentum must evidently coincide, thus resulting in a null total value.

Isodirac's equation (20) permits a quantitative interpretation of Limr_o^« = via L3 = S3 2 and L? = S from Eq.s (21) (for L = 1, L = 0 being unallowed for L3 = S3 f 0)

1 2 2 2 г = \b\bl бГ'ЬГ + ЬГ'ЬГ + ЬУ'ЬГ = (1/4)(ОГ + ъ?Ь°3 + Ь°3 Ь?) (24)

with numerical solution for the simple case of spherical symmetry

bf = bf = bf = y/2 a 1.415 (25)

which confirms a fundamental prediction of the isopoincare symmetry, that the nucleon is an isominkowskian medium of Type 9 [19], p. 103 ф < /?, 7 > 7, Aver.(6J) < 6J). Isorenormalization of magnetic moments. Yet another prediction of the isopoincare symmetry it that, when ordinary electrons are immersed in the hyperdense medium inside protons, they experience a deformation-isorenormalization of both their orbital and intrinsic magnetic moments, first generically studied in ref. [3], p.803, and then studied, nonrelativisti- cally, for cold fusion (le) in ref. [12], and studied relativistically here. The isodirac's equation permits a quantitative, simple and direct treatment of this aspect too, via Eq.s (22) which yield for cold fusion (le) (for L — 1 (see Fig. 1, p.5'25, of [12] for orientations)

pn = —1.9|e|/2mpCo = p„ + ^ + Лг"", Hn = +2.7|e|/2mpCe, (26a)

3 li? = -4.6|e|/2mpco = -2.4 - 10- |e|/2meco, (26b) ft' = = (1.41/1.65)/i>ntr = 0.8545/4"", (26c) ц?ь = (-0.8545 + 0.0024K"'r = -0.8521/j°rb. (26d) The latter numerical values should not be considered as final because of the need to study the general model n = (p+ f, !)нм with isorenormalization of the electromagnetic properties of both the proton and the electron (including the charge which is not isorenormalized in this first treatment). Nevertheless, the latter study is only expected to yield adjustments of numerical values (26) without structural changes in the theory. We can therefore conclude by saying that 'he isospecial relativity does indeed provide a quantitative representation of the cold fusion of protons and electrons into neutrons (plus

5There is a clear misprint in [15], Eq.s (2.45) with 64 = 13.5 rather than 1.65. As we shall see, the experimental value is precisely 1.65 also accounting for the binding energy. '440 neutrinos), with ail needed, specific, numerical predictions of the quantities involved in a form suitable for experimental tests. The extension of the results to other cold fusions of particles is here left for brevity to the interested reader [4]. Even though preliminary and in need for independent re-runs, a number of direct and indirect experimental verifications are today available in support of cold fusions (1): Direct verifications. The first direct experimental verification of the isotopic origin of electron paring in superconductivity has been, provided by Animalu [8] with rather impressive agreement with experimental data. The best verifications of cold fusions (lb)-(ld) are given by the uniqueness of the represented energy levels via models (4). The first direct experimental verification of the cold fusion (le) was done by don Borghi et a/. [15]. The experiments essentially consist in forming a gas of protons and electrons inside a metallic chamber (called klystron) via the electrolytic separation of the hydrogen. Since protons and electrons are charged, they cannot escape the metallic chamber. Nevertheless, numerous transmutations of nuclei occurred for matter put in the outside of said chamber. The measures can then be solely interpreted, in the absence of any other neutron source, by the cold fusion of the protons and electrons into neutrons which, being neutral, can escape the chamber and cause the measured transmutations. Verifications via the Bose-Einstein correlation. The most important indirect verification of cold fusion (le) has been recently achieved via theoretical [19] and experimental [21] studies on the Bose-Einstein correlation. These results are important because they confirm, not only the isominkowskian laws underlying cold fusion (le), but also their numerical values. In essence, studies conducted via the full use of the isominkowskian geometrization of the p — p fireball result in the two-point Boson isocorrelation function on Mq(x,g,R), [19], Eq. (10.8), p.122,

2 Cm = 1 + ^S^c"'?/^, Tj = Diag.(b; ,6f,if,-&f), (27)

where qt is the momentum transfer and К = 4° 2 + 4|2 + 6|2 is normalized to 3, under the sole approximation, also assumed in conventional treatments, that the longitudinal and fourth components of the momentum transfer are very small. Phenomenological studies conducted in [21] via the UA1 experimental data at CERN confirm model (27) in its entirety, and identify the numerical values

4J = 0.267 ± 0.054, 4J = 0.437 ± 0.035, = 1.661 ± 0.013, bj = 1.653 ±0.015. (28)

These measures have the following implications for cold fusions (le): A) They confirm the nonlinear-nonlocal-nonhamiltonian origin of the correlation, which is the expected origin of the cold fusion; B) They confirm the isominkowskian geometrization of Type 9 ф < 0, 7 > 7, Aver.(4£) < b\) for the p - p fireball which, having the density of the order of that of the proton, is directly applicable to cold fusion (le); C) They piovide a numerical confirmation of rest energy isorenormalization (23) predicted in [12], beyond the best expectation by this author. D) They permit a ditect representation of the nonsprerical shapa of the fireball and all its possible deformations; and E) They permit the reconstruction of the exact Poincare symmetry under nonlinear-nonlocal-nonhamiltonian interactions. Also, experimental value 6J = 1.653 yields the isorenormalized rest energy Ei = 1.36 MeV, thus implying the existence of the binding energy E = —0.072 MeV, which is small, also as predicted. Additional experimental verifications. Phenomenological calculations of deviations from the Minkowskian geometry inside pions and kaons were conducted in [22] via standard gauge models in the Higgs sector, resulting in the deformed metric rj = diag.((l — a/3),(l — a/3), (1 — a/3), -(1 — a)) which is precisely of the isominkowskian type (5) with numerical values PIONS тг* : 6J2 = if = 4°2 S 1 + 1.2• 10"3, 6J!S1-3.79• 10"3, (29a) '441

KAONS К* : if = ^ = 4Jai-2-10"4, bf ^ 1 + 6.1 • Ю"4, (29b) Piuns тг* are then isominkowskian media of Type 4 [19], while the heavier kaons /f* are of Type 9. This confirms measures (28) because all hadrons heavier than A'* are expected to be isominkowskian media of Type 9. Independent phenomenological plots [23] on the behavior of the meanlife of the (which, according to current experiments, is anomalous from 30 to 100 GeV and conventional from 100 to 350 GeV) via the isominkowskian geometrization yield the following characteristic values of the K% b? = bf = ti? 0.909080 ± 0.0004, b°7 S 1.002 ± 0.007, (30) which are of the same order of magnitude of values (29b). Measures (30) therefore provide an independent confirmation that the interior of kaons is indeed an isominkowskian medium of Type 9, and an additional independent confirmation of the isogeometrization needed for cold fusion (le). Plots [23] also computed the values Abf=i 0.007, 0.001. (31)

This confirms the yet another prediction in the range 30-400 GeV that the 6J quantity, being a geometrization of the density, is constant for the particle considered (although varying from hadron to hadron), while the dependence in the velocities rests with the 6fc-quantities. Physics is a science with an absolute standard of value: the experiments. Experiments themselves have their own standard of value: the more fundamental the law to be tested, the more relevant the experiment. In particular, experiments such as don Borghi's measures of the cold fusion n = (р+,ё~)нм, can °nly be dismissed via other experiments, and simply cannot be dismissed in a credible way via theoretical considerations or personal views. We therefore suggest the independent verification or dismissal of experiments [15], which can nowadays be re-run via a number of independent alternatives. 6 Numerous, additional chemical syntheses of hadrons can then be verified or dismissed. Similarly, we suggest the conduction of additional tests on the stimulated decays of (unstable) hadrons (which are the inverse of the cold fusions). The most important one at the basis of the possible hadronic technology [12] is the artificial disintegration of the neutron via the reaction у + n —* p+ 4- e~ -I- whose cross section has been predicted [loc., cit.] to peak at the frequency of the isoelectron ш = 3.5 • 1020 sec, nther than that of the electron ш = 3.25 • 1020 sec. This would permit a direct experimental test of the isorenormalization (23 of the rest energy itself. Similar artificial disintegrations are possible to verify cold fusions (lb)-(ld) and their internal isorenormalizations. Additional classical experiments have been proposed [18] for direct tests of the fundamental the isominkowskian geometrization of physical media, such as the prediction that a portion of the redshift of sun light at sunset (or a portion of quasars redshift) is due to an isodopplcr shift caused by the inhomogenuity and anisotropy of Earth (quasars) atmospheres (which arc media of Type 4), and numerous others.

References

[1] J.V. Kadeisvili, Isotopies of Lie's theory, Proceeding of the Workshop on Symmetry Meth- ods in Physics, JINR, Dubna, 1993.

[2] D.F.Lopez, Origin and axiomatization of ^-deformations, Proceeding of the Workshop on Symmetry Methods on Physics, JINR, Dubna, 1993.

fWe would like to thank Y. Oganesaian of the JINR for enlightening comments on these alternatives. '442

[3] R.M.Santilli. Hadronic J. 1, 578 (1978).

[4] R.M.Santilli, Elements of Hadronic Mechanics, Vols. 1,11,111, to appear.

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[7] B.Goss Levi, Physics Today, May 1993, p.17.

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[9] R.M.Santilli, Hadronic J. Suppl. 4B, issue no. 2 (1989); and contributed paper in the Proceedings of Deuieron 1993, JINR, Dubna (1993).

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|11] J.Chad wick, Proc. Roy. Soc. A136. 692 (1932).

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[18] R.M.Sa.ntilli, Isotopic Generalization of Galilei's and Einstein's Relativities, Vol.s I and II, Hadrohic Press (1991).

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ALGEBRAIC AND ANALYTICAL FEATURES OF (3 + 1) DIMENSIONAL TOPOLOGICAL SOLITONS Valerii I. Sanyuk

Experimental Physics Department People's Friendship University of Russia 6, Miklukho-Maklai Str., 1J 7198 Moscow, Russia

Abstract

Topological (3+l)-dimensional soliton approach looks like quite reasonable in description of a huge variety of spatially localized structures, arising in different physical applications. Therewith we would like to list the up to date available mathematical "machinery", as well as to discuss the abilities and limitations of listed methods in order to study properties and features of topological solitons in the real space-time. The special emphasis would be on algebraic, symmetrical and analytical properties of chiral solitons, which one can obtain by means of group-theoretical methods, the Liouville-Tresse algebraic invariants and the Painleve analysis of singularities.

1. Introduction It is widely recognized that a further progress of the soliton theory and its numerous applications should be related with an advance into the many-dimensional nonlinear problems. Here we deal only with a particular type of many-dimensional solitons, known as topological solitons, and we specify them as regular solutions of nonlinear evolution equations with all finite dynamical characteristics (momentum, energy, angular momentum and so on), and endowed with a nontrivial topological index (degree of mapping, Hopf index etc.) If we restrict ourself to Minkowskian space-time as the variables domain, then in allmost all possible applications, one faces with the common problem: how to investigate (3+l)-dimensional nonlinear functionals. Among others the following important appplications should be listed: Nonlinear Field-Theoretic Models (skyrmions in the Skyrme model, and towns in Faddeev's model. In a common fashion we regard '444

field variables as mappings of the kind: ф(х, t): R3 ® R} Ф, where the domain Л3 0 Rl is the Minkowskian space-time, and Ф is the field manifold. In the Skyrme model Ф = S3 ~ SU(2), while in Faddeev's model - Ф = S2. Omiting some details, which can be restored from Rcf. 1 and citations there in, one can write the Skyrme model Lagrangian density

rs^-^jTrV + ^Tr^,^]2, (1)

r in terms of the left chiral currents Lu = U~ dJJ, where V = Фо + 1таф" is the so-called principal chiral field, parametrized with the self-interacting mesonic fields ф" : ф„ + ф°фа = 1. In the Faddeev model, nicknamed also as n-field model, the Lagrangian density has the form

2 2 2 CF = -jFv„ + \ (d»n') + ..., (2)

where n" • na — 1, a = 1,2,3 and dots stand for terms, providing the desired asymptotic behaviour at spatial infinity. An auxiliary field is introduced through

b c Fuu = 2tabcdIIn"dl,n • n = — д?ам. For models (1) and (2)we have different topological charges. In the case of the Skyrme model this is the degree of mapping ф : S3 —» S3, expressed as Q = -ОЦ1/Л л ~48J (3) which takes a definite integer value for each homotopy class of fields, thus realizing the isomor- 3 3 phism Q : я3(S ) —> Z. Here x3(5 ) is the third homotopy group and Z is the abelian group of integers. In its turn field configurations in the Faddeev model are endowed with another topological invariant - the Hopf index

= jd3xa-b\ Ь = roto, (4)

2 realizing the isomorphism Q : r3(S ) —» Z. For both models it takes place the estimate of the energy functional from below through the topological charge, namely

Est. > 6УД*2Ш 2 3 8 3 4 EF > «A(4t) \/2 3 ' |QHI ' , indicating the possible existence of stable topological solitons in these models (see Ref.l Chapter 3 for details). The consistent reductio:; scheme for nonlinear functional of the type (1) and (2) includes the following procedures; I. Reduction to Static Field Configurations as achieved by direct minimization of the functional with respect to time derivatives. It looks to be valid the following statement: If derivatives of fields with respect to time enter the functional in quadratic form, then its minimum would be realized among static field configurations. Therewith one can ignore in futlier consideration the dependence of functional on the time derivatives, as well as on temporal components of fields and currents, and search for minima of static functionals. For the Skyrmc model it means that instead of Lagrangian (1) one has to investigate further the energy functional

Esk = /^{^[(vef + sin'eavflr + sin2/^)2)] 2 2 2 2 2 +t sin 0{(V0 x V/?) + sin /3(V0 X V7) (6) + sin20sin2/?(V/? x V-jr)2} J. '445 where it was used the representation of chiral field U(x,t) through chiral angle variable 0(x,t)

l/(i,O=exp{irano0(i,<)},

(ra are Pauli matrices), along with the polar variables for a parametrization of the unit vector n : n1 + in2 = sin /»(£, t) cxp{i7(f, t)}; n3 = cos P(x,t).

II. Reduction to Invariant Field Configurations one can perform in accord with the statement of the Coleman-Palais theorem2,3, which in brief reads: symmetric Junctionals should have symmetric critical points. It means, that in search of minimal energy field configurations it is sufficient to reduce the variational problem SE[U] = 0 for a functional of the type (6) to that for invariant (or symmetric) fields only: i£[f/inv] = 0. To get an explicit form of such invariant fields (ansatze) U,ny one needs to find the maximal compact group of symmetries of the functional. For the functional (6) in the first homotopy class (Q = 1) this group is

Gi = diag [50(3), ® SO(3)s]. (7) where subscripts I and S indicate isorotations and spatial rotations, relatively. For the Skyrme model functionals in higher homotopy classes (Q > 1) and the Faddeev model functional for all values of QH we have

Gs = diag [50(2), ® 50(2)s], (8) Therewith the fields invariant with respect to G\ are determined by the equation

-i[rV]t/ + i[?,tf] = 0 and are known as the "hedgehog" ansatz

й„У(^) = ехр{^0(г)} (9) and further, this symmetry group allows for separation of variables. The latter makes it possible to prove, that the invariant field configuration (9) realizes the absolute minimum of the energy functional4-*. It should be also mentioned that abovelisted results might be enforced or checked up 'by Direct Minimization in Extended Phase Space Procedure, combined with Spheri- cal Rearrangement Method, which one may consider as a generalization on functional spaces of Gelfnnd-Zetlin "Valley" Method, which was developed for minimization of functions of several variables. An exposition of these methods in application to the functional (6) one can find in Ref. 1, Chapter 5. In the cases of higher homotopy classes in the Skyrme model and for any QH in the Faddeev model the energy minima are reached on axially-symmetric field configurations. Using spherical

coordinates r, i?,a, one can write the equation for the G2-invariant fields in the form

-idaU + ^[r3,U] = 0, k€Z.

The corresponding axially-symmetric configurations are described by

0 = 0(r,tf); 0 = /?М); i = ka\ к 6 Z. (10)

The symmetry group (8) does not allow for separation of angle variables, therefore the reduced equations of motion would be a PDE, and its further study is still an unsolved problem, which merits an extention of the presented scheme. '446

3. Symmetry and Singularity Analysis of Reduced Equations Thus after performing the reduction in the subsequent way at most (as it happens for Q = 1 configurations in the Skyrme model, called also skyrmions,) one obtains ODE of the Newton type у = /(x, y,y'):

0" (x2 + 4 sin2 0) = sin20[l+ -2x0', (11) with boundary conditions 0(0) = ж; 0(oo) = 0. There is no any regular way to solve this equation explicitly. Therefore, one has either to follow Klein's program: to prove Existence Theorem, i.e. to get convinced that equation (11) admits regular solutions1, or to study algebraic and analytical properties of the obtained equation to get information, useful in searching for explicit solutions. The proof of existence of regular solutions from C°°(0,oo) for equation (11) with Q = 1 was given in Ref. 4, for axially-eymmetric field configuration in the Skyrme and Faddeev models in Ref. 7 (see also exposition in Ref. 1 for more technical details). Among methods of finding first integrals and explicit solutions for ODE of the second order in application to Eq.(ll) we tested I).Lie's Infinitesimal Method , 2). Liouville-Tresse Algebraic Invariants Approach, and 3). Painletie Singularity Analysis. The results of thest studies were reported already in previous papers8,9, therefore we discuss here in brief, mainly, some limitations of listed methods. I. Lie's Infinitesimal Method.In the standard manner, to get the symmetry group of ODE (11) one has to consider infinitesimal transformations of the type:

0(x) 0(a) = 0(x) + eV(x, 0);

x -• x = x + e£(x,0), where с is an infinitesimal parameter, {,17 are unknown functions, which determine the infinitesimal generator of the symmetry group in question. In general, the explicit form of the group functions is obtained from the invartance condition of Eq.(ll) against the action of the second extension of the symmetry generator. For Eq.ll this condition might be written the rm f° о я

+ = (13) where т) = dr\fdx denotes the total derivative, and /(x, 0,0') is the right hand part of Eq.ll presented in the Newton form. For a clear exposition of the Lie methods one may consult Ref. 10, 11. Next step is to rewrite the determining equation (13) as a polynomial in powers of 0', and regarding the latter as independent to derive the PDE system for unknown functions ( and r), which, as a rule, is more difficult then the initial one. For Eq. 11 one has, for instance, 2 £(sin5 20 + sin2 © cos 20) - 27 cos 20 + g4 ""2^'"1'8

+Ш + sin2Q) + sin20 ('""ffij,1"'8 + Mfp)

8 +2xr)z + MJ,„ + (2i)f - 36) ('"У + sin 20) = 0;

*f (w) ~ 2* + V sin 20 («J) + 2хЧ» - Цхх - 2Мщх

2 2 +MU + 56 ( "" e»"le + sin2©) + 4sin20jjx = 0;

+ + 4i? cos 20 + 8 sin 207)9

—2(x sin 20 + 8x(e - 2M($r + Мщ, = 0; M&D- 6{к sin 20=0, 'Physically meaningful solutions should be smooth enough, zt least, to be from the class of twicely differentiable functions C'(0,oo). '447 where it was denoted M = x2 + 4 sin2 0; = д£/дх and so on. As it has been demostrated" admisable solutions for the above system are functions £ and rj of the form:

2 f = a,0' + a0; v = k©' + SO' + bo, where n;, = ф(х, 0) are coefficient functions, and for their definition one has to perform rather tedious calculations. As a consequence this allows one to predict, that the first integral for Eq. 11 (if any) should be at least a polynomial of the third degree with respect to 0'. But this is the only prediction one can get out of the listed PDE system (exept those who is too clever). II. Liouville-Tresse Algebraic Invariants. Another grain of useful information on the structure of the solution manifold for Eq. 11 one can get from the Liouville-Tresse theory of algebraic invariants12. In particular, the theory provides with the equivalence criterium, which tells one when it is possible to reduce (by an appropriate change of the independent variable) equation of the form 0" = /(x,0,0'), to a more simple one 0" = /(x,0). It proves to be sufficient for this12, that the fifth absolute invariant (in the Liouville-Tresse classification)

i/6 = 0. (15) The latter invariant is defined for a general type of the second order equation 3 2 У" + <ЧУ' + 3a2y' + 3o3!/' + a„ = 0, by relations

"5 = H(HHX - MIX) + hWiy ~ II'J,) - atlf + 2 +3a,l l3 - 3aj/Jf2 - ЗазМз + a4l%, where д д h = + 3a2a4) - ^(2a3„ - a2l + aia4)

—Заз(2аз„ — a2z) — ataix д д lз = ~ За|аз) + — 2ejx + ацц)

-За2(а3у — a2l) +

In particular, for the Skyrme model at = 0, h = 0, thus ps = —atlf = 0, and from the geometric point it means that the solution manifold for Eq. 11, endowed with the normal projective structure, is imbeded into the projective space RP3. Another consequence was exprec;?d in the Dryma hypothesis12, that if for a second order equation the condition (5) holds, then such an equation should possess the Painleve property and by an appropriate change of variables can be reduced to one of Painleve transcendents Pj — Pvi- Thus the next step was to check, whether the Eq. 11 is of P-type, and then try to find the corresponding change of variables. III. Painleve Singularity Analysis for the Skyrme equation (11) as well as for its possible modifications (arising in the Skyrme-Manton S3(L)-model, and "super-symmetric" extension of the original model for the hedgehog fields) traditionally starts from the rewriting of ODE of the type (11) in the algebraic or polynomial form. For Eq. 11 it reads

+г»(1-»") (j + Ц^) =o '448

The corresponding solution might be expanded into the Laurent serie

z J — = —— 1- + —(z — z0) + ~\( — *o) + А z — z0 ZQ 20 3 + (£ + r0)log|z - 201 (z - Z„) + ..., where 7 2 3 2 5 n - „ _ (3 20 — 2 x 7) „ (3 х13г0 + 2 x 7) — 12' (2' x 32) 1 (2® x 33) * where a1 = 8j and z

References

[1] Makhankov V.G., Rybakov Yu.P., Sanyuk V.I., The Skyrme Model: Fundamentals, Meth- ods, Applications (Springer-Verlag, Heidelberg-Beriin-New York, 1993) 265 pp.

[2] Coleman S.,//Neui Phenomena in Sub-Nuclear Physics, fed. by A. Zicliichi (Plenum Press, New York, 1977) p. 297.

[3] Palais R.S.,//Commun.Math.Phys., (1979), 69 p. 19.

[4] Rybakov Yu.P., Sanyuk V.I.// Niels Bohr Inst, preprint, NBI-HE-81-49, Copenhagen, 1981.

[5] Rybakov Yu.P., Sanyuk V.1.//Int.J.Mod. Phys., (1992), A7 p. 3235.

2It should be noted, that topological models of magnets, based on the Landau-Lifshitz approach, do not possess the consistent reduction. They are reduced to ODE just on appropriate ansatze, which one finds from the physical consideration '449

[6] Gelfand I.M., Zetlin U.b.//Dokl. AN SSSR, (1961) 137 p. 295.

[7] Rybakov Yu. P. // Itogui Nauki i Tekhniki. Classical Field Theory and Theory of Gravi- tation. Vol. 2, (VINITI, Moscow, 1991), p. 56.

[8] Ptukha A.R., Sanyuk V.I.//Group Representations in Physics, Proc. of Workshop, Tambov-89, (Nauka, Moscow, 1990).

[9] Ptukha A.R., Sanyuk V.I.//Proc. NEEDS-92, (World Scientific, Singapore, 1993), (in press).

[10] Ibragimov N.Kh. Groups of Transformations in Mathematical Physics, (Nauka, Moscow, 1983) 280 pp. (in Russian).

[11] Zhuravlev V.F., Klimov D.M. Applied Methods in Oscillation Theory, (Nauka, Moscow, 1988) 326 pp., (in Russian).

[12] Dryuma V.S.//Moldavian Acad.Sci. preprprint, Kishinev, 1986.

[13] Conte R.//Painleve Transcendents: Their Asymptotics and Physical Applications, NATO ASI series. Series B, Physics; Vol.278, (Plenum Press, New York, 1992), p.125.

[14] Hurtubise J., Kamran N./f Painleve Transcendents: Their Asymptotics and Physical Ap- plications, NATO ASI series. Series B, Physics; Vol.278, (Plenum Press, New York, 1992), p.271. '450

LOOP SPACE AND W-TYPE ALGEBRAS Peter Schaller

/nstitut fuer Theoretische Physik, Technische Universitaet Wien Wiedner Hauptstrasse 8-10, 1040 Wien, Austria email: [email protected]

Abstract

The algebra of ad-invariant polynomials on a simple Lie algebra g is emplo; ed to form a set of constraints on the space of loops in a target space containing g. The classical theory is outlined. Aspects of the quantisation of the system are discussed.

Starting from the string action

(1)

we may (locally) use the invariance under diffeomorphisms and Weyl transformations to write

the metrit in the form g++ = g = 0; = 1. Naively inserting this into (1) we obtain the action of a (1+1) dimensional Klein-Gordon field:

(2)

The choice of the metric, however does not completely fix the gauge: (2) is invariant under conformal diffeomorphisms

8ф = t-д+Ф + i+д-ф] 3±e± = 0 (3)

acting as a global symmetry, as the conditions on the parameters c± indicate. . This global symmetry allows, to establish the connection between (1) and (2) also in the oppsite direction [1]: Starting from the Klein-Gordon field and gauging the global symmetry (3) will remove the restrictions on the parameters of the transformations and will thus yield a theory invariant under general diffeomorphisms. The latter is given by the string action. In this framework the entries of the metric in (1) are the gauge fields corresponding to the transformations (3). In our case, we know the result of gauging the symmetries (3). If we would not know the action (1), the (in my opinion) most simple way to obtain the gauged theory is provided by the Hamiltonian formalism [2]: The Hamiltonian density H, = (p2 + фа){а) of the Klein Gordon field generates diffeomorphisms in the time direction, diffeomorphisms in the space

direction are generated by the functions on the phase space Gf = р(а)ф'{а). Thus we obtain the diffeomorphism invariant theory by regarding G and H as constraints. To go back to the Lagrangian level we could introduce Lagrange multiplier fields AG(

Hamiltonian of the gauged theory as //,„„,«{ = J dtr(XaG + A;/#). The Lagrange multiplier fields become the gauge fields on the Lagrangian level. The transformations (3), however, are not the only global symmetries of the Klein-Gordon action: (2) is also invariant under transformations of the form

< 8ф = A=F°TDIh, in1 •••Д±Ф " I Д±ХПР = О (4) 451 where the tensors d obey the relations of a Jordan algebra [1]. Again, a gauge theory based on global symmetries of the form (4) is most easily realized in a Hamiltonian formulation. To this end let me remind you that the identifacation (ф' ±p)(±

ng=L,/g, L„ = {Y : S* (6) Js> and the constraints G, H, defined above may be comprised in the expression u2(cr) =< Y'(. (< ... > denotes the Killing metric on g.) The resulting constraint algebra is invariant under the adjoint action of the Lie group associated with g. It is generated by the homogeneous polynomials

Pi : g --» R, Pi{x) = fr(i'), 1 = 2,3,... щ = yw(KV) (7) where the trace is to be taken with respect to a faithful (e.g. fundamental) representation of g. Not all the UF, however, are independent: For a Lie algebra g with гапк(д) = к one may find a set E = {/,,...,ft}, which allows to express the pi,l g E as functions of p/,,...,p(|i. (fi — 1, ...Ik — 1 are the exponentials of g). We will denote the algebra between the {u,-, i 6 E] by iog (The system given by (5),(6) and wg is precisely equivalent to the free boson realization of W-gravity given in [1]. The formulation presented here, however, is much simpler then the Lagrangian formulation to be found in the literature usually. We use small w to indicate that we are dealing here with classical rather than quantized algebras). Let us illustrate our approach by a few examples: For g = gl(N) we may reformulate the symplectic form in (6) as шд = fs, tr(dY Л dY'), where we understand the trace to be taken over the defining representation. This expression for u> allows to calculate the algebra of the и,-, even though gl(N) is not semisimple. One obtains

The Fourier components of the ui then form the w„ algebra [...).

K,u<4 = ((* - 1)" -С- l)mK#a (9)

If one tries to truncate (8) by expressing u„ for t; > N in terms of the {u,-,i < N} as described above one will find expressions containing trY'(cr) on the r.h.s. and thus the resulting algebra is not a W-type algebra in the ususal sense. Nevertheless, to obtain the W-type algebra to,i(jv) we may start from gl(N) and add to the above system the constraint

trY'(

{Г(<Т1),П^)} = «'(^!-*«) (П) (10) is a second class constraint and thus the Poisson bracket is to replaced by a Dirac bracket, yielding

[u*(«ri),u|(trj)] = {(k - l)u*+,_j(o-l) + (l~ l)ut+|_2(«J2)- (12) - 1)(/ - IK-dMu,-,)^))^. - v?) '452

One has rank(g) = N — 1 and may choose {/I,.../JV-I} = {2, —,N} to truncate (12). The resulting algebra may be regarded as classical Wn algebra. (Of course, a direct calculation of wiHN) would give the same result.) The difference between (9) and (12) is a subtlety which seems to be somewhat neglected in parts of the literature on the subject. This is not crucial as long as one speciaializes to the case of W3: In the sl(3)-algebra one has tr(g4) = jtr(g2)tr(g'2) and thus obtains

[u?,u3] = (n-2mK+" (id;

r The nonlinear terms on the right hand side of (12) contribute to the value of a in (13) only. The latter, however, may be changed by a trivial reseating. For N Ф 3 the nonlinear extra term in (12)might be significant. We want to emphasize, however, that not necessarily the first К = гапк(д) of the u' are independent. For g = 50(4,1), e.g., u3 vanishes, uJ and u4 are independent and with u6 = uV - 1/6 (u2)3 we obtain

[U2(fi),u2(a;)] = - ^XMo-,) + u3(cr2))

M<7l),U4(ff2)) = £'(oi-a*)(MCTl) + 3U<(ff2))

2 3 3 [и4(<Т,),и4((Г2)] = й'(<*1 - ^)(3u4u2(

Let me finish my talk by some remarks on the quantisation of the classical system described here (i.e. the formulation of a W-string theory), a work, which is still in progress. Enlarging the taiget space by adding a factor m -f n = d we have to add J Xa and JdX' A dX, 1 X € Л" '" to u2, and u>, respectively. In a BR.ST quantisation we have to add a ghost field a and its canonically conjugate momentum 4, to the phase space for any constraint field u,-. The BRST-charge will have the form [4]

Q = J{W2 + - lHcJc, + с»ЗД]} + ... (15)

where the terms indicated by ... do not dependend on u2. The Fourier modes on a loop space induce a natural complex structure. Using the latter to define a polarization in the complexified tangent bundle (we neglect here the subtleties arising fom the fact that this complex structure is not reparametrization invariant) we will find on the quantum level

2 D + dimimg-A g — A f ,„ Q — J Сг c2 + ... (16) where 4 = -£2(6s2-6s + l), seB (17) л Fot a consistent quantization a necessary (but not sufficient condition) is that the first term on the r.h.s. of (16) vanishes. Following [5] a modification of the model is obtained by replacing the Lie algebra g by the corresponding Lie group G leading to the modification ,. к dim a , . dimg -4 Д (18) к + a[g) v ' where к is sn integer number and ct(g) is the dual Coxeter number. '153

References

[1] C.M. Hull; "Classical and Quantum W-Gravity", Queen Mary and Westfield College, January 1991 and references therein

[2] A. Mikovic; Phys.Lett. B287 (1992) 51

[3] M.J. Bowick, S.G. Rajeev; Nucl.Phys. B293 (1987) 348

[4] J. Thierry-Mieg; Phys.Lett. B197 (1987) 368

[5] A.D. Popov, S.G. Sergeev; JINR publications E2-92-261 '454

A NONLINEAR SCHRODINGER EQUATION AND SOME OF ITS SOLUTIONS

W. Soberer* and H.-D. Doebnert

t Institute for Theoretical Physics A, TU Clausthal, Leibnizstrasse 10, D-38678 Clausthal - Zellerfeld, Federal Republic of Germany, t Arnold Srmmerfeld Institute for Mathematical Physics

Abstract We analyze a family of nonlinear Schrodinger equations derived from a quantization scheme on manifolds M. Several interesting properties of these equations are mentioned and explicit solutions of the nonlinear equation for M = Ш3 and without potential (free equation) are constructed.

1. Introduction Investigations of the possibility of a nonlinear character of quantum mechanics, in particular of its evolution equation have been quite numerous (we quote only a few of them [l]-[6]). In most cases a nonlinear term is added to the usual linear Schrodinger equation. While the choices for this additional nonlinear term are often made to obtain certain properties of the full nonlinear equation they are still "ad hoc" in character. Recently, however, Doebner and Goldin [7,10,11] derived an imaginary nonlinear addition (proportional to a new real quantum number) to the usual Schrodinger equation from first principles via a quantization scheme on manifolds M — the Quantum Borel kinematic — [9] or, for M = IRn, equivalently via a representation theory of the current algebra on ffl." [8]. The nonlinear Schrodinger equation obtained in this fashion takes the following form ihi> = + + Щф, ф]ф (1) 1 р

where 2 + (2)

is the linear Schrodinger operator and p = фф denotes the probability density. As we will mention in Section 2 only the imaginary nonlinearity is determined from first principles whereas the real nonlinearity is still a matter of choice. We assume

, Ар , j1 J • Vp (Vp)3 R№J] = ]TCJRJ- hD ci— + C1-Z + c3— + c,—5— + 2 7 (3) j=l P P P Г P

where J := yj = (ф^ф — фЧф) and the parameters с,- are dimensionless. Equation (1) represents a five parameter family of real parts which endows the full nonlinear equation with a variety of interesting properties some of which maintain "cherished" principles of linear quantum mechanics as is reported in Section 3. Nevertheless, the nonlinearity induces some interesting deviations from the linear case one being the existence of square integrable solutions of the free equation (equation (1) with V = 0) including solitary waves as shown in Section 4. We conclude with a brief discussion in Section 5. '455

2. Origin of the Equation Th" nonlinear equation is rooted in a general symmetry (-algebra) of a manifold M, the quantization of a system 'living' on M, and is based on the assumption that any time evolution should respect local probability conservation. We sketch the different steps of the derivation of (!)• 1. Step. We consider a system localized in Borel sets В taken from a Borel field В (Л/) on M. To make localized region В move use the flow ip? of a vector field X 6 X0(M) on M. For technical reasons we choose X to be complete. The How with differential X has a group property (flow group), the parameter i could be the time, it acts on В as

The tuple K(M) := (В(М),Л"о(М)) is denoted as the Borel kinematic with В and X as generalized classical position and momentum observables. S. Step. To quantize K.(M) we construct a map Q = (IE,IP)

IE : B(M) -» SA[H), IP : Xa(M) SA(H) into the set SA(H) of all selfadjoint operators on a Hilbert space H with the following properties:

• Because of the Borel field and because of the interpretation of matrix elements of 1E(B) as probability densities to find the system localized in B, IE gives a pov-measure on M. We specify this for physical reasons (see [9], section 3.2.2), as a projection valued measure. Hence we assume E(B) as an (elementary) spectral measure.

• Because of the Lie algebra structure of Л"о(M) we want IP to be a Lie algebra isomorphism. We assume that the quantum analogue of ;he flow group

У*ЩВ)У* = Е(У,* (В)).

Hence, it is plausible to take for P(X) the differential of V*. Furthermore for physical reasons we want that IP(X) is a local operator (see [9j, section 3.2.2).

Q(IC(M)) is the Quantum Borel kinematic. We mention some of its interesting properties. Because of the spectral theorem H can be realized as L2[M,p) with a Borel measure p such that ЩВ) acts on L2(M, p) via the characteristic functions of B. One can 'quantize' the abelian algebra C°°(M, IR) via

Q = IR) - SA(H), f « Q(/) := /. (4)

Q(C°°(M,1R)) and Р(Д"0(Л/)) form an infinite-dimensional Lie algebra, i. e. the semi-direct sum S(Af) := Q(C°°(M,]R}) «£ lP(.Vo(A/)) which is a generic general symmetry algebra on M. 3. 5(ep.Constructing all Q{K(M)) up to unitary equivalence is equivalent to constructing all inequivalent representations of S(M) where the representation of C°°(M, IR) is given by (4). To have a classification theorem we need an additional very natural physical assumption. We want that IP(X) is a differential operator on a dense domain in L2(M,p). What does this mean? The closed linear space of ©-valued square integrable functions on M, L2(M, p), is a measure theoretic object. It contains no notion of differentiability. So one has to define a differentiable structure on the point set MxC such that its restriction to M is the differentiate '456 structure given on M and its restriction to

/ e C°°(1R3, IR) Q : Q(/)<4 = {ф, X=g(i)6Ao(IR3) 1PD : = i;[ff-V + V-a]v + I>(div^. (5)

Different D yield non equivalent representations. The physical interpretation of D is not yet fully explored. The next step gives some hints in this direction. 4. Step. In Steps 1,2 and 3 we described the kinematical situation. A time dependence has to be introduced in a further step. For simplicity we choose M = IR3. To postulate local conservation of probability in order to restrict the dynamics we use for H a Fock space HF generated cyclically from a vacuum |fl) through distribution valued operators iP'(x) and Ф'(х). D The Q and 1P act on the one particle subspace Нрл (see (5))

p(x) = Ф'(х)Щх),

IPc,(ir)!P"(x)|n) = Jg(x')' JD(x) = J°(x) - DVp(x) J°(x) = i (lP-(£)W(i) - • p is a position and JD a momentum density. J" is just the usual quantum mechanical current density (ft = 1, m = 1). Note that for matrix elements in HF,i

(ф,р(х)ф) = ф(х)ф(х) = p(x) (ф,30{£ф) = i (ф(£)Щ(£)-ф(х)Щ(х)) - ОЦф(х)ф(х) = JD(x) (6) holds. Now we are prepared to formulate the above mentioned postulate. (For a more general and fundamental derivation see [19, 20].) We introduce a time dependence via ф(х) —» ф(х, i), i.e. in the Schrodinger picture, and restrict the time evolution of ф[х. t) such that

J jtp{x,t)

Because of this we assume that there exists a generic vector field density such that p is its negative divergence. There is only one such generic density in our theory, that is JD(x). Hence we introduce the time dependence of p via

p(x,t) = -V- J°(2,t). (7) '457

5. Step. Equation (7) is a coupled partial differential relation for ф(х, t) and ф(х, t), first order in time. As a continuity equation it results from a nonlinear Schrodinger equation of the form

д,ф(х, 0=(4Л+К(х)) ф(х, t) + Г{ф, ф]ф(х, t). (8)

Неге Р[ф,ф] is a multiplicative functional of ф, ф, \ф, etc. Inserting (8) into (7) we get

ImF ReF = Я[ф, ф] not restricted by (7) where p is given by (6). It is reasonable to choose for the functional Я an expression similar to ImF, i.e. proportional to D, complex homogeneous of order 0, a rational function with derivatives not higher than second order in the numerator and euclidean invariant [10, 11, 15]. Introducing ft and m we get as a result the five parameter nonlinear Schrodinger equation (1).

3. Properties of the Equation In order to discuss equation (1) it is convenient to use the following notation: 2mD a n 7 —g—i '•— 2сг, b:=cu c:=4(c2 + cs), d := Ci + <ц, e := c3.

Let

A. By construction the overall probability to detect the particle is preserved by the nonlinear time evolution: ^№=0. (9)

B. If one generalizes the 1 particle equation (1) to an ЛГ-particle equation different subsys- tems do not interact through the nonlinear terms. Take e. g. N = 2 and f,j 6 IR" as

coordinates for the two particles. If the potential separates V(x, у) = Ц(х) + V2(y), then an initial state ф(х,у, 0) = ф1(х,0)фз(у,0) in which the x-and y-system are uncorrelated remains uncorrelated throughout its time evolution ф(х,у, I) = 0i(x, <).

C. The equation is complex homogeneous, i. e. if ф is a solution with initial data tp then for all A € С we have Хф as a solution for the initial data Xtp. Because of this property the type of experiments proposed in [3] to decide on the existence of a nonlinear term in the Schrodinger equation will give results independent of D.

D. We construct stationary solutions, i. e. solutions with dtp = 0, which are in L2(R3,

- -^Дф(х) + У(х)ф(х) = Еф(х). (11)

with '458

and a settled mass m*

(l-a-y-Jr,»)* "» (13) provided £ and m* are finite. In order to maintain square integrability for the solution ф one must require in addition 5 > 1. If the diffusion coefficient D vanishes, solutions (10) reduce to ordinary stationary solutions of the linear Schrodinger equation (11).

E. We find for the time dependence of the matrix elements (i) and (p) of i and —ikV

fr-4 <»> for arbitrary с,- as expected (first Ehrenfest relation) and

I(S) = -(W) •

-fiZ>{V[(c, - + (<*+ 2с5)Л2 + СзЯз + (в, + 1)Я5]),

i. e. for Ci = 1 = —c4 and c% + 2c5 =0 = c3 the second Ehrenfest relation cf linear quantum mechanics is valid.

-1 F. Equation (1) is clearly invariant under Euclidian transformations g : ф t-t Т3ф := фод . For с, + = 0 = сз the equation is even invariant under Galilei transformations g where 1 the action is the usual one ф i-t Т-Яф := exp{ — jj(mi> • x + о g' . Note that if the coefficients Cj are such that the Ehrenfest relations are satisfied the equation is also Galilei invariant.

4. Nonstationary Solutions of the Free Equation In the following we give some nonstationary and square integrable solutions of (1) for V = 0. At the moment there are no specific results on the solution variety. Our solutions are obtained from a Gaussian wave ansatz [17] ' = 7^Ьгр +i(A(t)xl+я(01+С(г,)) (16)

where т,s,A,B,C are real functions of time whose time dependence will be determined upon substitution of (16) into (1) and comparison of real and imaginary parts as well as different powers of x. This ansatz has been applied to particular cases of equation (1), see e.g. refs. [12, 13] in which the imaginary part of (1) is absent but certain real parts are retained, i. e. D = 0,c, = J. In refs. [14, 16] the case CI = —1, с; = 0 (j > 1) and in [17, 18] the general case including a harmonic oscillator potential is discussed. Here we shall give a treatment for V = 0 for all parameter values с,-. Because of the separation property (cf. item B. in Section 3) it suffices to to consider only the one-dimensional case. The ansatz (16) yields the following expectation values in terms of the time dependent functions

Upon substitution we obtain the following system of first order differential equations: 2 D 2 cr = — + ±Aa (17) am s = -+*-As (18) mm ' '459 and

• - hDc В — 2As В = — з + 2Ш — <т4 а2

С =

Equations (17), (18) give A (resp. В) as functions of cr, a,s and i. Replacing 0 where

?(')••= (22) with ^

l + 7e we obtain for q and « the following second order differential equations

q = aq~2rq + /V"4t (23) and 9-Jr —)rer- (24) '"[f 4 with

a := 27(d + 7e) 3 3 ^ = 1 - C7 — (1 + 2rf)7 — e7

Equations (17) and (18) give A (resp. B) as functions of a, a, s and s whereas the determination of С involves only integration once q and s are known. We now construct solutions of (23) using a function k(z), z := In q which has to be determined later (see (28)). With an ansatz

q=k(e')q1-2* (25) we obtain к~ = (2t — l)k2 + ak + f}. (26)

IfT = i,a = 0 = /3we have immediately

Otherwise we find (after using (22) through the t dependence of к (ко = k(q(0)))

( rkdk 1 „(*«)) = «.ехрЦ {2T_1)k, + Qk + l3). (27)

Actually, in order to integrate (26) or, equivalently, to evaluate the integral in (27) one has to distinguish various cases given by the values of r, a and /S In each case the integral can be given explicitly in terms of k. Since the resulting expressions for a are in some cases quite cumbersome we defer from writing them down here. It remains to calculate k(q(t)) = k(t). '460

Taking the time derivative of к and using (25), (27) and (22) we obtain the following first order differential equation (28) together with the initial condition , m «о = —r^o^o (29) rn Moreover, we can write down s(l), once k{i) an, hence,

s(t) » -о {jf (g^ - yek(p) (j^Y) *} Л. (30)

Finally, one has to insert a and s into (21) and integrate this equation where one more integration constant Co appears. Equations (27), (28), (29) and (30) provide a complete solution for tr,s,A,B,C in (16) and the five integration constants appear in the form of a0,ao,s0,satCo- Of course, the difficult part that remains, is to determine k(l), i. e. to solve (28) which is possible only in a few cases (see [17, 18]) since, as stated above, a as a function of к for general parameter values cj turns out to be quite complicated. It should be noted, however, that the ansatz (16) does lead to solitary wave solutions of (1) for certain values of the cj [17]. In order to construct the solitary solutions we start by requiring that a and thus q be constant which assures that the shape of фф does not change. This requires f} = 0 (see (23)). Moreover, (17) implies in this case A = —J^JJI = const (that the right hand side of (19) vanishes is equivalent to /3 = 0) and the the integral in (30) can be evaluated easily. To obtain solitary waves which move with constant speed one has to require in addition that a = 0 which yields in this case s(t) = s0 + sot. The integration of (21) is easy but cumbersome (see [17]). The solutions thus obtained are Gaussian solitary waves moving with constant speed. They are square integrable solutions of the free nonlinear equation and have the following (поп minimal) position-momentum uncertainty

АхАр=^у/Г+^ (31) which yields the classical limit: lim Дх Др = mD. (32)

5. Conclusion We reviewed a derivation of quantum mechanical evolution equations from first principles using a quantization method for systems localized on a manifold, e.g. M = ]R3. The method is based on a general symmetry algebra S(Af) of M and it depends, up to unitary equivalence, on ПДЛ/)" and a real quantum number D £ JR.1, which enforces under the postulate of probability conservation an imaginary nonlinear term

DhA (фф) 2 фф

in the usual linear Schrddinger equation. It is convenient to choose (an arbitrary) real nonlinear term correspondingly. One gets a 6 parameter (including D) evolution equation. The parameters should be universal, a minimal choice is Ci = 0, i = 1,... ,5, i.e. the nonlinear real term vanishes. Then only the quantum number D remains and has to be interpreted. But also if the С{ ф 0 the equation (1) is a somehow universal generalization of the usual linear Schrodinger '461 equation. This can be understood from the discussion in section 3. Square integrable stationary solutions are of the form \ф) where the phase factor Р(ф, ф) is a functional of a corresponding bound state ф of a linear Schrodinger equation with the same potential but with a scaled mass, i.e. the noniinearity gives 'only' a phase factor (and a D dependent exponent <5 of ф). We construct nonstatioiuuy solutions from a Gaussian wave ansatz, we prove their existence and again, they behave similar to the corresponding solutions of the linear Schrodingcr equation. Because the bound states and energy levels can be calculated from the corresponding linear problem, D (and also a, i — 1,...,5) is measurable from quantum mechanical precision experiments under the condition that it is possible to measure the mass m of the system independently of the nonlinear term. This is the case (see section 4). Furthermore one can get an estimate for D (an upper bound) through an analysis of a 2-level system (see e.g. also [5]); calculations for this are in progress. It is tempting to consider the solitary solution as a free "particle". There are, however, two serious obstacles to such an interpretation. First, the necessary condition 0 = 0 implies ICj si 1 for at least some Cj. This means that the nonlinear term in (1) is of the same order as the linear one which seems physically unreasonable. The second difficulty is that 0 — 0 is exactly one of the two conditions in which the transformations between stationary states of the nonlinear equation and the linear one with a rcscaled mass do not exist (see (12) and (13)).

Acknowledgements One of the authors (W. S.) would like to thank the organizers of the workshop Symmetry Methods in Physics for providing travel funds which made his attendance possible.

References

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[3] A. Shimony, Phys. Rev. A 20 394 (1979).

[4] N. Gisin, J. Phys. A 14 2259 (1981).

[5] S. Weinberg, Ann. Phys. 194, 336 (1989).

[6] J. P. Vigier, Phys. Lett. A 135 99 (1989).

[7] H.-D. Doebner and G. A. Goldin, Phys. Lett. A 162 397 (1992).

[8] G. A. Goldin, R. Menikoff and D. H. Sharp, Phys. Rev. Letters. 51 2246 (1983)

[9] B. Angermann, H.-D. Doebnfir and J. Tolar, in Nonlinear Partial Differential Operators and Quantization Procedures, Lecture Notes in Mathematics 1037 (Springer: Berlin, 1983), p. 171.

[10] H.-D. Doebner and G. A. Goldin, in 'Particles and Fields', World Scientific 1993, p. 116.

[11] H.-D. Doebner and G. A. Goldin, in Annales de Fisica, Monografias, Vol. 11, 199S, p. 442-445. '162

[12] A. G. Ushveridze "Special Doebner-Goldin Equation as a Fundamental Equation of Dissi- pativc Quantum Mechanics", to appear in Phys. Lett. A.

[13] A. G. Ushveridze, "Dissipative Quantum Mechanics. Special Doebner-Goldin Equations, its Properties and Exact Solutions", to appear in Phys. Lett. A.

[14] V. V. Dodonov and S. S. Mizrahi, "Doebner-Goldin Nonlinear Model of Quantum Me- chanics for a Damped Oscillator in a Magnetic Field", to appear in Phys. Lett. A.

[15] H.-D. Doebner and G. A. Goldin, "Properties of Nonlinear Schrodinger Equations As- sociated with Diffeomorphism Group Representations", to appear in Journal of Physics A.

[16] G. A. Goldin, Int. J. Mod. Phys. В 6 1905 (1992) .

[17] P. Nattermann, W. Scherer and A. G. Ushveridze, "Quasi Classical States and Solitary Waves for the General Doebner-Goldin Equation," preprint TU Clausthal 1993.

[18] P. Nattermann, W. Scherer and A. G. Ushveridze, "Exact solutions of the General Doebner- Goldin Equation," preprint TU Clausthal 1993.

[19] H.-D. Doebner and J. D. Hennig, " Borel kinematic, quantization and the second order Riemannian time evolution," preprint TU Clausthal 1993.

[20] H.-D. Doebner, "Borel Quantization and its generic time dependence", to appear in the Proceedings of the III. Wigner Symposium held in Oxford, 1993. '463

VARIETIES ON QUANTIZED SPACETIME Arne Schirrmacher

Max-Planck-lnatitut fur Physik, Miinchen"

Abstract To construct a quantum deformation of the Minkowski space one can start from different descriptions: Instead of vectors or spinors we consider Grassmann manifolds. A model for electrons in quantized epacetime is sketched.

1. Quantized spacetime The Minkowski space is a homogeneous space possessing a natural group of transformations, i.e. Lorentz transformations and translations. It can be defined as the R* with a metric g of signature (1, — 1,— 1, — 1). Using Pauli matrices it can also be described by hermitean 2x2 matrices. These can be embedded into generalized homogeneous spaces as the Grassmann manifolds. Understanding the Pauli matrices as'vector-bispinor objects, the Minkowski space can be found in the second rank spinor space. These three descriptions allow to realize the following transformations associated to the groups indicated:

AA A i" X = x "

vector hermitean matrix 2nd rank spinor

R4 H(2) S'^ffiS0'1 x> «XI

Grassmannian GIA(C)

x —» Ax + а X MXM+ + F x—• M ® M x

50(3,1) P С 5

For the g-deformation of the Minkowski space the spinor approach was used first [1, 2, 3]. In

the vector approach only the q-Euclidian space and the 50Pl,(4) was found [4, 5]. We will exploit in this paper the third approach via f-Grassmannian manifolds as known from twistor theory (cf. e.g. [6]). Let us start with the quantum group 5Л,(2) and its quantum plane •-Ci; ) -С»- :) (ч 'postal address: P.O. Box 401212, 80712 Miinchen, Germany; e-mail: ars at dmumpiwh. '464 with the relations xy — qyx and ab= q ba etc. (2) choosing q € R. In order to make the quantum group complex we have to introduce new variables and an antimultiplicative conjugation a —> {a, a} etc. and the relations

IJ = q~l yx and ab = q~l ba etc. (3) follow. The mixed relations can be constructed from Я-matrices if one uses the complex quan- ll2 ,,2 tum plane x= [x,y, -q y, q' x) that transforms with a quantum matrix T e SL{qii)(4): x-> x' = Tx. We have

V -q-lb

+_1 In particular with M also M' = M is a SLq(2) quantum matrix. Due to the conjugation properties T is pseudo-unitary with the quadratic form

ф=(г *) : Г+ФГ = Ф, (5)

i.e. it belongs to the quantum group 5£/{,0у(2,2). For Gi{9ij}(4) and its subgroups there exists a number of multiparameter quantum deformations. Let us analyse Л-matrices of flag type:

П> - • • r\2 a12 . Г21 . . • Г13 013 b 13

• • - r31 C13

П4 «14 tu Cl4 du en ггз a23

r24 a24 bJ4 • • • C24

Г34 034 . Г43 . Г44

The indices are chosen as (1M2,21,13,31,22,14,41,23,32,24,42,33,34,43, 44) w.r.t. the indices 1...4 of T. The matrix R = PR, with P the permutation matrix, has two different eigenvalues, e.g. 1 and q7, iff гц = 1, = 95/r,-j (j > t) and a,-,- = 1 — no matter the other entries. Analysing the Yang-Baxter property, we get the following three cases:

Case(I) For bl3, Ci3, Ьи, сы, du, e14, k>4,c2l = 0 one finds the multiparametric version of the

standard GL,(n) Я-matrix [7, 8, 5]. The embedding of SL,{2, C) requires rla = r2i = Г34 =

Г43 = q, as the compatibility with the conjugation requires ТЦ = r13. The mixed commutation

relations of the quantum group then depend essentially on two parameters p = тц/г13 and t = 143/газ independent from q. For illustration we give the commutation relations for the complex plane: '465

xy ~ q yx xy - g"1 yx

xy = ri3yx ух = гкху (7)

xx ~ Гц xx у у = ra3 у у .

Case (П) For bl3,ci3,blt,c24 = 0 and i>u, ец ф 0 we find a new Я-matrix. То embed Si,(2, С) we need: Г12 = r2i = Г34 = r« = q. The compatibility with the conjugation requires here r13 = = s, r14 = r23 = дл. We can choose bH = — qs\, el4 = gs~'A. The ill'T-relations give the quantum group commutations. We find those already used in [2, 3]. Restricting.a = d and b = —qc a S(/,(2) subalgebra can be constructed. The complex plane reads in this case

xy = q yx xy = q~' yx xy - q*s~' yx yx = q^s-1 xy (8) xx - qa-1 xx-qXyy у у — qs_1yy, which generalizes the complex plane of [3].

Case (Ш) For 6,3,013, b,4,c14,d14, еи, b2i,c24 ф 0 we get the one-parameter solution of [9]. It is, however not possible to embed SL,(2, C). We summarize the results in the following table:

embedding of S£,(2, C) conjugation # par.

'standard' multiparameter rn = Г34 = q I"J4 = Г13 4

extended Г12 = '"3-1 = 9 T24 ~ Г13 =: s = ^э = 4s 2

c GLf ( 4) not possible — 1

We introduce the «/-deformed Grassmannian manifold as a 2 x 4 matrix ['£] that has the same commutation relations as the two last columns of а 5У{,0}(2,2) quantum matrix

As the Grassmannian is a projective space we have the following equivalence relation.

iY iYA' Z ZA and in particular for A = Z~x we have the chart on the Minkowski coordinates X = YZ'1. Hence in this approach the quantum Minkowski space relations follow from a product of noncommuting quantum submatrices of SU{Vj}(2,2). Writing

x=(A,B,C,D) — Х = , (11) 466 one finds for the length ||z||m = gijx'xJ = detqX Case(I) yields

AC = qpCA AD = qtDA ВС = (qv)~lCB BD = ((/i)"1 DB • (12) CD = DC AB = pt BA+tqXCD with the invariant length ||г||м =qw (CD - q~2t~x AB) , where ш = q+q~l. The corresponding metric shows that the conjugation structure of the Euclidian an theJMinkowski quantum planes are different: For the q-Lorentz group the hermiticity of X gives A = B^ В = А, С = С, D = D, whereas for the deformation of 50(4) one has (4,5] A = В, В = А, С = D, С = D.

Case(II) gives

AC = CA-q~l\AD AD = q2 DA ВС = С В + q\ BD BD = q~2 DB (13) CD = DC AB = В A + q\ CD - 9"1 A D2

with the invariant length ||X||„ = qu(CD - q'2 AB) . This case agrees with the results of [1, 3] (using q «-» l/q, since we choose X -» MXM+ instead of X -» M+XM). In this approach it is clear how to enlarge the q-Lorentz group to a

Consider SU{,tl)[2,2) matrices of the form

(M N\ (14) V M'J that contain in the undeformed case Poincare and scaling transformations. In general the determinant of M does not automatically commute with the N part and it can not be set to unity unless we require further conditions for the deformation parameters. E. g. for the case (I) we must set q = 1 and t = 1/p as discussed by Demichev in this volume (cf. also [10]). A M different way is to start from T = ( € f/{,0)(2,2) and consider *-{*-(? £-) with (is)

By definition det

1 1 det4(KZ" ) = det,(K) deti/,^" ) (16) 9 Г13

from which follows that the q-Minkowski distance is invariant.

l|x'-v'|pM = detq(A7 — VJ = detq (МХМ+ + F- (MVM+ + F)J (17) + = det,(Af) detQ(X - V) det,/,(M ) = II*-vIIM •

with F = iNM+ = iNM+. The full Hopf-*-structure remains to be discussed. '467

2. Free electrons in quantized spacetime Using the relations (13) for the momentum algebra (pA,pB,pc,p°) with g <-> 1/? to conform with [11], i. e.

A C A D pV = + P P = pV + qX p p 2 с в pV = T PV pV = р р -ч-'Хр»р° (18) 2 В С pV = Я Р°Р Р Р° = PV- and the metric ( 0 —gJ 0 0 -10 0 0 (19) 0M = 0 0 0 1 \ о 1 -qX) we get p2 — 3up'pJ = ~(pcpD — я2рлрв) • (20)

The conjugation is p* = ps, pfl = /И, pc = pc, p° = p°. We seek for a Dirac operator

i := = gu'i,pJ s. t. ^ • j/ = pJ . (21)

Calculating j/ • rf and using (18) we find the q-deformed Dirac algebra

1AtB + 1B1A = 7'57D + 7Z>7/1 = 0 7*7° + 7°7B = 0 1A1° + 4~21C1A = ~qX-rDtA (22) 7B7C + 9а7С7Я = q3XfDiB Iе 1D + 7D7C = qXfBiA + (7Д,В,0)2 = 0 (7C)2 =

The deformed 7's can be identified with the undeformed ones as follows ^ -

7* = + (23) = vl^0-^- With 70+ = 70, 7l+ ~ —7* there is an element q s. t.

A + 2 B (77B)+ = q2 П1А (m ) = 4~ ni (24) [V7°)+ = (V7°)+ = mD. using the identification one finds

V = 7° = 1C + О ~ ?A)7° + £ 7Л7В7С - Ч2~ 7*7*7° (25) w ш The Dirac equation reads f - m)rb = 0 . (26) 468

For the Dirac spinors we have the scalar product (ф,ф) := ф*т]ф implying hermiticity of jj The g-Dirac matrices have in the chiral representation the form '-U")- (27) where {a'} und {5'} are two sets of g-deformed Pauli-matrices. The Dirac operator reads

PC IP* ЯРВ Ч2Р° (28) PD -qpB pc—qXpD

Clearly j/ • j/ = ps. The covariance ot this operator under q-Lorentz transformations can be proven [12]:

(29) \7тр M-)\Sp Д М+)-\Э-Кр )•

In the center of mass frame we have

л в С D р =р = 0, р = у/ф^ЧЕ, p = y/jJZE/q (30) with the energy E = (pc + pD). The Dirac operator reads

qE l' H = Щ = E/ql (31)

q~*E ф = m x в2ф = тп2ф 2 (32) q E x = rn ф E x = The states have to have the energies E = ±m which are twofold degenerate. We can take as solutions \ 0 l V4(o) = (33) y/2q ±1 «го = ^ \ I о; ±1. that are covariantly normalized on the energy sign e = ±1:

(ф'Ж^) = = £ (34)

since = Х+Ф+ Ф+Х Х+М-гМф + ф+М+М'х = . (35) The general solution can be obtained by boosting

i-e. Po = °'Pei=(m (36)

W?/ The Lorentz transformation po~* P = Лро yields

(12) '469 and consequently

*l(o) = 4=f °

\TqbJ

(38) qb \ qd 1Ш = y/Zq— I' =F«c \±c J The positive quantity Щч\Ф) - + X+X (39) is invariant under rotavons 5(/,(2) but not under boosts of C). It can be interpreted as the probability Tn order to characterize the states we can use the energy sign and the spin. Iri the ihe energy sign operator reads U Ч-^+лч (40) with Е\ф) = с |0) l. - i тп (41) In a general frame we have £ = ra-1 jf. The spin operator in the rest frame is

-9 E = = ^hV"7V)(7C + 7D) (42) 1/9 -1/9 with = ф) [S,£] = 0. (43) To characterize a state we can take appropriate orbital quantum numbers of simultaneously commuting operators, in our case рг,рс, p° and the three-component of the angular momentum [13], and the internal quantum numbers с and a (or 5 = to):

c D Ф = \m,p ,p ,L3\t,a) . (44)

The non-diagonal operator-! pA, pB, L± ... commute with 17-factors with the diagonal ones, and hence act as raising and lov.'ering operators generating spectra proportional to powers of q.

References

[1] Carow-Watamura U, Schlieker M, Scholl M, Watamura S: Int. J. Mod. Phys. A 6, 3081 (1991).

[2] Podles P, Woronowicz S L: Commun. Math. Phys. 133, 381 (1990).

[3] Schmidke W B, Wess J, Zumino B: Z. Phys. C52, 471 (1991). '470

[4] Reshetikhin N Yu, Takhtajan L A, Faddeev L D: Leningrad Math. J. 1, 193 (1990).

[5] Schirrmacher A: J. Phys. A24, L1249 (1991).

[6] Ward R S, Wells R О jr.- Tmstor Geometry and Field Theory, Cambridge University Press (1990).

|7] Reshetikhin N: Lett. Math. Phys. 20, 331 (1990).

|8] Schirrmacher A: Z. Phys. C50, 321 (1991).

[9] Cremmer E, Csrvais, J L: Commun. Math. Phys. 134, 619 (1990).

[10] Chaichian M, Demichev Л P: Phys. Lett B304, 220 (1993).

[11] Ogievetsky 0, Schmidke W B, Wess J, Zumino B: Lett. Math. Phys. 23, 233 (1991).

[12] Schirrmacher A: Quantum groups, quantum spacetime and Dirac equation, in: Low dimensional topology and quantum field theory, H. Osborn (Hg.), Plenum Press, New York, 221 (1993).

[13] Pillin M, Schmidke W B, Wess J: Nucl. Phys. B403, 223 (1993). '471

SYMMETRIES, RECURSIONS AND LINEARIZATION FOR TWO-COMPONENT SYSTEMS OF HYDRODYNAMIC TYPE M.B.Sheftel'

North-Western Correspondence Polytechnical Institute, 191065, St.-Petersburg, Russia

Systems of hydrodynamic type (SHT) have a number of important applications in physics. They include gas and fluid dynamics, nonlinear elasticity and phase transition models, Born- Infeld nonlinear electrodynamics, relativistic string equations, nonlinear acoustics, soliton theory etc. [1,2]. Here we study symmetries of a diagonal two-coinponent SHT with the unknowns s,r and two independent variables t,x, which may explicitly depend on t:

S, = tp(s,r,i]Sx, r, = ^(s.r.t.Jr, (1)

(subscripts denote partial derivatives). The results can be easily transferred to a system with an explicit dependence on x:

т St = ^'(s.r.iJS,, г, = ф (з,г,х)гх. (1") Hydrodynamic symmetries of the system (1) are generated by Lie equations:

S, = f(x,t,s,r,sI,rI), rT - fi(x,t,s,r,sc,rx), (2) which do not depend on higher derivatives. Here r is a group parameter and x,t are independent of г in a canonical representation [3]. The essence of our approach is to single out a class of systems (1), which are rich in symmetries and hence can be linearized. More specifically, we select the systems with an infinite set of hydrodynamic symmetries, which depends on arbitrary smooth solution of a linear system. Then any smooth solution of such a system (1) reduces to a solution of the linear system with variable coefficients. For solving the linear system we use its symmetries. They turn out to be recursion operators for the system (1), which map symmetries into symmetries. Thus we obtain recursion formulas, generating new solutions of the linear system out of known solutions. We can obtain also an infinite countable set of exact solutions of the system (1), which are invariant with respect to higher symmetries (Lie equations depend on higher derivatives). Define functions Ф(я,г, f) and Q(s,r,t) by the equations

фг = vr/(v - Ф)> Q, = Ф.;{Ф ~

i{s,r,t) = 6(«)Ф. + d(r) Фг + Ф0(з), (4)

6(s,r,t) = Ь(з)0, + d(r)0r + 0o(r). Theorem 1. In a generic case a diagonal two-component system (1) of hydrodynamic type has an infinite set of hydrodynamic symmetries, parametrized locally by two arbitrary functions of one variable Ci(s),C2(r), iff the two conditions are satisfied: 1. coefficients ip and ф of the system (1) satisfy the equalities: '472

Ф„(з,г,0 = for, e„(s,r,1) = /Зф„ (5) where /3 is a constant and partial derivatives with respect to t are taken for constant values of s and r; 2. there exist such four functions of one variable 6(a), d(r), Фо(з), 0o(r)> which satisfy the equations:

Фг = ФГ(Ф-Э), 6. = 0.(0-Ф), (6) where Ф and 0 are defined by formulas (4). These symmetries are generated by Lie equations

St = v[s,r,t,x)Sz + b{s), rT = ф(з,г,t,x)rx + d(r), (7) where ф, ф are defined by the formulas: rt ^ ф = а(з,г)ехр[0(х + J

1 - ф = с(з,г)ехрЩх + t(s,r,t)dt)} + ^Q(s,r,t)

for 0^0,

ф = o(s,r)+ f v{s,r,t)dt-b(s,r)[x+ f

ф = с(з,г) + f ty(s,r,t)dt — 0(s,r)[i + f 0(s,r,t)

ar(s,r) = Фг(*,г,0)(а-с), C,(s,r) = 0.(s,r,O)(c - a) (10) and ф, ф are defined by the formulas:

Ф = b(s)Vl + d(r)v5r, ф = Ь{з)ф, + Л(г)фг. (11) Here we restrict the definition (3) of functions Ф(s,r,t),Q(s,r,t) by the'condition:

Ф„ = 0rl = Рф„ (12) which is compatible with the equalities (5). The second condition (6) of theorem 1 has a trivial solution:

6(л) = d(r) = О, Ф = Ф0(л) = 0 = 0o(r) = C0 = const. (13)

It corresponds to symmetries with Lie equations linear homogeneous in derivatives sx, rx:

ST =

ф = a(s, r) exp[0(x + j ф, r, t)dt)} + C0, (15) '473

•ф = С(з,г)ехр[Дх+ f ^(a,r,t)dt)] + C0, Jo for 0^0,

ф = n(s,r) + C0[z+ f

ф = C(s,r) + Co[i+ / Jo for /8 = 0. For nontrivial solutions of equations (6) Lie equations (7) are linear inhomogeneous in derivatives.

Corollary 1. The condition (5) of theorem 1 is necessary and sufficient for the system (1) to have an infinite set of hydrodynamic symmetries (14), (15) or (16), which are linear .homogeneous in derivatives. In particular, the condition (5) witli /? = 0 is satisfied for any system (1) with the coefficients

Sf = v(s,r,t,x)S, + l(s), rf = $(s,r,i,x)rx + d(r). (17)

Here

St = ip(s,r,t,x)Sx + !(«), r* = ij>(s,r, t,x)rx + d(r). (18)

Here ф and ф are again of the form (8) or (9), where a(s,r), c(s,r) are substituted by a(s,r), c(s, r), which are defined by the following formulas:

a(s,r) = b(s)(a,(s,r) -4>sa(s,r)) - b(s)(a,(s,r) - Ф,а(з, r)) +

, + Фг(<*(г)С(л,г)-(7(г)С (я,г)) +Фо00а(*,г) - Ф0(а)а(а,г),

6(s,r) = J(r)(aO>,r)-0rC(s,r)) - d(r)(C,(s,r)-0,C(s,r)) +

+ 0,(6(s)5(s,r) — b(s)a(s,r)) + 0o(r)C(s,r) - 0o(r)C(s,r), where Ф = Ф(з,г,0), 0 = 0(s,r,O). Corollary. Let a(s,r), c(s,r) and a(s,r), c(s,r) be solutions of the linear system (10). Then the functions fl(.',r), c(s, r), defined by the formulas (19), satisfy system (10) iff the functions

which are homogeneous in derevatives. For Sx • rz Ф 0 and Co ^ 0 the invariance conditions

take the form: (

a(s,r) exp[jS(x+ / ф,г, t)dl)] + C0 = 0, (20) Jo '474

C(s,r)exp[0(x + f ф(з,г,1Щ] + Co = О Jo

for /3 = 0, (

a(s,r) + C0(x+ [ 4>(s,r,t)dt] = 0, (21) Jo

c(s,r) + Co[x+ f 0(з,Г,*)Л] = 0 Jo for /3 = 0.

Theorem 3. Let the condition (5) of theorem 1 be satisfied for the system (1). Then any solution of the system (20) for j3 ф 0 or (21) for 0 = 0 turns out to be a solution of the system (1), if the conditions of the implicit function theorem are satisfied. Any smooth solution s(x,t), r(x,t) of the system (1) may be obtained from the system (20) or (21) in the vicinity of any

regular point (i.e. where Sx-rz ф 0).

Remark. Formulas (20) and (21) define in an implicit form a linearizing point-like transformation for the system (1), explicitly dependent on t: solving system (1) reduces to solving linear system (10). A search for solutions of system (10) is facilitated by the use of recursion relations (19).

References

|1] P.J.Olver and Y.Nutku, "Hamiltonian structures for systems of hyperbolic conservation laws", J.Math.Phys. 29, 1610-1619(1988).

[2] O.F.Men'shikh, "On the Cauchy problem and boundary value problems for a class of systems of quasilinear hyperbolic equations", Soviet Math. Dokl., 42, No.2, 346-350(1991).

[3] N.II.lbragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht-Boston (1985). '475

LEVEL CLUSTERING IN THE VIBRATIONAL-ROTATIONAL SPECTRA OF THE ICOSAHEDRAL HAMILTONIAN N. A. Smimova 1

Institute of Nuclear Physics, Moscow State University, Moscow, 119899, Russia

Yu. F. Smirnov 2

fnstitulo de Ffsica, Universidad Nacional Autonoma de Mexico, Mexico 20, D.F., Mexico

Abstract

The qualitative theory of the level clustering of high-/ levels in the rotational spectra of the icosahedral symmetry molecules is developed using approximate methods of solution of three-term recurrent relations (TRR)

If the matrix of the Hamiltonian in an appropriate basis is tridiagonal the eigenvalue problem for such a Hamiltonian can be easily solved using the discrete quasi-classical approximation (DWKB) [1, 2]. The Schrddinger equation for a tridiagonal Hamiltonian is reduced to a three- term recurrent relation (TRR)

= + (Hn,n — E)Cn + #п,п+дС„+д (1) where Д is the step of TRR. Following DWKB one considers C„, u>„ = H„t„ and p„ = //г,,„_д as smooth functions C(n),w(n) and p(n) of the variable n and constructs potential curves (PCs),

U±{n) = u>(n) ± p(n + Д/2). (2)

PCs are the boundaries of the classical admissible region (CAR) for discrete variable n. Namely, for the energy E, CAR corresponds to the values of n for which the inequalities,

U~(n)

C(n) = exp <±i jf arccosl^m^pj dK] (4)

and decreases exponentially beyond the CAR. This method has been applied to the analysis of the vibrational-rotational spectra of the icosahedral molecules. One of such molecules is Geo- Ceo has a shape of cut off icosahedron (see fig. 1) and possesses six fivefold symmetry axes (C5), ten threefold symmetry axes (C3) and fifteen twofold symmetry axes (Ci). At small angular momentum J the molecule can be considered as spherical top and the energy levels

' Supported in part by a grant of the Moscow Physical Society. 2On leave of absence from Institute of Nuclear Physics, Moscow State University, Moscow, 119899, Russia. '476 are (2J + l)2-fold degenerate. However at large angular momentum the energy levels split due to the effects of non-rigidity of the molecule which arise microscopically from the rotational- vibrational interaction. The effective rotational Hamiltonian allowing for non-rigidity in the lowest-order perturbation theory can be expressed in terms of irreducible tensor operators T*3:

H.JJ = + ^-(T* + 5Г_\), (5) here the fivefold symmetry axis of the icosahedron is chosen as quantization axis (z || C\).

Fig. 1: The icosahedron and its symmetry axes.

Using symmetrical top eigcnfunctions | JMК > as basis functions we obtain TRR with a step Д = 5. Following DWKB [1, 2] we introduce smooth parameter к = Kj, where j = . 1 1 at large J. In this case the PCs are of the form:

{/*(£) = ^=(23U6 - 315fc4 + 105к2 ± 42fc(l - fc2)5/2) (6)

(see fig. 2). The icosahedral symmetry of the PCs will be evident if one notices that in the classical limit к — cos 0 , where в is the angle between the quantization axis and the axis of rotation. The extrema and the crossings of the PCs correspond to the rotations about symmetry axes as pointed out in fig.2. The level energies are obtained from the DWKB quantization rules [1, 2]

1 fk- *L v/(l - B2) dk 4 ' ' 2'Д 2

where N = 0,l,2,..,ff = 1,2,... ,Д — 1 ,B = and kr • the classical turning points,

s/AJ+i)

From this formula one can see that the levels where kT and к/ correspond to the same PC are Д-fold degenerate. So we conclude that the levels in the regions of central maxima are fivefold degenerate. Eventually the symmetry of the molecule gives rise to the coincidence of the energy levels with each other in all maximum regions of the spectrum. As the spectrum has two central regions where the energy levels are 5-fold degenerate and two crossing regions where the levels are not degenerate then the energy levels in the upper part of the spectrum appear to be 12-fold degenerate. In turn the levels in all minima of the PCs are 5-fold degenerate and there are four such minima in the spectrum. As a result we obtain 20-fold degenerate levels in the lower part of the spectrum. '477

-1 о ' A a--) Fig. 2: a)The potential cloves (PCs) and the schematic picture of the energy levels, b) The rotating spectrum of the icosahcdron (level clusters).

Due to tunnelling effects the energy levels split and form the groups of almost degenerate levels - level clusters. Thus the levels in the upper part of the spectrum are 12-fold quasidegenerate and the levels in the lower part of the spectrum are 20-fold quasidegenerate. These results can be interpreted from the consideration of the non rigid spherical top rotation. 12-fold quasidegenerate clusters form if the molecule rotates about one of its six equivalent fivefold symmetry axes (in two possible directions) with the smallest momentum of inertia. 20-fold quasidegenerate clusters appear if the molecule rotates about one of its ten equivalent threefold symmetry axes (in two possible directions) with the largest momentum of inertia. The energies of the levels can be easily calculated in harmonic approximation [1, 2]:

E = Eo -r hoj(n + 1/2), (8)

+ where n = 0,1,2,... ,Ea = U(k0),ka - the extreme point of the PC (U(k) = U (k) or U(k) = t/"(*)),

S[lU"(k)p(k + 5/2)]I/a |*=fa for the CAR in the vicinity of all extrema of the PC, 2 j[U'(k)U'{k)]V |1=*„ for the CAR in the vicinity of the crossings of two PCs.

In the maximum regions of the PCs the energy of the levels is a decreasing function of п(.йш < 0) and in the maximum regions it is an increasing function of n(hu> > 0). The results of calculation are given in the table I. The analytical results coincides with the results of numerical calculation [3] and the appearance of level clusters is confirmed. Using the method mentioned above one can obtain not only the analytical results for energy levels of the rotational spectrum of the molecule, but it is also possible to find the wave function and to calculate the probabilities of electromagnetic transitions. '478

Table 1

The upper part of the spectrum: -0.0545 < EA < 0.1750

the central maxima k0 = 0.447 E0 = 0.17457 fiu> = -3.666j the crossing

regions *o= 1 E0 = 0.17457 Д ш = -3.666.7 The lower part of the spectrum: -0.097 < Eq < -0.0545

the central minima k0 = 0.188 EO = -0.09699 Лш = 2.04j the right and the

left minima fco = 0.795 E0 = -0.09699 hui = 2.03j

References

[1] P.A.Braun, Teor. Mat. Fiz. 37, 355 (1978).

[2] P.A.Braun, A.M.Shirokov, Yu.F.Smirnov, Molec. Phya. 56, 573 (1985).

[3] D.F.Weeks, W.G.Harter, J. Chem. Phys. 90, 4727 (1989). '479

ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLE ASSOCIATED WITH QUANTUM

ALGEBRAS SU,(a) AND SU4{i, 1) Yu. F. Smirnov1 and A. Del Sol Mesa

Institute) de Fisica, UN AM Apartado Postal 20-364, 01000, Mexico, D. F. MEXICO

Abstract

The main characteristics and properties of q analogs of Hahn polynomials are obtained using the theory of orthogonal polynomials on the nonuniform grid x(s) = q2'. The connection of these polynomials with the Clebsch-Gordan coefficients for the su,(2) and ли,(1,1) quantum algebras is established. It is demonstrated that each property of f-Hahn polynomials (finite difference derivative, recurrent relation etc.) is in correspondence with the definite relation for the g-Clebsch-Gordan coefficients. In such a way a group theoretical interpretation of the properties of various orthogonal polynomials on nonuniform grids can be found.

Introduction This conference is devoted to the memory of the famous theoretician Professor Ya A. Smorodinsky. He had interested deeply and worked successfully in various areas of the modern theoretical physics. One of problems, that was elaborated by him at last years, was a problem of orthogonal polynomials of the discrete variable (see for example Ref. [1]) and the connection of these polynomials with the representation theory of quantum algebras [2]. In this talk some aspects of this problem, which were discussed by the first author with Prof. Ya. A. Smorodinsky at October of 1992, will be presented. Namely, the close connection of the orthogonal polynomials of the discrete variable on nonuniform grids [3] with the various elements of Wigner-Racah algebra for quantum groups ли,(2) and

'On leave of absence from Nuclear Physics Institute of Moscow State University '480

The necessary formulae and relations, concerning the q—analogs of the Clebsch-Gordan coefficients (CGCs) are summarized in the Section 3. A comparative analysis of the properties of the q—Hahn polynomials and q—CGCs is given in the Section 4. Thus the interpretation of the q—Hahn polynomial properties in terms of the su,(2) and (,su,(l, 1)) representation theory will be obtained in this Section.

1. General formalism In Ref. [3] the solution of the second order finite difference equation of the hypergeometric type is considered. This equation can be written in two forms:

УуЩ i ?ГАу(<.) Vy(s) + Ay = 0 (1.1) Дх (S - 1) Vx(s)J + 2 LAI(S) VI(S) 'д^йШ-Ш""-» м where y(s) is a function of the discrete variable s = о, a + 1,..., b and the finite differences are of the form Д/М = /(* + 1)-/(а), V/(s) = /(s)-/(s- 1). (1.3) As for the functions t(Sф)) ==&(j)-}r(*)A»(j-J f (s) ) . ^ the first of them is a polynomial of second (or less) power on x, the second one is a polynomial of the first power on x. The equations (1.1), (1.2) are equivalent to the recurrent relation.

A.J/(s + 1) + B,y(s) + C,y(s - 1) + Ay(s) = 0 (1.5) where Ф) + T(S)AX(S - I) A.= Ax(s)Ax(s - 1) '

С = (1-6) * Vx(s)Ax(s-l)'

B, = -(A. + Ct). The solutions of these equations are the polynomials y„(s),n = 0,1,2,..., satisfying the orothogonality relation 22 »»(<)»»wpwm« - 5) = (i.7) я where the weight function p(s) is determined by the recurrent relation

p(s + 1 )Ma) = + ф)Дх(* - Ms + 1). (1.8)

The NUS method allows to find: 1) eigenvalues A„, 2) the normalizing factor d„, 3) the particular values of the polynomials y„(s) in the end points a and b of the interval [a, 6] where the argument s is determined, 4) the explicit analytical form of corresponding polynomials, 5) the finite difference derivatives of these polynomials, 6) the coefficients a„,j3„,7„, of the recurrent relation

Ф)Уп(л) = a„yn+1(s) + у8„у„(л) + 7„y„_,(s), (1.9)

7) the coefficients a„ and bn at two first terms of highest powers in the polynomial

l y„(x(5)) = anx"(s) + bnx"- (s) + ... (1.10) '481 under consideration. The details of the calculaation of the quantities 1), 2),..., 7) listed above are described in the books [3] and are not reduced here. This formalism was used by us for the detailed investigation of q—Hahn polynomials. Its results are given below. A part of them can not be found in the books [3j.

2. q—Hahn polynomials

The q—Hahn polynomials Л°3(х(.?); N) are orthogonal polynomials on the nonuniform grid

x(s) = q3'(q is real number, a = 0,1,... TV - 1) (2.1) with the weight

N 2 .fr) _ Aola+tN.Ц,-ЗЦ-1/»«М.2.-1) + ~ ~ ЩР + ,,, pW~q2 [jv-s-l]![s]! • (г-г>

ThHeree orthogonalitand below thy econditio standarnd isnotation of the fors fomr (1.7)uq—numbers". and "q—factorials" are used

(2.4)

[A]! = [*][*-1]...[1], к = integer. (2.5) The difference equation for q—Hahn polynomials has a form (1.2) with

Ф) = q°+N+"(q - q-'mia + N - s], (2.6)

r(a) = (q- tf-1)?*-^!^!/? + 1][ЛГ - 1] - q'[s][a + 0 + 2] J, (2.7)

0+ +г A„ = ? " [гг][п + a f f) + 1]. (2.8)

Thus q—Hahn polynomials satisfy the following recurrent relation of the type (1.5)

Q0-2.+N- 1[ДГ _ A _ L][0 + 4 + 1]A(E + 1)+

+ qN-fi-u-l [,][„ + д. _ _ 1)+ + {A„ - qa-2'+N'l[N - s - IP + 3 + 1]-

- f-fi-*-* (5][„ + N - ,]},(,) = 0, (2.9) A„=[JV][n + o + 0 + l], n = 0,1,2,...,N- 1. (2.10)

The normalization factor is of the form

7 N ! -in- dr?* МЧ" + "К" + + P + + "1 «n-W Я ) »[AT _„_!]![„ +^ + „]![Q + /3 + 2n + i]!

X92<.+2W+W(;V-l)+(N-l)(2£i+/J+N)+i*J(/)+l)+n(o+<'+2) (2 11) where

B - = _9-l). In the •'classical" limit q —» 1(A —» 0) q—Hahn polynomials coincide with usual Halm polynomials on uniform grid x(s) = s [3]. The difference analog of Rodriguez formula connects yn(s) polynomial with the n—times difference derivative of the (modified) weight function p(s). In particular for the q—Hahn polynomials it gives the following explicit analytical formula

a0 h ts No)- ( IV n«b>+0+N+Un-nn [N-s- l]![s]l )q [a + N-s-!]![/?+sj! '482

;1уЧ [a + ЛГ-s + m-!]![/?-fs + n-mj! ' ££ ' 4 [m]!|n-m]![i-m]![JV-s-n-l+m]!' У ' The numerical values of this polynomial at the end points of the interval [0, N — 1] are of the form

4 [n]![fl![JV-n-l]!« ' (213) ЛГ 1 „ (N _ n - [ - ]-'[" + "]' ,(а.-ЦУ-1(п-н)) [nl![JV-n-l]!M!ra]![JV - n - 1]![q]9 ! The symmetry property

(_1)-уп(°+в+А0ЛЙ<>(Л? - s - 1 ,N,q-*) = h°"(s,N,q) (2.14)

is valid for q—Hahn polynomials. The coefficients of the recurrent relation (1.9) are given by the following expressions

_ (14 '1 In + liltt + jy + n + lh-fr+'+Ц . , К Л0) [Q + ^ + 2n + 2][a+^ + 2n + l] ' ' (q - + n][0 + n][a + 0 + N -f n][N - nig'"""-1 7n = " - ,[a -- +, 0а +,o-u 2n][.a +, 0^o-^ + 2n + n1 ] (216)

a +,+JV 0n = 7 о ST(? ° ([Ar - n][n][a + 0 + 2n + 2]-

— [N — n — l][n + l][a + 0 + 2n]) + ^[q + 0 + N + n + l][n + l][a + 0 + 2n]-

- [a + 0 -V + n][n][o + 0 + 2n + 2]) j. (2.17)

The formulae of difference derivatives connect the h„+i(s) polynomial A„(s) and h„(s — 1) or

h„(s) with hn+i(s) and /i„+i(s — 1). They were found using the Rodriguez and look as follows

[n + l)h°+\P+1(s,N + V,q) = = - s][/3 + s)hf(s-, JV; q) + q°+N-l-«[a + N- - 1, N;q), (2.18) +e+ + 1) - 'Wl'-'M = 9" '-"[a + P-r (2.19) In conclusion the expressions for the coefficients of the equation (1.10) are presented

„ _ [a + 0 + 2n]\ a"-4 [n]![a + 0 + n]!(? — 4~*)n' 1 '

&n 9 ~ [n-l]!(9-«piy-la+ /3 + n]l

As an example two lowest q-Hahn polynomials are given below

УМ = 1, yi(s) = a tq2' + bi = +w la+ N+, = 9» -'[a + N - s][sj - q ^ [e + s + lJfjV - s - lj. (2.22) '483

3. su?(2) quantum algebra and q—Clebsch-Gordan coefficients It is well known (see for examaple [7]) that the su,(2) algebra is generated by the operators

J+,J-, J0 satisfying the equations

[Jo, j±] = ±j±, [j+, J-] = [2Л1 =

04)+= (Jo)+~Jo. (3.2) Its irreducible representations (irreps) D' with the'highest weight j = 0,1 /2,1,... are determined by the highest weight vector |jj >, satisfying the equations

J+\jj >= 0, J,\jj >= j|jj >,< jj\jj >= 1. (3.3)

The general basis vector of this irrep is of the form (— j < m < j)

The explicit form of the irrep is given by the formulae

< jm'\Jo\im > = ^mm',

< jmV„| J± > = Vli T rn][j ± m + l]«m.,m+„ (3.5) its dimension is equal to (2j + 1). The Casimir operator is determined by the expressions

J C3 = J_J+ + [7.+ 1/2] , (3 d) C,|jm> = (i + i]J|im>. (3.7)

The expansion of the tensor product of two irreps is given by the usual ClebscVuordan series ji+h £ . (3.8) j=lii -л I Its generators (coproducts) are of the form

J0(l,2) = J0(l) + Ja(2), •4(1,2) = J±{\)qj'™ + 9-л(1)^±(2). (3.9)

The definition of q—Clebsch-Gordan coefficients (CGCs) is similar to the classical case 7=1:

m m m liija : j'm >,= £ Ui ih i\j ) I]гтг >, (3.10)

Сг(12)\]аг : jm >,= \j -f : jm >, . (3.11) The orthogonally conditions are valid for the dGCs

m m X) (Ji tJi 2 (л т,>2ш2 |/m'), = 6jji6mmi, (3.12) m,mj (3.13) '484

Also they have a symmetry property

J1+i J (iim^'jmjjm), = (-l) ~ (j2m2jimi|jm),-i. (3.14)

The following recurrent relation for CGCs

m r'^'Vbi - "4Щ1 + "4 + + i\\j-i - m, + 1] (jimi + lj2m2 - l|jm),+ 4+n,,+ + 9"" V[ii + m,][i, - m, + l][ji - m2][j2 + m2 +1](;,m, - lj2m2 + l|jm),+

2П> 2 + - ™i][j'i + + 11 + Я' l?2 - m2l(j2 + + 1] + [m + i]

a ~li + il )(ii"4J2m2|jm), = 0 (3.15) can be obtained calculating the matrix element < j\mij2m2\C2(l, 2)|j1j2j'm >, of the Casimir operator at first in direct manner and then using the Eq. (3.11).

The calculation of the matrix element < jim1j'2m2|J„(2)|jij2 : jm >, in two ways gives the following recurrent relation [8j on parameter j:

(jim,j2m2\j - Цщ*

U - m]U + m]\ji + j2 + j + !][; + j2 -j + l][j + ji - j2][j - j, + j2] *ЛJ' [2j-l][2i+l]

+ {jimij2m2\j + lm),

l{j + m + l}\j-m + l][ji + ja + j + 2][ji + j2 - j][j + ji - j2]\j - j, + j2j " (2j + l)(2j+3)

+ o.^-b--^'"-^]^'''" ml (ftw. +

- (2toi + J2 - J + lib' - Л + л]) - ^ (q-'^Ui + ~ - »»,]) } = 0 (3.16) At last the relation

< ji"»ii J2"i2l(1,2)|jij2 : jm >= y/\j ^ m][j ± m + l](jim,j2m2|jm), (3.17) follows from (3.5). Thus all the necessary for the comparison of the properties of q— Hahn polynomials and q—CGCs is prepared.

4. ij-Clebsch-Gordan coefficients and q—Hahn polynomials The relation between q—CGC and q—Hahn polynomial is similar to the classical one [3].

(hmmmtim),-, = q) (4.1)

where a = ji — n»i,N = j\ + j2 — m + 1,a = m + ji — j2, fi = m — jt -(- j2, n = j — m. Using this identification we can establish the connection between the weight function p(s) and the particular case of CGC

цтияпааь-. - (-ir^g'-P- '485

= ( 1 JJi —"Ч —)—5

bi + + + !]'[?• + к -jI' M21

[ji - "ЧШ2 - ma]![j, - j2 + Л!H, + j2 + j]\\jx + к + 3 + 1]! ' ' in agreement with the result of Ref. [7]. Also it easy verify that Eq. (2.19), i.e. the finite difference equation for g—Hahn polynomials is equivalent to the recurrent relation (3.15) for the CGCs with respect to the projection mi (or m2) of the angular momentum ji (or The substitution of the expression (2.12) for q—Hahn polynomials into Eq. (4.1) gives the q—analog of the Rach formula for su,(2) CGC [9]

(jimij2m2|jm), =

= |_iyi-mi9m,(m+l(-j(j(j+lHjl(jl'H)-ij(i3+l))

x AChjd) lii — к + j']![~ji + к+зУ-

m ! 12i + 1) b' - + ~ i]![j2 - ™г] 171 \j\ + m\)\\j2 + m2]l

where Mi ,• л j]![ii - к + j]![-ji + h + j]! Ц34*) = [ji + k+j + 1]! *

The formula (2.13) can be transformed into the explicit expression for the particular case of CGC [7]

(iiiii2m2|;'m),-. =

_ ^Ji (i -m) -10'l +» -3)(j'-jl +« -1) x

v Ml tj + - m2]!U -t- к - ill'- V W J1? - т]!Ь» + mi]!l7. ~ Л + j]!L>i + к - i]l[j. + Й + j + 1]!" 1 '

The orthonormality relation (3.12) for CGCs is equivalent to the orthogonality condition (1.7) for q—Hahn polynomials. The relation (3.13) means the closure property of q— Hahn polynomials

u The symmetry (3.14) of CGCs is a direct consequence of the properly (2.14) of 7— Halin polynomials. The recurrent relation (1.9) for the last ones takes a form of the recurrent relation (3.16) for CGCs. The finite difference derivatives (2.18) and (2.19) are equivalent to the property of CGCs

m! m \J{ji ±m,)(ji 4F + l)<7 (ii i =F lj2m2|jm), mi + y/{ji ± m2)(ja =F + l)q~ (jim,j2m2 T 1 |>m), =

= у/(j T m)(j ± m H-1 )(j'imij'2m2|jm + 1), (4.6)

following from the Eq. (3.17). Also the analog of the Regge symmetry for q—CGCs can be found for q—Hahn polynomials. However we omit this point because of shortage of place. '486

Thus the comparative analysis of the properties of q—CGCs and q—Hahn polynomials, given above, shows that each relation for q—Hahn polynomial has the corresponding partner among the properties of su,(2) CGCs and vice versa. This parallelism appears useful and fruitful for the study of both these objects. Similar program of investigations was realized also for the su,(l,l) CGCs (discrete series) and q—Hahn polynomials on the grid (2.1); for su,(2) and su,(l,l) CGCs and dual <7-Hahn polynomials on the grid i(s) = [a][s+l] and q—Racah coefficients [10] and q- Racah polynomials on the same grid the results of this study shows that a close connection exists between elements of the Wigner-Racah algebra (3j, 6j,... symbols) for su,(2) and su,(l,l) quantum groups and orothogonal polynomials on q—linear (х(ч) = 51*) and q—squared (x(s) = [s][a +1]) grids. It gives also the group theory interpretation of the properties of these orthogonal polynomials. It allows to think that the theory of orthogonal polynomials of the discrete variable can be an useful tool in the representation theory of more complicate quantum algebras (U,(n) [II] etc.) too. The authors are thankful to A. U.Klimyk, A. F. Nikiforov, S. K. Suslov and V. N. Tolstoy for fruitful discussions.

References

[1] Ya. A. Smorodinsky, S. K. Suslov Yad. Fiz. 35, 192 (1982), 36, 1066 (1982).

[2] V. G. Drinfeld Son. Math. Doki 32, 254 (1985) M. Jimbo Lett Math. Phys 10, 63 (1985). N. Yu. Reshetikhin, LOMI Preprints E-4-87, E-17-87 (1988). N. Yu. Reshetikhin, L. P- Faddeev, L. A. Takhatajan Algebra i Analis 1, 178 (1989).

[3] A. F. Nikiforov, S. K. Suslov, V. B. Uvarov "Classical orthogonal polynomials of a discrete variable", Moscow, Nauka, 1985 (in Russian). Springer series in computational physics, Springer Verlag, 1991 (English translation).

[4] Т. H. Koornwinder Proc. Koninkl. Nederl. Acad. Wet. Sev. A, 92, 97 (1989): CWI Report AM-R9013, 1990; in "Orthogonal Polynomials: Theory and Practice. P. Nevai (Ed) NATO ASI, Series C, vol. 294, pp. 257-292 Kluwer Acad. Publ. 1991. A. S. Zhedamov Mod. Phys. Lett. A7, 1589 (1992). Ya. I. Granovskii, I. M. Lutzenko, A. S. Zhedanov Ann. Phys. 217, 1 (1992). R. Fioreanini, L. Vinet Ann. Phys. 221, 53 (1993). 1.1. Kachurik, A. U. Klimyk Preprint 1TP-93-3E, Kiev 1993, Bogolyubov Institute for Theoretical Physics.

[5] R. Askey, and J. Wilson SIAM J. Math. Anal. 10, 1008 (1979), Mem. Amer. Math Sc. 54, 1 (1985).

[6] G. Gasper, M. Rahman Basic Hypergeometric Series, Cambridge Univ. Press, 1991.

[7] Yu. F. Smirnov, V. N. Tolstoy, Yu. I. Kharitonov Yad. Fiz. 53, 959 (1991).

[8] A. N. Kirillov, N. Yu. Reshetikhin LOMI Preprint E-9-88 (1988).

[9] Yu. F. Smirnov, V. N. Tolstoy, Yu. I. Kharitonov Yad. Fiz. 53, 1746 (1991).

[10] M. A. Lohe, L. C. Biedenharn Basic Hypergeometric Functions and the Borel-Weyl Con- struction for U,(3). Preprint. Department of Physics, Duke University (1993). GENERAL DEFORMATIONS OF BOSONS AND THEIR COBEERENT LEASES1 Allan I. Solomon and Roger J. McDermott Iteaft/ о{Sgrthetaaiks md Gaenputiag, ffee Opm Itovaraitji, ШШг Кеугкз, MKT 6A A, UK

Abstract Ife dktiise a gcntraj <5eftam*tk»t of lb" тамлк»! «чю»«Шк>п rekltasa wfe&b i»- dede tbe itifldard defcnnitkmi м tjtse&I cm We ёрдаЗиг {fet sumxjatwt eefaneat «tales, Md «valmte the tqaweiog («ore- reduction) рпздсШеЕ, Wo sbaw tfcat, la a>*- tridiiiiccOoB to tize canneatrosii esse of OTheresl suit», Kjateiiflj («cm-, Ftetber, ftaSla ike Wbvesttoaa! tfcawf, fof partkaiit deforwatfora U» coherest "Ut «odaba. тге&ятш ia 4oiH cojapcrjiMMs o$ lb»

1. Introduction It raioterestio g to *рос«1«Лс «щ tbe rtwoas «faith base lead to the feecrish activity is thfc дек fieJd of quanta» grouj?s over tfee !»st few уе*г». Рямп the purely аж&етв&Ы (Most of »««, it is true that ibese r&tbcr complicated tystcin» do щенкМ. лот* iBarest. Mcwrthdess, must of Uw pebiicatioo» агаян to have «massked fiocctphysicists , wfekk U a Sttfe surprising given the psu»dty of physical results of the thewy. iter»®!», if i»ae wete ш a rtfieqtiie fr«m« of mied, one could rmigbly Mtegorrae the literaiid*: м fclbwe: Exactly sohable- mtxkfa 3. AppraeJnjate modek 3- "WJurtif" physic». ТЬвд, i» Уж first ottegory we took! place tfee work «ж autistic a! mechanic*! lattice roodefcs fij, where * quwti,»«n gptap arise хяЙЧхеЯу jji the solution. fa tbe teeofid c&iegafy we ficd fO*»y «ррЯойей» ш wfckk, ihtieid of i (яои-Зяевг) system to ho dmsSbtA by « ЬкйШошш *»ЫсЬ сдажзи of as dement «f «. Ые t'lsbra. tfligetber with «Sow tw the toury of cooventHMia] (Beear L>t «j^Mafe) ootmrntatkei ruks, for exempte, bet *m reî^* se^&st the гэПег more corapRcated, more general and вда*Вяеад, reiatious of » cjuiHtum graop. i It Ь *ШЯеаЕ1 to dirtsogaafe this category, rnihas occ were « phikisophy, {roat the one s£ quantum te îve л нкяее accsmte пихМ cf a. «at! зуйедз; see, for tficaopie j3j) tfeat satfe cossJ9t«a»t mat&eroaiicaistaietutesesi*!, howavtr, it.>«»ot uertatonabb W the nuithettiatka! phîe«t (o ccnstrwt physical tjKxSfel» described Цу *uch strwdcrea, ho^c&Hy ieeding U> fcatrnr» which may be «crified eKperhaecialiy, or imbed «ited tmt ов еяреэткндаЫ groKods. й Is In (fee last spirit №«t the jsnaeet note b vmttea. We pnasst a gimemi deforntAifea» ef the ce»ofl«c4l coBinsutstioo гЙ«1йт(Ц g«at:t»{ o&aiigh to n:.riw!e both of types of bons

Jaw Ptofae»» >V A. Stmwsxtedss, ЬоЫиц Jlu»ft S-tO ialy, 1Ш. '488 deformation prevalent in the current physics literature. We then describe some of the quantum optics associated with this deformation; the coherent states and their quantum noise properties. These results are counter-intuitive in so far as they are qualitatively different from the predictions of conventional theory, which we recapitulate in the next section. The advantage of the approach adopted here is that we deal solely with states; we nowhere invoke a hamiltonian, which would involve a choice and additional ambiguity. Further, a consistent theory of the time-evolution generated by a q-deformed hamiltonian has not yet been developed, to our knowledge.

2. Conventional Theory We review some of the elementary properties of the coherent states associated with conventional (non-deformed) bosons, satisfying the canonical commutation relation

[a, a'] н aa' — a'a = 1 • [лг.в*] = «' (1) where N = a'a. Defining the vacuum state |0 > by a|0 >= 0, the (normalized) n-photon states satisfying N\n >= n|n > , are given by

|п>=Уг|0>- (2) Coherent states [4] are defined to satisfy o|A >= A|A > (3) and are given by

|A> = (ехр|А|2)-> ехр(Аа')|0 >

= (exP|Ap)-*£^}|„> (4) where |A > is normalized. Among the many interesting properties possessed by coherent states, we shall here be concerned with their "noise" reduction attributes. We define the quantum noise (ДХ)2 (variance squared) of the operator X in a given state by

(ДХ)'=<Х'> -l (5) which is a real positive number for a herrnitian operator X. The physical (hermitian) operators are the electric field component x and the magnetic field component p associated with the single mode a, given by

x = [a + a1)/^ p = (a - a*)/iV§. (6)

These satisfy [x,p] = i from which it is easy to show that

(Ax)(Ap) > \ (7) for any state. A state for which (Д=)(Др) = j is said to satisfy the Minimum Uncertainty Property (M.U.P.). It is easy to show that for the vacuum state |0 >

(Дх)о(Др)о = \ (8) '489 and for uie coherent state |A >

(Дх)л(Др)л = i (9)

So |0 > and |A > are M.U.P. states. Further, the variances (Дя)а and (Др)2 in the states |0 > and |A > are equal to i. For the number states |n > the (equal) variances are given by n -f j. The following question then naturally poses itself: Are there states for which the variance is less than the vacuum value? This would necessarily occur in one component only, due to Equation (7). Such states, which have been described theoretically [5] and produced experimentally are called squeezed states. We may define the degree of squeezing by the ratio (AI),,,./(AI)O and a considerable degree of squeezing has been obtained experimentally - states being produced which exhibit of the order of 20 % of the vacuum noise in one component (time-varying). However, in the present note wc are only concerned with number states and coherent states; and it is clear that, in the conventional (non-deformed) case at any rate, such states do not exhibit squeezing, or quantum noise reduction properties, relative to the vacuum.

3. Deformed Commutation Relations We now consider the deformation of the commutation relations of Equation (1) above given by [6] aa* - f(N)a?a = 1 (10) where a* and a are now the generalized creation and annihilation operators, and N is the corresponding number operator such that Л'|п) = n|n). The number operator N still satisfies

[Af,a<] =a' but is no longer equal to a'a. We take / to be a real function, aiiJ the vacuum |0) is still defined by a|0) = 0. We define a (normalized) one-particle state by a'|0) = |1). This formalism incorporates the deformation schemes previously encountered in the literature as special cases.

Examples:

1. f(N)=l. This is the usual commutation relation of the Heisenberg-Weyl algebra of conventional theory as given in Section (2) .

2. f(N) = q. This is the original form of the so-called q-oscillator ( or Maths boson), first suggested by Arik and Coon [7]. It has sin-o been studied in detail by several authors [8], Kuryshkin [9], Kulish and Damaskins'iy [10].

з = This fives a deformed commutation relation equivalent to that of the q- boson (Physics boson), first utilized by MacFarlane [11] and Biedenharu [12] in connection with the representation theory of quantum groups.

4- /(")=*! where F(N) is an analytic function and ДО) = 0. This form of deformed commutation relation can also be related to the extensive work of Bonatsos, Oaskaloyannis and others [13], and to the generalized oscillator formalism in recent work of Jannussis [14]. '490

Building up normalized eigenstates of the number operator N by repeated application of the generalized creation operators in (10), we obtain

W = (U)

where the function [n] ("box ") is defined recursively by

[n + l] = l+ /(«.)(») (12)

with initial condition [0] = 0.

An explicit expression for [n] is

[n] = l + /(n-l) + /(„-l)/(n-2) + /(n-l)/(n-2)/(n-3) + "-+/(n-l№-2)-/(2)/(l) (13)

n-l

" S~7wr- <14>

The functions [n] can be thought of as generalizations of the basic numbers of q-analysis [15]. They obey a highly non-linear arithmetic but for appropriate choice of the function /, they tend in some limit to the ordinary integers. We define coherent states of the deformed bosons given by Equation (10) in precisely the same way as we did for the conventional case; namely, as eigenstates of the annihilation operator. To achieve the alternative definition given by Equation (4) it is necessary to introduce a generalized exponential function, analogous to that introduced by Jackson [15] in the case of the basic numbers corresponding to the second example noted above. We do this by first

defining a generalization of Jackson's q-derivative operator 4DX .We define an operator Dz such that (15)

This generalizes the action of the ordinary derivative in the expected way, for example,

n Dx® = [nlx—(16)

The eigenfunction E(x) of DT which is equal to one for x = 0 is given by

= (17) n=0 1 J

and is well-defined provided the function / satisfies the appropriate convergence criteria. We are using the standard definition for [n]l,

[l»]l = [n][n-l]l [01 = 1 (18)

If /(n) > 1 as n —+ oo then E(x) converges for all real values of x. If /(n) < 1 as n —» oo then convergence is ensured for a certain range of x dependent on the precise nature of the function /. When / = 1 we recover thf usual exponential function. '491

Since а £(Аа')|0) = ЛД(Аа')|0), we can use E(x) to construct generalized coherent states as normalized eigenstates of the generalized annihilation operator.

J |A> = {B(|A| )}-^(AOt)|0> = . (19)

4. Noise Reduction Properties If we now consider the generalized bosonic operators given by (10), using the same definitions for the the field quadratures, x and p, as in Equation (5) we find that, just as in the conventional case, the vacuum uncertainty product (Дх)а(Д/>)о = J is a lower bound for all number states. However, unlike in the conventional case, it is not a global lower bound. Consider the quadrature values in eigenstates of the generalized annihilation operator. Then

<*)л = <А|^(а' + а)|А) = -1=(А + А) (20) and

Ил = (A| 1((а»)2 + а2 + а'а + аа»)|А) (21)

= |{(А + А)2 + 1-еу,л|АП (22) where

е/Л = 1-

Hence

(Дх)'= -{1-е/.л|АП (24)

Evaluating the variance for the other component, we find that (Д p)\ = (Д x)2 so

(Дх)л(Др)л = i{l-eW|Ap} <\ (25)

However, it can also be shown that

2 5{1-£/.л|А| } = ^|([х,р])л| (26)

(Дх)л(Др)л = ^[х,р])л| (27)

Thus we see that these generalized q-coherent states satisfy a restricted form of the Mini- mum Uncertainty Property (M.U.P.) of the conventional coherent states. Additionally we see that there is a general noise reduction in both quadratures compared to their vacuum value. In conventional coherent states there is no noise reduction relative to the vacuum value. In conventional squeezed states, there can be noise reduction in only one component. '492

5. Special Cases We can apply the preceding analysis to the q-deformed bosons recently studied in connection with quantum groups. (a) 'Physics' q-bosons First consider the q-bosons first described in references [11] and [12]. These use the deformation given in Example (3) above, and which we have termed 'physics' q -bosons. They are characterised by the commutation relation

aa* — qa^a = q~N. (28)

As noted above, this can be rewritten [6] as

aa,-/(A)e1a = 1. (29) where f(N) = In this case, for normalizable eigenstates, the function е/,л is negative and so noise reduction does not take place. This is in agreement with the findings of Katrie) and Solomon [16]. (b) 'Maths' q-bosons We now consider the q-boson described by Arik and Coon [7]. This formalism leads to the "basic" number [n] of classical q-analysis and will therefore be termed a 'malhs' q-boson. It is characterised by the deformed commutation relation

aai-qaia = 1 (30)

For q 6 (0,1), the Jackson q-exponential B,(|A|2) converges, provided

e,|A|3 = (l-g)|A|3

Given this condition on A, we have normalizable q-analogue coherent states satisfying Equa- tion (3) in which

(Д»)5 = (Ар)! = (Дг)д(ДР)л = i{l < I (31)

Hence, for this type of q-boson, we do obtain noise reduction in boih quadratures with respect to the vacuum value.

6. Acknowledgements One of us (A.I.S) would like to thank the organizers of the workshop for their kind invitation and generous hospitality.

References

[1] H. J. de Vega, Nuclear Physics В (Proc. Svppl.) 18A, 229 (1990)

[2] D. Bonatsos et al. , J.Phys.A.:Math.Gen. 24 , L403 (1991)

[3] J. Katriel and A.I. Solomon, "Non-ideal lasers and deformed states", ( to be published).

[4] R.J. Glauber Phys. Rev. 131 27766 (1963).

[5] H.P. Yuen Phys. Rev. A. 13 2226 (1976). '493

[6j A.I. Solomon and J.L. Birman Symmetries in Science VII ed. B. Gruber, (Plenum Press, N.Y., 1993).

[7] M. Arik and D.D. Coon J. Math. Phys. 17 524 (1976).

[8] A. Jannussis, L.C. Papaloucas and P.D. Siafarikas Had. J. 3 1622 (1980)

[9] V.V. Kuryshkin Ann. Fond. L. de Broglie 5 111 (1980).

[10] P.P. Kulish and E.V. Damaskinsky J. Phys. A. 23 415 (1990).

[11] A.J.MacFarlane, J.Phys.A:Math.Gen.22,4581(1989)

[12] L.C.Biedenharn, J.Phys.A:Math.Gen.22, L873 (1989)

[13] C. Daskaloyannis J. Phys. A. 25 2261 (1992).

D. Bonatsos and C. Daskaloyannis J. Phys. A. 26 1589 (1993).

[14] A. Jannussis J. Phys. A. 26 L233 (1993)

[15] F.HH. Exto. Jackson q-Hypergeometricn Mess. Math. 3Functions8 62 (1909) and. Applications, (Ellis Horwood, Chichester, 1983)

[16] J. Katriel and A.I. Solomon J. Phys. A. 24 2093 (1991). '494

METHOD OF VARIATIONAL PERTURBATION THEORY A. N. Sissakian1,1. L. Solovtsov2

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Head Post Office P.O. Box 79, Moscow 101000, Russia and O. Yu. Shevchenko3

Laboratory of Nuclear Physics, Joint Institute for Nuclear Research, Dubna, Head Post Office P.O. Box 79, Moscow 101000, Russia

Approximation of a quantity under consideration by a finite number of terms of a certain series is a standard computational procedure in many problems of physics. In quantum field theory this is conventionally an expansion into a perturbative series. This approach combined with the renormalization procedure is now a basic method for computations. As is well-known, perturbative series for many interesting models including realistic models are not convergent. Nevertheless, at small values of the coupling constant these series may be considered as asymptotic series and could provide a useful information. However, even in the theories with a small coupling constant, for instance, in quantum electrodynamics there exist problems which cannot be solved by perturbative methods. Also, a lot of problems of quantum chromodynamics require nonperturbative approaches. At present, a central problem of quantum field theory is to go beyond the scope of perturbation theory. A. great amount of studies is devoted to the development of nor.perturbative methods. Among them is the summation of a perturbative series (see reviews [1,2] and monograph [3]). The main difficulty is that the procedure of summation of asymptotic series is not unique, which is generally a functional arbitrariness, and the correct formulation of a problem of summation is ensured by further information on the sum of a series [4]. At present information of that kind is known only for the simplest field-theoretical models [5]. In refs.- [6-10] approaches are proposed which are not directly based on the perturbalive series. Thus, the method of Gaussian effective potential has recently become rather popular [11-14]. Many of nonperturbative approaches make use of a variational procedure for finding the leading contribution. However, in this case there is no always an algorithm of calculating corrections to the value found by a variational procedure, and this makes difficult to answer the question how adequate is the so-called main contribution to the object under investigation and what is the range of applicability of the obtained estimations. However, even if the algorithm of calculating corrections, i.e. terms of a certain approximating series, exists, it is not still sufficient. Here i f fundamental importance are the properties of convergence of a series. Indeed, unlike the case when even a divergent perturbative series in the weak coupling constant approximates a given object as an asymptotic series, the approximating series in the absence of a small parameter should obey more strict requirements. Reliable information in this case may be obtained only on the basis of convergent series. It is more reliable to deal not wiih an arbitrary convergent series but just with the Leibniz series (an alternating series with terms decreasing in absolute value). Then it will become possible to compute upper and lower estimates for a given quantity on the basis of first terms of the series. In case of additional free parameters influencing the terms of the series, these estimates may be made as close as possible to each other.

'E-mail address: sisakianSesd. jinr.dubna.su 5E-mail address: Bolovteotatheor. jinrc.dubna.su 3E-mail address: ahevchenko® mainl.jinr.dubna.8u '495

In this paper, we consider a method of variational perturbation theory (VPT) [15-17]. Despite the word "perturbation" being present in the name of the approach, the VPT method does riot use any small parameter. The additions in the VPT method are calculable because this method employs only calculable Gaussian functional quadratures. Besides, a VPT series can be written so that its terms are defined by the usuai Feynman diagrams. In this case, the VPT series will surely differ in structure from the conventional perturbation theory, and diagrams will contain a modified propagator. Here we will apply the VPT method to Green functions of the i^4-model in the Euclidean rf-dimensional space. To this end we write the 2f-point function in the form

G2„ = J iWJexp (-.%]), (1) where

and the functional of action looks as follows: 2

S[y>] = 50[И + + AS,M,

SoM = \ j dx(d

We shall construct a VPT series by using the following Gaussian functional quadratures

J /)<рехр{ - [i < tpK

+ < ] ] =

The VPT series for the Green functions (1) is constructed in the following way: oo

G2„ = (4) n=0

G2„.n = J - 5'М)"exp(-SoM - ^M - • (5)

The variational functional will be taken to be dependent on certain parameters, but the total sum (4^surely will not depend on these parameters. Their choice can be such as to provide the expansion (4) being optimal. The functional Sjv] should be defined so that the terms of the VPT series (4) be calculable, i.e. the form of £[¥>] should be such that the functional integral in (5) can be reduced to the Gaussian quadratures (3). This requirement does not mean that the functional must be quadratic in fields. We can pass to the Gaussian functional integral by using the Fourier transformation. We choose here, for example, the sum of harmonic and anharmonic functionals being SM.i-e.: = + (6) where M and в are the certain parameters through which the VPT scries is optimised. We obtain n n к ^

x вг1 (mj - тгУ'к~'{--^У+1~к9\кЛх1), (7) '496

where

2 S^(X ) = р{-[ЗД+у£2М]}. (8)

The latter expression can be written as follows

-1/2 -SP + x2 ) = del I 1 2 (9) "-a •>• m

where 3^!п(х2) are calculated on the basis of diagrams of the k — th order of conventional perturbation theory with the propagator Д(p,X2) — (p2 + X*)-1. A new mass parameter xJ is depended on и and variational parameters M2 and 9. Thus, the N — th order of the VPT expansion (4) can be constructed with the same diagrams as the conventional perturbation N — th order is made up. Let us consider a case of the quantum-mechanical anharmonic oscillator ( AO ) as an example of exploiting the VPT method. The AO from a point of view of the path integral 4 formalism is a one-dimensional

For calculating Green function G4 we will use the two-parameters anhannonical VPT functional

2 5M = [fl50M + ^5.-M| . (11)

The application of the asymptotic optimization that requires the contribution of the remote terms in the VPT series to be minimal allows one to find the relation between the parameter 0 and x '• 160x3 = 9. The remaining variational parameter is fixed on the basis of a finite number of VPT expansion terms. For the ground state energy in the first order of VPT we get the strong coupling expansion

= A,/3[0.663 + 0.1407W5 - 0.0085u>4 + •••], (12)

where the dimensionless parameter u2 = mJA~3'3. We have to compare the oftained result with the exact value [18]

E'"ct = Al/3[0.668 + 0.1437ш2 - 0,0088w" + •••]. (13)

We can also calculate the mass parameter ft1 connected with the two-point Green function: 1 = Gi[p = 0). In the strong coupling limit we obtain p3 = 3.078A2'3, whereas the exact 2 3 value is = З.ООЭА ' . We can estimate the energy of the first excited level Ex. Defining the energy shift pi = Ex — E0 and using the spectral representation for the propagator we arrive at the following estimate for Ц\ :/ii < ц[+\ where

+) /I = 2G3(a = 0)/GJ(p = 0). (14)

By analogy with the sum rules, we may expect a sufficietly rapid saturation of the spectral representation, which brings /ii and closer to each other. In the first order of the one- parameter VPT in the strong coupling limit we get p(,+1 = 1.763A1'3 , whereas exact value is fi'zac' = 1.726A1'3 [18]. The effective potential and corresponded numerical characteristics for AO was computed in [16]. m

Sow Reminder * tnassien i^j thexy in Itw fwir-rtiaiensioOBi Eedideati space. Йог tin? gemmating functional of the Gzcea fuwtions W\J\ with the variational atMiiifin tabes in Lite fwro § 1obtain tfc» following VPT «ri«

W\J\ - (IS) * ««6

H«e wc useef the somUnt С, ~ 4!/{16*)1 bom the SoboJev inequality (sw, for inntanre, refe. [3,19J): (16) and set A = In the given case we m interested »a the problem nf tonveqgeiwc of Ibe series. Asymptotic estimate of гетав1сгас|ншмон lenne can be made by (he ГшнЛпмв! Siuldfcpoiul iuet-Ыи! (l-3Jfflj. f tbe To this «id, we represent Webt l

mm = (-*)'j - ntfi&> + <-»«>}, (151

where

Щ<р\ = f^-tsiH- П8)

ТЬс main wntribaLton lo the integral (if) the le&dSttg ofdet in the laigj; sinlrflc-fKHat parallel»* я CPBK3 from the ftinrtkms obeying the eijualiun ~

= {is)

an leaving the action fttnclienai to be finite. Their explicit form w as iblfnu-s

АгЫи^ purajnetfls p w {20) reflect the iranstatnKwl ш^вЫб^штпсеоС^Ье theny under cooa&niticxf. From (20) «wi (21) it ЗЫЬеш thai «* =s- gi {32*-*}* * ss & result we obtain

ИУАЦ-МГ ~ (V)*- \fft+ n>/* < (32)

Prora ibis expression it is dear that nnspcelive of Utc та|ыя of tfae coupling constant g, the VPT series {15} аЬво|и1к1у tanvergei when 1 > t/2 and wbea { > 1, « follows from the Sobolev inequality {16), that seri« i' of poiitivT sign. In the intend 1/2 < I < I at large n Ibe series (15) is the Inabetx series. Km again the «aits i = i corrtMponds both to the diangt of the regime of the VPT series and to its asymnioijc optimbsttiira. Note is to be made tfwl (be exprcssioi! (22) determine* oaiy the lading twftribotion to the functioeai dependence of so the large parameter n. tn particular, in (22) we da not reproduce a certain multiplier that вррешз in the mart, to hading order in к. However, the properties of amwrgenee of the series еяв be quite wcil anoly?«xi in the leading orrtrar in n. '498

Let us consider the VPT method in view of point of connection between the VPT and the method of Gaussian effective potential ( GEP ) [11-14]. In the following, we shall have in mind the dimensional regularization setting n — d — 2e, where d is an integer number. We separate the classical contribution in the generating functional of Green functions W[J] by writing

where

and the function ipc satisfies a classical equation of motion SS/Spc = — J- In the standard classical approximation one would retain only the addends quadric in fields in expression (25) for the quantity P[p]. In this case the functional integral for D[J\ becomes Gaussian and for W[J] the ordinary one-loop representation arises. Let us now calculate the quantity £>[J] by using the anharmonic variation of the action functional. We choose the VPT functional in the form 5jv>] = Л3[у>], where

The space volume Л appears here because Vej/ is derived from the effective action by using the constant-field configurations. Thus, the parameter x, optimizing the VPT series, does not depend on fi. Any power of Jt'lv] in the VPT expansion can be obtained by the corresponding number of differentiations of the expression exp(-iefl'[vs]) with respect to e with putting e = 1 at the end. As to the addend Я3[у>] in the exponential, giving rise to a nongaussian form of the functional integral, the problem is easily to solve by implementing the Fourier transformation, due to which only the first power of Я[<р] emerges in the exponential. As a result, the VPT series takes the form

D[J)

det -oo X (26)

where

Д(р) = (p2 - M2 + tO)-1,

w и * 'Ш'

M2 = m2 + 12A

The integral over v in (26) contains the large parameter ft and, hence, can be evaluated by using the method of a stationary phase. Then, the effective potential in the first nontrivial '499

VPT-order looks as

VCJ, = VI + VO + VI,

V0 = (27)

V, ^Д? + ЗАД?, where До is the Euclidean propagator &Eud{x = 0, M2) written with the help of the dimensional 2 regularization. Here M is the massive parameter taken at e = 1 and v = v0, where v0 is the stationary phase point in the integral (26). The corresponding equation reads

J J 2 2 M" = m + 12Ay> + Х Д0 (M ). (28)

One can apply now the following optimization versions (see [15,16]): (i) The requirement 2 min |Vi| (here there exists the solution to the equation Vi = 0); (ii) dVejj/dx = 0. It is easy to find out that these different versions give rise to the same optimal value of the parameter X2: X2 = 12A. As a result, the effective potential (27) with the condition (28) yields the GEP in n-dimensional space [21]. It is interesting, that the first order of the VPT for D[J] give us GEP by various ways of constructing the variational procedure. However, despite the same result in the first order, other properties of the series are different. In the case of the harmonic variational procedure the VPT series is the asymptotic series. In the case of the anharmonic VPT procedure we can obtain the convergence series. It is important in the point of view of the stability of the results obtained as the leading contribution. When the variational addition is harmonic, the VPT series is asymptotic and its higher- order terms behave like the terms of standard perturbation theory. Nevertheless, the harmonic variational addition produces a certain stabilization of the results for the further radiative corrections. In the regions where the partial sums of conventional perturbation theory suffer of oscillations specific for asymptotic series, the VPT series gives a stable result. However, the harmonic way of varying the action though rather making pass to large coupling constants does not lead essentially off the weak-coupling region. This can be achieved by passing to the variational addition of the anharmonic type, which can be explained as follows: For higher- order terms of the VPT series the major contribution to the functional integral comes from the field configurations that are proportional to the positive degree of the large saddle-point

parameter. Therefore, the effective interaction A54[v?] — S[v] is dominated by the conventional term AStH that, as in perturbation theory, leads to an asymptotic series. A different picture arises when the action is varied with the help of an anharmonic functional. Here the degrees of fields in A5<[^] and 5[ys] are the same and the variational addition greatly influences the asymptotic behavior of higher-order terms of the VPT series. In this paper we have shown that there exists a finite region of values of the variational parameters where the VPT series converges for all positive coupling constants. We would like to mention one more interesting possibility of the VPT method, namely, construction of the Leibniz series as the VPT series. When a searched quantity is represented by this series, even and odd partial sums of the series define the estimates of upper and lower bouuds. In other words, for a case under consideration we can obtain nonperturbative estimates of the upper and lower bounds; and which is more, when we have free parameters defining the VPT series terms, we can govern the accuracy of estimates. This investigation was performed with a financial support of Russian Found of Fundamental Researches (93-02-3754) Acknowledgement The authors are grateful to V.G.Kadyshevsky, D.I.Kazakov, L.D.Korsun and G.V.Efimov for interest in the work and useful comments. '500

References

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[3] J.Zinn-Justin: Quantum Field Theoiy and Critical Phenomena. Clarendon Press. Oxford, 1989.

[4] M.Reed, B.Simon: Methods of modern mathematical physics. Vol. 4 Analysis of operators. New York, San Francisco, London: Academic Press, 1978.

[5] J.Climm, A.Jaffe: Quantum Physics. A Functional Integral Point of View. New York, Heidelberg, Berlin: Springer 1981.

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[10] L.Polley, U.Ritschel, Phys.Lett. В 221 (1989) 44; M.H.Thoma, Z.Phys. С 44 (1991) 343; J.Soto, Nucl.Phys В 316 (1989) 141.

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INHOMOGENEOUS QUANTUM GROUPS IN PHYSICS E. Sorace

Dipartimcnto di Fiaica, Universita di Firenze and INFN-Firenze

Abstract The symmetry under a quantum group is completely realized from a physical system when the algebraic sector describes a single excitation and the coalgebra describes 2 excitations sectors. The usual phonons of the classical harmonic linear chain are then exactly described from the invariance under the 1 + 1D quantum Poincare. The Dirac equation associated to a 3 + ID quantum Poincare is presented and discussed together with its invariance group. Even the magnons of the XXZ Ileisenbcrg model share exactly a symmetry under a quantum group: the ID quantum Galilei. From of its irreducible representations one gets an explicit formula for the n-magnons bound states energy which exactly solves the corresponding Bethe ansatz equations.

1. Introduction. In his talk prof. Fairlie has proposed to apply the fourtienth century canonical sentence against metaphysics from William of Ockham to the use of Quantum Groups in physics. About this it seems reasonable that we consider well done the introduction of a new mathematical structure in physics when it happens that all its original mathematical -content plays a role in actual physical applications. If we apply this criterion to quantum groups it implies that we must find a physical meaning to all the Hopf algebra structure i.e. not only to the algebraic sector but to the coalgebraic one too. FVom the experience with the symmetries generated by the Lie groups, we argue that a Q.G. applies to a quantum physical system iff the "algebra" describes the elementary one object sector and the "coalgebra" the 2 objects sector, extented by coassociativity to 3,..many objects sectors. In the following I will show you some examples, done by a team of Firenze, which fulfill all these requirements. First of all the most ancient, many body, fundamental physical system, i.e. the phonons of Einstein, Debye, Born,... on the Id harmonic chain (just the one of Isaac Newton, Johan Bernoulli,...), which realizes the symmetry under an S,( 1,1) Hopf algebra [1] and after a model yet waiting for a conclusive phenomenological analysis, i.e. the deformed Dirac equation [2] covariantly associated to a quantum Poincare [3,4]. As a last example I will treat by means of the quanwm Galilei the well known usual magnons of the XXZ (but also XXX) Heisenberg magnetic chain [5,6] , whose 1931 [7] Bethe complete quantized solution for the isotropic case has been extended by the same "ansatz" to the non isotropic one [8]: we obtain by analyzing the irreducible representations of the quantum "symmetry group" the exact closed formula for the energy of the n-magnons bound state. This formula, at your knowledge, didn't exist before in literature: it solve the Bethe ansatz constraints of the bound states and in an appropriate limit gives the n-bodies bound state energy of the ID boson gas. I recall that the inhomogeneous Q.G. have been mainly obtained by a contraction technique

[9,10], although a way by Lie-Poisson straightforward quantization to get Funq(G) together with duality calculations to obtain U,(g) can be also devised [11,12]. As you know the contraction of a Lie algebra g is realized by a linear transformation of the basis, depending on a parameter £, singular for e —» 0. Taking the dominant terms a new algebra can be obtained. The same contraction can be extended to the Hopf algebra associated to the given semisimple Lie algebra. The limiting procedure must be done on all the Hopf relations: if the results converge we get a constructive definition of the contracted quantum group. The crucial point '502 is that, to avoid exponential singularities or trivial results in the elements kj = e'"i and к'1, with Hj in Cartan subalgebra and q = e', г too must be scaled, and in a reciprocal way to Hj. So, if i is dimensional, the new parameter w derived by contraction from z acquires dimensions: in the kinematical groups it appears as a fundamental length or inverse mass. Moreover you can deal now with genuinous finite difference operators realizations of the quantum algebra.

2. The £(1,1)/ Symmetry of the Phonons. £(2) can be defined as the group of matrices

r=(oi) ("> with v unitary and n complex. It is then easy to verify that the group combination laws of the parameters (and those of the inverse)

vi пЛ fvj n2\ _ /v,v} «1П2 + ПЛ ( 0 1 ^ V° «2/ \ 0 1 J

(extended to 0,n) conserve the С'-algebra defining relations [13]:

tm = qvn; nv = qvn\ vn=qvn; vn = qnv; nfi = qnn

but also the following ones [14]:

vn — nv = ш(иа — v); [n,n] = —ш(п + й) (with their conjugates). The first relations are connected via duality to the quantum algebra £,(2) given in [15],in which only the coproduct differs from the classic structure, while the second ones are paired to the quantized £(2)/ in which both algebra and coalgebra changes [9], where 1 means lattice. The further quantization of the 2-dimensional Euclidean algebra comes from the rescaling:

% XJ,Pv,Pz,w) = diag(l,£,£,£"'} {J\,Jt,Ji,z) ,

The quantum algebra resulting in the limit e 0 satisfies the relations

IP*,P,] = 0, (J,Pr] = iP„ [J,Pv] = -^ah(wPx) . (2.1)

Moreover

wP 2 шР 2 APX = 1 ® Px + Px ® 1 , APy = t~ 'l + е -/ , AJ = e-»'P*l2®J + J®ewP*/2 ,

S(PX) = -Px , S(PV) = -Pt , S(J) = -J+~wPy,

Let us introduce the immginary 1 = iw and then write the l-Poincare 1 + ID in terms of the

generators P0, J, P,in place of tPv, iJ, iPx, and the consequent changes. So the algebra and the non trivial coproduct actions are

[P, Po\ = 0; [J, P] = iP0 [/, P0] = i/lsin(lP)-,

A P„ =e-^/2®P0 + P0®e^/2

AJ =e-^/2®J + J®e^/2 '503

Wc see that P is primitive and it is defined modulus an integer multiple of 2тг//, it is then useful to introduce к = e.xp(ilP)- The Casimir operator is given by:

, , , C = PD -(2/iy «n (iP/2). (2.4)

Л differential realization of the algebraic relations (2.1) yields

P0 = (i/v) 0,, k = exp(/9x), J = i(r/v)d, - (t>*/a) sin(~ildt). (2.5)

By putting С = m2v2, the expression (2.4) gives rise to the following I'DE for a field z(x,l) :

2 J (9? + (2v/1) sin (-i!0I/2) + mV) z(x, t) = 0 . (2.6)

For v = u>/l, с — mv2 this equation describes exhaustively an harmonic chain of equal masses, of pulsation ш and a force —c2z(x, t). Indeed eq.(2.6) is a finite difference equation in x

((Of + mV) 2(1,1) = v2l~2 (z(x + /,<) + z(x -1,1)- 2z(x,t)))

in which one can imbed the ODE system:

5j(0 = J («i-i(0 + WO - 2^(0) - Л,-(0 , (2.7)

which describes the dynamics of such a chain where Zj(t) is the displacement of the j'-th point (j = 0,1,..., N) and initial conditions z,(0), ij-(0) must be specified. Putting L = N1, the Born-Von Karrnan b.c. are z(0, t) = z(L, I), the Cauchy data for (2.5)

consist in giving smooth functions г(х,0) and d,z(x,0). When z(jl,0) = г;(0), Stz(jl,0) = ij(0) for all j , we gel the solutions of (2.5) as Zj(t) = z(jl, i), no matter of the points x ф jl. The continuum limit / —» 0 of this system generates the (1 + 1 )D Klein-Gordon equation with velocity и as a realization of the Casimir of the E( 1,1) algebra. Let us now put m = 0, the case of the usual physical phonons. A realization of the E( 1, I); in terms of the diagonal P and the position operator X — id/dp is given by:

P0 = (2/1) sin(/p/2), 0 < p < 2vr// ,

J = (l/i) {sin(/p/2),X}+, (2.8)

P = p.

The chosen domain of p allows the reduction to the first Brillouin zone, where Pa has positive

values (let us remark that the energy is t>P0). The expression for the boost J can be inverted in X : A' = (1/2) {/o~\ The time derivative of X is given by X = iv (/Jo, ЛГ| and the commutator gives the well known group velocity of the phonons X = u, = v cos(IP/2) . All the properties of the system of one single phonon are thus exactly recovered from £7(1, l)j. ,pin 2 For the many excitations sector the coproducts induce the global variables PD = e~' / x ! 2 2 2 y.P™ + P^e ^' , J = e-"*"/ J' ' + JW е^П and k = Thus from the last equation we have P = P'1' + P'2' + 2irn/f, where n is an arbitrary integer number used to keep P in the fixed Brillouin zone: the umklapp process is a consequence of the quantum symmetry. To discuss the composition of the energies we must have two phonons with the same direction

of propagation, velocity parameters v, and v3 and dispersion relations:

W m П, = (2vi/l)MiP /i\ ns = {2v2/l)s\n{lP !2) . '504

which can be produced in a 3D cristal owing to the possible different polarizations. The P0 are (1) 2 then given from P0 = fii/ui and PQ ' = HJ/UJ. So that from the coproduct rule for Po we get

2 1 P0 = (2//)sin(/Pl )/2) + (2//) ain(/P<4/2) с '"*"'*

= (2//)sin(f(P"» +. (2'9)

The exchange symmetry together with the reality of the result is straightforward. Therefore in the physical processes where the energy is conserved two phonons can fuse if and only if a global velocity v exists such that, if fi = |P„|v = (2v/i)sin(JP/2), then fi = + П2. We have thus shown that the coalgebra rules generate the fundamental nonlinear fusion law of the phonon physics. About the boosts it must be said that their composition produces a non symmetric expression: one can symmetrize getting at the same time a real form which still closes the algebra and gives rise to a symmetric position whose time derivative is again the correct group velocity. The procedure for any number of phonons is then given from the coassociativity rules.

3. The Dirac equation associated to the к deformed Poincare group. The first indisputable quantum 4 D Poincare group has been obtained by Lukierski et al. [3,4] by contracting the a priori known De Sitter 50,(3,2) quantum Hopf algebra, while its dual has been only recently conjectured [16]. The contracted algebra is a real form under an involution having the standard properties at the level of algebra and coalgebra. The simplest [4,17] final structure of the algebra, where to maintain the space isotropy the deformation has been chosen directed along the time axis, reads:

[д,РЛ = 0 , [Я, Яо] = о,

[Mit Р,] = ic,jk Рк , [М„/у = 0,

[/,;, Р0] = iP< , [Li,Pj]= iSij ksinh{PBlk), (3.1)

[Mi,Mj] = t£i;ifc Mk , [M^Ljl = itjjk Lk

2 [Lu Lj] = -ieijk(Mkcosh(P0/k) - U(4к ) Pk P, M,).

Where Рд = {P0,P,} are the deformed energy and momenta, the Mi are the space rotation generators (they close an undeformed Hopf subalgebra), the Li are the deformed boost generators and the deformation parameter к has the dimension of the inverse of a length. The coalgebra and the antipode result:

&Mi = Mi<8>I+I®Mi , AP0 = Р0®/ + /®Яо,

APi = Pi ® exp(^) + exp(-^) ® Pit p p AL( = Li® exp(^) + exp(-jjr) 0 L{ (3.2) (p' ® ^exP(§) + exp(-§M ® ъ)

S(Pll) = -P„ , S(Mi) = -Mi

S(Li) = -Li + ^Pi . '505

The deformed Casimir operators are:

C, = (tk sinhtg)) - Pi Pi

(3 3) ) P P P \ c2 = fcosh(^)-^-j w;-mwi where [4,17] Wa - Pi Mi and IV, = к sinh(P0/k) Mt + cijk P,Lk. It can be observed the identity of C\ with that used for the phonons and that of E(2,1), [9]. It is then obvious to search for a fc-dcformation of the Dirac equation: we have const ructed the fc-Dirac operator invariant under the spinorial representation of the k-Poincare, it differs from that written in [4] which result? invariant on the mass shell: both the operators in the classical limit к —> oo produce the usual Dirac equation. Anyway this same invariant i-Dirac operator has been recovered [18] in the framework of "finite fc-Poincare tranformations" for fields of any spin. To start let us notice that a representation of the Lie algebra 0(3,1) is a P„ = 0 representation of the fc-Poincarc llopf algebra (sec also [19]). So the 4—dimensional representation of Of3,1) built with the usual 7 Dirac matrices is a PM = 0 representation of the fc-Poinearc. Now let us recall that "classically" the spinorial s = 1 /2 representation of the Poincare group is obtained by summing up an orbital (spinless) representation and the P^ = 0 one built with the 7 of Dirac. Thus to construct its quantum counterpart we must combine the quantum spinless and the quantum zero momentum representations by substituting the sum with the coproduct rules. By denoting in calligraphic the global generators, in capital italic the generators of the orbital "spinlcss" part (first tensor factor) and in small italic the "spinorial" operators, we get:

V, = Pi V0= P0,

Mi = Mi -I- in,. p £3= X.3 + exp(-2j) CM)

P 1 C+ = /,+ + exp( —2f) <4 + 2f< ГПз p- •

P j C-- L-+ expf-jl) 1- + 2£ ,Пз ' where m, — i/4 с-yyyk and - -г/2 707, , (/± = lt ±il?) closn an usual 50(3,1) algebra thus furnishing the Pfl = 0 representation of the к- Poincare. The final form of the global boost generators is:

P 1 = Li + exp(-ф li - Ujk mi pk • (3.5)

These Ci together with the M, and the P, , Va close the algebra (3.1); in the limit к oo we recover the usual "spinorial" representation of the Poincare algebra. A reali?ation with differential operators on diagonal = p,, can be given for the spinless P0, P,\ Mi and Li and we define

Po =Po , P, = Pi ,

м 9 M, - P, (3.6) ( д 9 \ '506

They close the algebra (3.1) with W0 = = 0 and therefore Ci = 0. The Casimir Ci written in terms of the global generators does not change from Си and the second Casimir C% is obtained by substituting for Wo and Wj in Ci the operators

Wo = Vi Mi = гт Pi

Wi = к s\nh{Va/k)M, + cijkVj ,

Ck = m, к sinh(P0/fc) - tyl ljPk where J, = exp(-P0/2fc) U - l/(2fc) eijk mj Pk . Using the 7-matrices algebra one gets:

in agreement with the classical limit к —» oo. One can now write a fc-deformed Dirac operator invariant under the global generators •Po, P{, Mi, Ci. From the commutation relations (3.1) it is not difficult to show that the operator: P i V=- exp(-^) nPi + itok sinh(P„/A) - ypo Pi Pi ,

verifies [Z>, £,] s= 0, and trivially [25, M,} = [V, Vt) = \D, V0] = 0. In the к -» oo limit we recover the Dirac operator. The square of V results to be

3 then the se'cond invariant takes the form: Cj = — т 231 . The operator T> is therefore the square-root of the second Casimir Ci and not of Cj: however we remark that a given value for Cj induces one and only one positive value for Ci- The ft-Dirac equation is then: V ф = m (l + V ,

where m2 = 4k2 sinh2(^) — PiPi •

Using the expression )or the Casimir Ct we can express the operator D in the following form

v = ехр(-Й) (" 7i Pi +'10 2k sinh(fi)) +ilo%- (3-7)

Thus for the massless case the A-Dirac equation coincides with the one proposed in [4]. Assuming to be on "mass shell", C\ = m2, the fc-Dirac wave equation is linear with respect to the space derivatives while it is a finite difference equation in time with the fundamental shift given by i/(2k). An evidence of the deformed Dirac equation will be very important, owing to the fact that in its derivation both the sectors of the Hopf algebra are involved. A first order analysis of the fc-Dirac equation coupled to the Coulomb field has been done in [18,20] to search a contribution to the Lamb shift from the deformation . The result has been negative, both in the nonrelativistic approximation [18] and in the relativistic regime [20]. In this last case to the exactly zero first-order contribute it follows a singular second order perturbative effect, from '507 which it is heuristically possible by giving to the scale it-1 the status of cut-off to produce an estimate of the order of the fermi or less. I send to the previous talk of prof.J. Lukierski for a discussion about the general implications of the A-Poincare.

4. Magnons and deformed Galilei Symmetry. We want now to show how a quantum symmetry works in the non relativistic analogous of the previously discussed system. Let us then consider the XXZ Heisenberg magnetic model whose Hamiltonian with periodic conditions Sjv+i = Si, (S = (Sr, S", S*), S2 = s(s + 1)), is given by [21]: N

W = a)(SfSf+i + stsUi) + s;s;+1); (4.1) t—1 a 6 (0,1): for a = 0 one gets the XXX and for a = 1 the ID static Ising model. Let | 0 ) be the state with all the spin directed downwards. This is an eigenstate of Ti with an energy 2 given by Eq = 2JNs . It is well known that in terms of the states with one spin deviate, ф = 2 Si 0 ), the eigenvalue equation for H translates into the algebraic system i

2je((l - «)(/,_, + /<+,) - 2f.) =(E- £„)/; , (4.2)) which, diagonalized by means o', fj = e'k' leads to the disppy"' л relation of the wave vector к of one magnon:

E-Ea = -4 - (1 - a) cos it) . (4.3) Even in this case, the solutions of the chain motion are obtained by evaluating the solutions of differential equation (actually a finite difference equation)

(-4Ja(l - (1 - a) cos(~iadt)))f(x) = {E - Ea)f{x) (4.4) at integer multiples of the lattice spacing a. In the continuum limit a —» 0, the 10 free 2 -1 Schrodinger equation with the effective mass m = lima_0(—47s(l — a)a ) is obtained, which therefore shares the symmetry of the ID Galilei group. It has then been natural to search for a deformation of the extended Galilei algebra, whose Casimir generates the discretized Schrodinger equation of the magnons. This Quantum Galilei group, we called Г(1),, has been found [5]: it is generated by four elements В, M, P, T, with commutation relations [В, Г] = iM , [fl,T] = {if a) sin(aP) , [P,T] = 0, [M,.] = 0. The coproducts and the antipodes read

А В = e~iaP ® В + В ® е'°р, AM = e~iaP ® M + M ® eilP,

ДР = 1®Р + Р®1, ДТ = 1®Т + Т&1,

7(B) = -B-aM, 7(M) = -M, 7(Я) = -Р , у(Т) =-Т.

Its Casimir operator is then C = MT- (1/a3) (1 - co-i(aP)) (4.5) The algebra admits the following differential realization :

В ~ mx, M = m, P = —idc,

J AT = (mo )"'(l — cos(—ia3x)) + c/m, '508 where с is the level value of the Casimir. By putting (mo2)-1 = —4 Js(l —a) and c/m = —4Jsa, one sees that the expression of T coincides with the left hand side of (4.5). Moreover the algebra is invariant under P »-» P + (2ir/a)n, the position operator can be defined as X = (\/M) В and eventually the group velocity and all the properties of the one magnon states are obtained from the Г,(1) symmetry. The two-magnon states ф - J2i>j fij St^jl 0 )' w't,h f'i = Ли mllst satisfy the following algebraic system in the coefficients fij:

(E-EO + BJS) f^ - 2s( 1 - A) (J„j fin + JIN f„j)

(<, 6) = -Jij((l - a)(fu + fjj) - - fr) , ' where the bonds Jy are equal to J when the label (ij) are nearest neighbor pairs and vanish otherwise. For s = 1/2 the amplitudes fa have no phvsi' jl meaning, indeed in this case they cancel between right and left members so that they are completely free. It is this situation which can be treated by means of the "Bethe Ansatz" and which the Quantum symmetry applies to. The Bethe procedure [7] for solving the system (4.6) in the case of spin 1 /2 imposes the separate vanishing of left and right sides respectively. The general solution of the equation generated from the left member can be written as fij = c|c,a(p,'"fwj)+c;e'°); the vanishing of the right member constrains c, written as с'*'2 to satisfy the Bethe boundary conditions [8]:

cot^ = sinn (pi — p2) /2 v 2 (1 - a)cosa(pi + p2)/2 — coso(pi - p2)/2 '

To these constraints one has to add those for pi and p2 arising from the periodicity conditions /«una = /njn,+jv . Thus the Bethe ansatz gives us correctly all the solutions of the model, subdivided in a set of the continuum, for which the energies are the sums of the energy of the constituents and the momenta are real number, and one solution - the so called "bound state" - of complex conjugate momenta with asymptotic conditions (N very great) . In this last case Pi "h P2 Pi — P2 for и = —2—1tv — —2— from eq.(4.7) one gets e~"v = (1 -a)cosau, which inserted in the expression of the global energy gives Еь = —2У(1 —(1 — ») cos3 au), which in the ferromagnetic case (J < 0) is always lower than the continuum energy. We show now that all the results about the bound states for any number of magnons can be easily recovered in the framework of the Г(1), symmetry. From ДP one gets straightforward the conservation of the momenta, with Umklapp, and from ДT we recover the two magnon energy T\2 : 1\2 =r,+r2

s _1 = (JWja ) (l - cos(aPi)) + (M2aV( 1 - cos(aP2)) + C./M, + C2/M2 .

2 For a fixed value of the spin s, Afi = M2 = M ~ (—47s(l — a)a )'' and C2/M2 ~ C,/M, — -4 Jsa. In the differential realization Pi = -idand P2 = —idXJ the action of Ti2 is

Tl2f(xux2) = -8Jsf(xux2)-2Js{l -a)(-f(x,,x2 + a)

+f{xi,ха - o) + f(xi + a,xj) + /(i, - а,1з)) its eigenvalue equation is thus equivalent to the vanishing of the first member in (4.6) . The dispersion relations and the energy of the continuum reads

E - Eq = -iJs(2 - (1 - ot)cos(op[) - (1 - a) cos(ap2)) . '509

The two magnon state eigenfunction and the explicit spectra can be obtained by imposing the periodicity and the Bethe boundary conditions. Let UB now consider the law for composed mass: iaPl Ml2 = M,e + M2c~ • and then write the energy for the two magnon system in the form

2 1 TX2 = (а Ми)" (l - cos(aP,2)) + Uu , (4.9)

where (Af„ - ih - M7)' Ъ-Ъ + и,-^»"^ • (4.10) 2 a2 ii,7 Mi Mi It is then possible to recover the coproduct of the Caiimir

СЦ = M\2 Uii .

The operators C\2 and M\2 label the irreducible representations of the composite system and therefore must assume the same value over each state of a given irreducible representation.

Moreover the relation between C]2 and Ml2 is one to one only for dCi2/dM12 = 0, where it is

2 M12 = M1+M2 + a M1M2(Ux + U2). {4.11)

For two s = 1/2 magnons, in the above notations, eq. (4.11) reads

M\2 = 2M / (1 — a). (4.12)

Defining 2iv = P\ — P2 the last condition yields just the Bethe Ansatz for bound states in the limit of an infinite number of sites

е-"" = (1 - a) cos(aP/2).

By substituting equation (4.12) into (4.9) and (4.10) we get the quoted form of the energy of bound states: T„ = -2j(l - (1 - A)2 cos2(aP/2)) . We now give the generalization to the n-magnon case by exploiting the coassociativity. The total energy obtained from the quantum group can be written as n 1 Ti2...n = £ 3* = (а^-.пГ (l - cos(aPls...„)) + U12...„ (4.13) it=i

where P]2...„ =- £LI Pk and

1 2 it ^ - Mi2..,{k-\) - Mk) l/i2...n = > ^ Uk ~ r-j > r; J7 T7 • (4.14) M tei k^i i2...kM12...lk-\)Mk

ia{p + + M,...k = c * " 'V + Mh...k , l

Cl2...k — M\2...k U11...к • The bound states are obtained from (4.13) and (4.14) by imposing the vanishing of the

sequence of the derivatives of C\2...k with respect to M\2...k for fc = 2,... n . The conditions determining M\2...k can be written as:

2 Мп..м = Мц...,,,.,) + Mk + a M,,...(*_,) Mk (l/,2...<*_ 1) + Uk) , к = 2,... n . (22) МО

Wc have been able to solve them, obtaining

I 1 M!2...* = -(2J(l-a)a )" tt*_1(l/(l-a)), k = 2,...n, (23)

- 2Л1 - a) / 4 T"- n = 1/(1 - a)) (T"U/(1 ~ a)) ~ cos

, M(l_,)t = 2iW/(l-Q) = -(j(l-a)V)" > Jfc = 2...n .

Equation (24) give then,- for the first time at your knowledge -, an explicit formula for the energy of the n-tnagnon bound states: we can check it by deducing the analogous formula for the bosons gas. Let us then observe that by putting 1/(1 -a) = 1 + aC/2 and then sending a to 0 in the XXZ we get the Bose gas model where С is the strenght of the 6 potential. In this limit 3 then the TI2...„ become exactly apart an infinite additive costant -n(n — l)(n + 1)C /12, which is exactly the energy of the n-bosons bound state [22] To this 1 add that even the continuum states can be described very compactly in terms of the Г,(1) elements. We can thus reasonably conclude that not only we have gained an undoubtful demonstration of the necessity and utility of introducing Quantum Groups in physics by the application to longtime known well established lattice models, but it has been also shown their role as symmetries of quantized spaces. So we could say that these new mathemathical entities not only survive the philosophical razor but satisfie also the other prescription about scientific concepts "nomina sJnt consequentia rerum

References

[1] F. Bonechi, E. Celeghini, R. Giachctti, E. Sorace and M. Tarlini; Phys. Rev. Lett. 68 (1992), 3718.

[2] A. Nowicki, E. Sorace, M. Tarlini, Physics Letters B302 ,419 (1993).

[3] J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. В 264, 331 (1991). J. Lukierski, A. Nowicki, H. Ruegg, Phys. Lett. В 271, 321 (1991). J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, "Quantum Deformations of D=J Poincare and Conformal Algebras", In press in "Quantum Groups" eds. H.D. Doebner and V. Dobrev, Springer Verlag (1992).

[4] J. Lukierski, A. Nowicki, H. Ruegg, Phys. Lett. В 293, 344 (1992).

[5] F. Bonechi, E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini; "Quantum Galilei Group as Symmetry of Magnons.", Phys.Rfiv. В 46, 5727 (1992).

[6] F. Bonechi, E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini; "Heisenberg XXZ model and Quantum Galilei Group J. Phys. A 25, L939 (1992).

[7] H.Bethe, Z.Phys. 71, 205 (1931).

[8] R. Orbach, Phys. Rev. 112, 309 (1958). Champaign, III., 1991). '511

[9] E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Math. Phys. 31, 25<18 (1990), and "Contractions of quantum groups" in Quantum Groups, Lecture Notes in Mathematics n. 1510, 221, (Springer-Verlag, 1992).

[10] E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Math. Phys. 32, 115.5 (1991), and J. Math. Phys. 32, 1159 (1991).

[11] L. A. Takhtadzhyan, " Lecturcs on Quantum Groups ",in Introduction to Quantum Groups and Integrable Massive Models of Quantum field Theory, 69, (World Scientific).

[12] P. Maslanka, "The Eq(2) group via direct quantization of Lic-Poisson structure and its Lie algebra", preprint 1993.

[13] S. L. Woronowicz, Lett. Math. Phys. 23,251 (1991), and Comm. Math. Phys. 144, 417 (1992) S. L. Woronowicz, Comm. Math. Phys. 149, 637 (1992). P. Schupp, P. Watts and B. Zumino, Lett. Math. Phys. 24, 141 (1992).

[14] A. Ballesteros, E. Celeghini, R. Giachctti, E. Sorace and M. Tarlini, "An R-matrix approach to the quantization of the euclidean group E(2)", J. Phys. A in press.

[15] L.L. Vaksman and L.I. Korogodski, Sov. Math. Dokl. 39, 173 (1989).

[16] W. Zakrzews!.i, "Quantum Poincare group related to k-Poincarc algebra, Warsaw Univ.preprint, March 1993.

[17] S. Giller, J. Kunz, P. Kosinski, M. Majewski and P. Maslanka, Phyb.Lclt. В 286, 57 (1992).

[18] J. Lukierski, H. Ruegg, W. Ruhl, " From k-Poincare algebra to k-Lorentz quasigroup: a deformatiom of relativistic symmetry ", Kaiserlautern preprint KL-TH-92/22, Physics Letters B, in press.

[19] S. Giller, J. Kunz, P. Kosinski, M. Majewski and P. Maslanka, "On q-covariant wave functions", Lodz University Preprint, August 1992.

[20] L. C. B'.edenharn, B. Mueller, M. Tarlini, " The Dirac-Coulomb problem for the k-Poincnre quantum group", Phys.Lett. В in press.

[21] D.C.Mattis, "The Theory of Magnetism l-Statics and Dynamics"Springer Scries in Solid State Sciences 17,(1981).

[22] ll.B.Thacker, Rev .of Mod. Phys. 53, 253 (1981). '512

INFINITE SOLITON SYSTEMS, QUANTUM ALGEBRAS, AND PAINLEVE EQUATIONS Vyacheslav Spiridonov1

Centre de Rechercbes Mathcmaliqucs, Universite de Montreal, C.P. 6128-A, Montreal, Quebec, H3C 337, Canada

Abstract This is a. short review of the recently discovered relations between: (i) the particular rcflectionless potentials with infinitely many bound states (infinite soliton systems), (ii) the 17-analogs of the Heisenberg-Weyl, sl(2), and more complicated polynomial algebras, and (iii) the Painlcve transcendental functions. The lattice version of the scheme unifying (i)-(iii) is outlined as well.

It is well known the KdV equation can be solved with the help of inverse scattering method for two classes of initial conditions u(x, t = 0). The first one consists of the potentials u(x) satisfying the restriction /^(i + |x|)|u(x)|rfx < 00, which garantees that the number of bound states of the auxiliary one-dimensional Schrodinger equation,

Ьф(х) = -ф"{х) + и{хЩх) = Щх), (I) is finite. Refiectionless potentials with N discrete eigenvalues are the simplest exactly solvable examples from this family. Since they generate JV-soliton solutions of the KdV equation they are called the soliton potentials. The second class is related to the non-singular periodic functions, u{x + T) = u(x), characterized by the presence of N gaps of finite width in the spectra of equation (1). These finite gap (hyperelliptic) potentials are reduced to the solitonic ones in the limit T —» 00. They can be thought as superpositions of infinite number of solitons ("periodic solitons") but there is no scattering problem for such objects, i.e. the solitary character of the ingredient waves is completely lost. Recently, the reflectionless potentials with infinite number of discrete levels 1 ave been considered in [1-7]. For x —» 00 such potentials decrease slowly and the standard inverse scattering method does not produce constructive results. These potentials deserve to be named the infinite soliton systems since they do not reflerl and may be approximated with some accuracy by the iV-soliton potentials (TV < 00). Moreover, this class absorbs the finite gap potentials as well. Despite the complex behavior of the functions entering the definition of these systems, many their properties have been uncovered. Among the infinite soliton potentials one can find an interesting family characterized by a purely exponential form of the discrete spectrum [1], the dynamical symmetry (or spectrum generating) algebra of which coincides with the ^-analog of the Heisenberg-Weyl algebra [2]. The general class of such self-similar potentials showing discrete spectra composed ifrom multiple geometric series and obeying more complicated (polynomial) symmetry algebras have been described in [5]. It appears after the g-periodic closure of the dressing chain for (1) (see below). In the limit q —> 1 one gets ordinary periodic closures which are related to the finite gap potentials [8] and nonlinear ordinary differential equations, known as Painleve equations [9, 10]. The spectra of nonsingular potentials taken from the latter class consists of independent arithmetic series [9]. When the basic parameter q is a primitive root of unity the self-similar potentials

'On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia. '513 are reduced to the specific (hyper)elliptic potentials obeying additional discrete symmetries [6]. The technique of Darboux (dressing, Backlund, etc) transformations is in fact applicable to a great variety (if not all) of integrable nonlinear evolution equations [11], including the lattice models. The recent consideration of the finite difference analog of the dressing chain and its self-similar reductions in [12] gave some unifying scheme for the orthogonal polynomials of a discrete variable, which appear as rational solutions of the (ordinary, or q-deformed) discrete Painleve equations. Let us describe briefly the technique that allows to find potentials with the 7-deformed syminetry algebras. One takes a set of Hamiltonians, Lj = —tP/dx2 + Uj(x), and represents them as products of the first-order differential operators,

+ = ^+ /;(*). (2) up to some constants Li = AfAj + Aj, (3) i.e. uj(x) = fj(x) — fj(x) + Aj. Then one imposes the following intertwining relations:

LjA+ = AfLj+t, AJ Lj = Lj+1 A~, (4) which constrain the difference in spectral properties of Lj and Z

/;(*) + /;+.(*) + fj W - fj+,W = A>+, - Xj А Щ, (6)

which is called the dressing chain [1, 9]. Rewritten in the form

f2(x,a) + f'(x,a) = f[x,a + 1) - Дх, a + 1) + (7)

it is known also as a shape-invariance condition [14]. It should be noted that any spectral problem with a nontrivial discrete spectrum can be represented in the form (6), or (7) with ordered discrete eigenvalues given by the constants Aj. The functional dependence of the solutions of (6) (or (7)) on the parameter j (or a) may be complicated but can always be established. Since all tractable potentials can be put into the "shape-invariant" form this term does not have constructive content. However, this does not mean that the particular examples considered in [13, 14], which correspond to the simplest Ansatz fj{x) = a(x)j + +'({x)lj, arc not distinguished from other problems. The correct name for the solutions of such type (with separated variables, or with their particular dependence) is similarity, or self-similarity solutions - a term widely used in the theory of solitons. The regular procedure for deriving a particular type of them for the differential-difference equations like (5) was discussed in [15], but the general theory is not yet developed. E.g., the complication associated with equations (6) consists of the fact that there are two unknowns, fj(x) and Hj, i.e. the system is highly underdetermi ned. (Self-)similar solutions are the solutions invariant under the subgroups of the general symmetry transformations of a given equation. The potentials we are interested in appear as fixed points of the combination of an affine transformation of the coordinate line, x —t qx + r, and a shift along the discrete lattice, j —* j + N. Indeed, the change in the numeration of solutions by an integer, maps solutions of (6) to the solutions, fj{x) —» fj+n(x), щ -» Цц.ц. The same '514 is true for the afflne group, fj(x) -> qfj(qx -f r), /j, -+ q2fij. We may look for the class of solutions invariant under these two symmetry transformations:

fj+Mx) = qfjil1 + r), Ih+N = (8)

This is the definition of functions relating the items (i)-(iii) in the abstract. The simplest example, defined by the reduction /,(1) = q'f(q>x), Aj = q7>, has been found by A.Shabat [1]. Actually, the general class of closures of the dressing chain (8) has been introduced by a different method of reasoning [16], namely, from the q-deformation of the (para)supersymmetric quantum mechanics [17]. At the operator level, the relations (8) lead to the Schrodinger operators with non-trivial q-deformed symmetry algebras. Let us consider the products:

Mf = AfAf» • • • Af+N_t, M- = ... Aj+1Aj, (9)

which generate the interwinings

LsMf = M+Lhn, M-L, = Li+NM'. (10)

The structure relations complimentary to (10) look as follows: N-1 N-1 Af/M- = n^i - -W)' MJMt = - Ai+0- (11) k=0 k=0 These identities show that if the operators Lj and Lj+м are related to each other through some simple transformation, e.g. 2 x Lj+N = q ULjU- (12) where U is a unitary operator, then the combinations Bj = MfU, B~ s U'1 M~, map eigenfunctions of Lj onto themselves, i.e. they are symmetry operators for Lj. The form of U is restricted by the requirement that the Lj s be of the Schrodinger form. Taking V to be the affine transformation generator, Uf(x) = f(qx -f r), fixing the indices and removing their irrelevant part, we get the symmetry algebra [5]:

LB+ -q7B+L = uB+, B~ L — q2LB~ = шВ~, (13) /V N 2 B+B- =П(£-А*), B-B+ = l[(q L + u-\k). (14) *= t *=i For N = 1 this is a q-analog of the Heisenberg-Weyl algebra which for special values of the parameters serves as the spectrum generating algebra [2]. For N = 2 this is a ^-deformation of the s/(2) algebra, and for N > 2 we get some polynomial relations describing the symmetries of self-similar potentials. Let us write out explicitly the system of nonlinear differential equations with deviating argument that one needs to solve in order to find the explicit form of the self-similar potentials;

^ (/.(*) + /*(*)) + /?(*)-/?(*) = Дь

^(Mx)+/3(x))+f2(x)~f2(x) = „„

2 2 — m^ + 4Mqx + r)) + f N(x)-q fi(qx + r) = ftp/. (15)

Note that the limit q —» 1 is not trivial. If the parameter r is not zero then we get the realization of the algebra (13), (14) at q = 1 which generalizes the one described in [9]. Let us '515 take г = 0 (for q Ф 1 this corresponds to the fixed point reference frame). Then the N = 1 case describes the deformation of harmonic oscillator potential appearing at q = 1. The N — 2 case corresponds to the deformation of the conformal quantum mechanics [16], sincc in the limit q —* 1 the potential oc ui2x2 6(4 — 1)/хг is generated. The N = 3,4 cases correspond to the q-deformation of the Painleve IV and V equations respectively, because these nonlinear ODli's appear at q = 1 (9, 10]. If the operators B± are well defined and have N normalizablc zero modes, then the spectra of the self-similar potentials, Ьф„ = Е„Фп, consist of the superposition of N independent geometric series: for n = N m + 1 (А,- TW". (Aj — for n = Nm + 2 (16)

(Ajv-• Т for 72 = N(m 1) where we assume that Aj < Aj+i. Note that for 17 > 1 the formula (16) does not apply due to the presence of singularities iti the potential [2]. I11 the limit q —» 0 there remain only the first /V levels, i.e. we get ЛГ-soliton potentials. An interesting situation takes place when the parameter q is a root of unity, i.e. q" = 1. Such cases happen to be related to the hyperelliptic potentials, the N — 1 system has been analyzed in detail in [6]. Depending on whether q is a primitive root оГ unity of odd or even degree, the solution may be unique or non-unique. So, the q = -1 system exists only when the initial condition /(0) = 0 is imposed and it provides a non-standard realization of the harmonic oscillator algebra. The general q3 = 1 solution exists for arbitrary initial condition and is given by the equi-anharmonic Weierstrass function:

3 ui(I)=2p(x + nj), (P? = 4(p -1), (17) where f!J+3 = fi,, and q'ilj =ui2- the real semiperiod of the doubly periodic function p(x). The analytical solutions at q* = 1 exist for special initial condition but they contain functional non- uniqueness. The particular subcase of the (x), (18) where a(x) and 6(1) are two functions of a discrete variable x. Taking a set of operators Ly.

+ Ls = flj(® + 1)T + аДх)Г~ + ЬДх), Т±ф(х) = ф(х ± 1), (19) and substituting factorization operators Af of the form

Л+ = рДх)Т- + Л(х), Aj = p;(x + 1)T+ + fj(x), (20) '516

into (3), one gets: aj(x) = pj(x)Mx - 1), bj(x) = p?(x) + /?(х) + A,-. (21)

The essential difference from the continuous space scheme appears in the possibility to alternate the sign in the chain of refactorization conditions: AJA+ + Aj = (-1ГЧ4+И7+» + (22)

relating the spectral properties of Lj and iJ+I. Substituting (20) into (22) one gets the discrete dressing chain [12]:

w(*)/i(*) = (-ir'Pj+.(*)/j+i(* -1). (23)

+ 1) + //(*) = (-1Г(Р?+1И + /?«(*)) + Mi, (24)

where ftj = (—l)">Aj+i— Aj. The derivation of the symmetry algebra is similar to the continuous case. The relations (13) can be preserved whereas (14) are changed to: N N + 5 s 2 В 0- = (-1) П(£B-B+ = (-l) l[(q L + u,-Tk), (25) fc=i _ t=t where N N

Tt = (-l)a,"Atl 5 = (26) k=1 l=k The sign factor (—l)5 now allows one to get a (/-analog of the su(2) algebra for yV = 2, 5 = 1 which was not possible in the continuous case. The unitary operator U entering the definition of the reduction (12) may be taken as a general element of the unitary transformation of the line. The simplest choice is the shift + operator U = Г{ , Т?ф{х) = t/i(x± fi), where <5 is a fractional number. The closure (12) is then equivalent to the conditions:

Pj+N = qPj(x + S), /л-л'Ы = qjj(x + 6), (27)

I'j+N = g2Mj, = o-j. (28) These conditions lead to the set of finite-difference equations which can be integrated for some choices of the parameters. So, the N = 2, 5=1 case is completely integrable and it is associated with the

References

[1] Shabat, A., The infinite dimensional dressing dynami.al system, Inverse Prob. 8, 303-SD8 (1992).

[2] Spiridonov, V., Exactly solvable potentials and quantum algebras, Phys. Rev. Lett. 69, 398-401 (1992).

[3] Gesztesy, F., Karowski, W., and Zhao, Z., Limits of soliton solutions, Duke Math.J. 68, 101-150 (1992).

[4] Novokshenov, V.Yu., Reflectionless potentials a.,d soliton series of the KdV equation, Tear. Mat. Fiz. 92, 286-301 (1992).

[5] Spiridonov, V., Nonlinear algebras and spectral problems, in: Proc. of the CAP-NSERC Workshop on Quantum Groups, Integrable Models and Statistical Systems (Kingston, Canada, July 13-18, 1992). Eds. J.LeTourneux and L.Vinet (World Scientific, 1993); Sym- metries of the Self-Similar Potentials, Comm. Theor. Phys. (Allahabad) 2, 59-73 (1993).

[6] Skorik, S. and Spiridonov, V., Self-similar potentials and the 7-oscillator algebra at roots of unity, Lett. Math. Phys. 28, 59-74 (1993).

[7] Degasperis, A. and Shabat, A., Construction of rellectionless potentials with infinite discrete spcctrum, preprint (1993).

[8] Shabat, A.B. and Yamilov, R.I., Theory of nonlinear chains, Leninqrad Math.J. 2, 377-400 (1991).

[9] Veselov, A.P. and Shabat, А.В., Dressing chain and spectral theory of Schrodinger operator, Funk. Anal, i ego Pril. 27, n. 2, 1-21 (1993).

[10] Adlcr, V.E., Resections of polygon, Funk. Anal, i ego Pril. 27, n. 2, 79-82 (1993).

[11] Lcznov, A.N., Talk presented at this Workshop.

[12] Spiridonov, V., Vinet, L., and Zhedanov, A., Difference Schrodinger operators with linear and exponential discrete spectra, Lett. Math. Phys. 29, 63-73 (1993).

[13] Infold, L. and Hull, Т.Е., The factorization method, Rev. Mod. Phys., 23, 21-68 (1951).

[14] Gendenstein, L.E., Derivation of exact spectra of the Sclirodinger equation by means of supersymmetry, JETP Lett. 38, 756-759 (1983).

[15] Levi, D. and Winternitz, P., Continuous symmetries of discrete equations, Phys. Lett. A152, 335-338 (1991).

[16] Spiridonov, V., Deformation of supersymmctric and conformal quantum mechanics through affine transformations, in: Proc. of the Workshop on Harmonic Oscillators (Col- lege Park, USA, March 25-28, 1992). Eds. D.IIan, Y.S.Kim, and W.W.Zachary (NASA Conf. Publ. 3197, 1993) pp. 93-108.

[17] Rubakov, V.A. and Spiridonov, V.P., Parasupersymmetric quantum mechanics, Mod. Phys. Lett. A3, 1337-1347 (1988).

[18] Atakishiyev, N.M., Talk presented at this Workshop.

[19] Suslov, S.K., Talk presented at this Workshop. '518

[201 Atakishiyev, N.M. and Suslov, S.K., A realization of the g-harmonic oscillator, Theor. Math. Phys. 87, 442-444 (1991).

[21] Papageorgiou, V.G., Nijhoff, F.W., Grammaticos, В., and Ramani, A., Isomonodromic deformation problems for discrete analogues of the Painleve equations, Phys. Lett. Л164, 57-64 (1992).

[22] Andrianov, A.A., Iofle, M.V., and Spiridonov, V.P., Higher-derivative supersymmetry and the Witten index, Phys. Lett. A174, 273-279 (1992). '519

INFINITE DIMENSIONAL LIE ALGEBRAS AND THE GEOMETRY OF INTEGRABLE SYSTEMS I. A. B. Strachan Dept. of Mathematics and Statistics, University of Newcastle, Newcastle-upon-Tyne, NEl 7RU, England.

Abstract A particularly interesting multidimensional integrable system is the self-dual Yang- Mills equations. This contains, via a process оГ dimensional reduction, many examples of lower dimensional integrable systems. Another important multidimensional integrable system is the self-dual Einstein equations, and recent work has shown that these are related by the use of infinite dimensional Lie algebras. In this paper some of these ideas are generalized to other infinite dimensional algebras (in particular, to the Moyal algebra, and to the algebra of Hamiltonian vector fields) while retaining the geometrical ideas central to the integrability of the self-dual Yang-Mills/Einstein equations.

1: The Geometry of Integrable Systems One important idea to emerge recently is the connection between the self-dual Yang-Mills equations and the theory of integrable systems. Before developing this idea it is first necessary to outline certain aspects concerning the self-dual Yang-Mills equations. Let G be a finite dimensional Lie group with associated Lie algebra g, and let the metric on C* be given by ds2 = 4(du dv — dr dw). The self-dual Yang-Mills equations are the field equations for the g-valued gauge potentials

Au, namely. (1.1) where F„„ = [D„, Dv] and = + . One crucial property of these equations is that they arise as the integrability condition for the otherwise overdetermined linear system [1]

IDU-(D,1 Ф=0, where £ 6 CP1 is a constant known as the spectral parameter. The connection with the theory of integrable systems came from the observation that many well-known integrable systems are reductions of these self-dual equations. Under this scheme one has the following procedure: • choose metric (e.g. ft4 or R2+s) and gauge group, • choose symmetry group, • choose normal form for the various constants of integration . The various choices at each stage leads to various examples of integrable systems. Thus one has the following conjecture [2]: "many, (and perhaps all?) of the ordinary and partial differential equations that are regarded as being integrable or solvable, may be obtained ifrom the self-duality equations (or its generalizations) by reduction." The list of integrable systems which fit into this scheme is now quite large (for a recent review see [3]), and include: • monopoles '520

• chiral and a-models • Toda equations • KdV and NonLinear Schrodinger equations (and more generally the AKNS hierarchy) • various integrable dynamical systems • Painleve equations The hope is that the apparently disparate properties, and the techniques used in the study of, these individual equations may be unified within this scheme.

Example 1.1

With ju(2)-valued gauge potentials

Ar=0, —l) '

а—{ф о )' А"~'\Фг -w) (Ф depending only on и and x — r + uj), the self-duality equations (1,1) become the NonLinear Schrodinger equations [4]

This, show that the NonLinear Schrodinger equation is embedded within the self-duality equations. In fact something stronger holds. With symmetries generated by d„ and d„ + dv the self-dual Yang-Mills equations are equivalent (not just contain) the NonLinear Schrddinger equation (ignoring a degenerate case where A„ = 0).

The integrability of the self-dual Yang-Mills equations (and hence the equations which are reductions of them) is established with the Ward correspondence [5]. This is a 1 — 1 correspondence between solutions of the self-dual Yang-Mills equations and certain holomorphic vector bundles over regions of the complex manifold CP3 (sometimes known as twistor space). Thus any solution to the self-dual differential equations are encoded entirely within holomorphic/geometric objects, from which follows the linear system (1.2) and hence the solution itself. There are well defined ways to construct such bundles (this being a geometric way to frame the Riemann-Hilbert problem) and hence to solutions of the self-duality equations themselves. One notable exception (at present) to the above list is the KP equation. Such equations (which have nonlocal Riemann-Hilbert problems) have been shown to have linear systems of the form (1.2) with gauge potentials in a particular infinite dimensional Lie algebra [6], but the geometrical correspondence - which hold for finite dimensional Lie algebras is absent. It therefore is important to see what infinite dimensional algebras may be used while retaining this geometrical description. In fact, there is already a well-known example of such a system, namely the self-dual Einstein equations [7]. These are the field equations for a four dimensional metric with: • Weyl tensor self-dual, • Ricci tensor zero. As in the self-dual Yang-Mills case, a solution to these equations are encoded within entirely geometrical/holomorphic objects. The rest of this paper is arranged as follows. In section 2 a general construction will be given for a certain class of infinite dimensional algebras, from which the self-dual Einstein equations emerge as a special case. These do have a geometrical description and hence are integrable. In section 3 a deformation of the self-dual Einstein equations is introduced, using an algebra known as the Moyal algebra. A direct geometrical description is lost, but the integrability of '521 the equations remain.

2: The Aigebi t, of Hamiltonian Vector Fields Let { , } be a generalised Poisson bracket acting on functions defined on some manifold ЛГ, satisfying the conditions: • {/.fl} =-{«./} (antisymmetry) • = U,g}h + {/. h]g (derivation) • l/i {Si Л}} + cyclic = 0 (Jacobi identity) With respect to a basis x', г = 1,... , dimAf, one may take

G'' + <7" = 0 (2.2)

(nuch generalised Poisson brackets were first studied by Sophus * Given such a structure one may define an associated Lie algebra Ham of Hainiltonian v<.. or fields. Let Lf £ Ham, where

• Regard Lf and Lg as differential operators, and define the Lie bracket for the algebra by \Lt>Ls] = LlLn ~ L3Lf,

• Regard Lf and Lg as vector fields on jV and define the Lie bracket for the algebra be the Lie bracket of vector fields [Lj, •£/„]*,,>.

In both cases [Lj,Lg] = . The fact that this forms a Lie algebra follows trivially from (2.1) — (2.3). The idea now is to study the self-dual Yang-Mills equations with gauge potentials taking values in this infinite dimensional Lie algebra. Let yAA' be spinor coordinates for С (or perhaps R2+2 etc. depending on a choice of reality condition). The self-dual Yang-Mills equations are the compatibility condition for the otherwise overdetermincd linear system:

л 4 1 £лФ = * '||^77 + Л/1/1,|ф, /1,Л' = 0,1, я" ' € CP .

In what follows it will be assumed that these take values in the Lie algebra Ham constructed above. Thus the AAA^S ™ represented by vector fields ЛДА' *-> L]/л,, where the functions JAA' depend on both the coordinates on С and on N. With this, the linear operators С A are now vector fields on С4 ® Л/",

Owing to the equivalent definition of the Lie bracket, the self-duality equations are a special case of the (Frobenius) integrability conditions for the distribution (2.4), i.e. [£O,£I]LIC = 0- '522

The integral surfaces Ф (satisfying САЧ — 0 /4 = 0,1) of this distribution may be regarded as curved twistor surfaces, and the space of such surfaces as a curved twistor space, fibred over tfie Riemann sphere. The converse construction involves studying an appropriate Riemann-Hilbert problem for the infinite dimensional group. Similar ideas have been applied to the SU(oo)-Toda equations in [8], which also develops the notion of a r-function for this eystem and its associated hierarchy. Example 2.1 Let the structure constants for the finite dimensional Lie algebra g, with respect to some 3 c k Fr basis e', i = 1,... , dim g be с' к, so [e', e>] = J2k V • i °m tbis one may define a generalised Poisson bracket by setting

к (the conditions (2.1) and (2.2) are automatically satisfied due to the properties of the structure functions), and let the associated infinite dimensional Lie algebra of Hamiltonian vector field be denoted by Ham(g). The original Lie algebra is now a subalgebra of ffam(g), since

к Thus any solution to the self-dual Yang-Mills equations with a finite dimensional algebra may be encoded within the structure of a curved twistor space by first embedding д in Hotn(g).

The next example, which shows how the equations for a self-dual vacuum metric fits into this scheme, will be considered in more depth, as this is needed in the next section. Example 2.2 The intcgrable system that will be studied here arises from the linear system [9]

[<д„ + и]ч< = о, [г-5) this being the linear system that arises a reduction of the Yang-Mills self-duality equations under two null translational symmetries. The Poisson bracket will be taken to be the standard bracket, i.e. i r . _ _ V,g)p«..on - gigg gidQ

(the suffix will sometimes be used to distinguish this canonical bracket from the more general brackets described above). The integr&bility condition for (2.5) is

and equating the various powers of ( yields

4„ = 0, = о. The first of these may be solved by introducing a single scalar function Q(x,y,i,j/) such that а Ь = П,,. The second equation then implies {Пх, = H(x,y), '523 for some function H{x,y). Assuming Л is a non zero function, a change of variable will reduce this to the standard form {fi ,г>П,»}ро<««оп — 1 ) (2.6) or, % 8 SL D-N _ CFSL Д'П = дхРх дуду дхду дудх This is just Plebanski's first heavenly equation. Solutions to this defines a self-dual metric

ds1 = Sltfdx'dx' where x' = x,y and x' — x,y. (2.7)

Solutions to (2.6) may be found using the nonlinear graviton construction [7]. This associates a curved twistor space T to such a self-dual metric, and conversely, starting with such a twistor space the corresponding metric may be found. The connection between this construction and the fi formalism of Plebanski was found by Newman et.al. [10]. Solutions of the wave equation Оф = 0 on such a self-dual space are related to elements of the cohomology group HX(T, 0(—2)), where the connection between the two is given by a contour integral

where Л £ //1(T, 0(—2)). Here и»0({),ш'(4) are coordinates on the twistor surfaces of the twistor space T. Further, the Green function for the • operator may also be given a cohomo- logical description [11] and hence solutions to the equation Оф = f may be found.

This second example provides a local construction for all cases where the rank оГ G'3 is even and constant. Then, by a theorem of Darboux, one can find local coordinates in any region U of M such that Ir 0\ ( ° C" = -Ir 0 0 J . \ 0 0 0/ Thus the self-dual Einstein equations arise in the very special situation where G'' has the above form globally on A(• These integral surfaces containing a lot more information than just the solution to the field equations. By expanding these as a power series in the spectral parameter and equating the different coefficients of various powers of the spectral parameter, one obtains an infinite number of recursion relations from which an infinite number of conserved currents and charges may be constructed. This was first done for a 2-dimensional chiral model [12], and then generalised to the full self-dual Yang-Mills equations [13], and the same idea may be applied to the systems here. Thus with ^

n=о one obtains, on equating different powers of £ in the linear system (2.5)

(2.8) (2.9)

Thus starting from a solution fi to Plebanski's first heavenly equations one may solve these recursion relation. The two lowest order equations may be solved with p(n=0) = x,p

j'n) = (0,0,p£>,p£>) and check the conservation law, which when written in terms of the metric (2.7) taken the simple form

+ [ n.vsj!"1 - SW<">] . - [fVj<"> - J2,«;<">] ..

Thus

Vj<"> = [a.vjp^ - fi,sip£>] г - [n, ^ - n,tip

(4 (N) = [{«., , P >}poiMDn] r - [{П.Х, TP }POI„ON] = p^-'-p^^O,

making use of the recursion relations (2.8) — (2.10). More details of the geometry behind this construction may be found in [14].

3: The Moyal Algebra The Moyal bracket is a deformation of the above Poisson bracket. It is defined by

OO ! . T, 2J 25+1 У _ I I \ if,9w, = £ g^T D-D' ( , ) т-^т-^-Цд] • (3.1) 5=0 ^ ' jsO *

where к is a 'deformation parameter', in the sense that

1>т{/,з}ро1лоп •

In what follows it will be convenient to reexpress the Moyal bracket in terms of the undeformed Poisson bracket:

{/,p)MO= (д1'-Щ/Лд1'~'д}ры^п- (3.2)

Given such a bracket one may define the Moyal algebra as above. Again, it is automatically a Lie algebra due to the properties of the Moyal bracket. Historically it was introduced by Moyal in his attempt to reformulate quantum mechanics in terms of Wigner distribution functions on phase space [15]. The bracket is essentially unique [16], and the associated algebra has a number of remarkable properties. In particular it contains, either as subalgebras or as special cases, all the finite dimensional simple Lie algebras, and a large number of other infinite dimensional algebras [17]. '525

Example 3.1 With the torus bases, defined by

Lm = ^exp(-im.x) exp(-«cm x j^)

(where m = (m]1mt)1 x = (i, y), and m x n = mi7i2 — m2n,), the Lie algebra commutator is

[Lm, Z„] = г sin к(т x n) Lm+n, with the quantity r depends on the particular normalisation of the Lm's. This is known as the sine algebra, and it was in this basis that it was shown that Moyal algebra contains all the finite dimensional Lie algebras. In the limit к —» 0 (and where the normalisation has been chosen so that r = 1/ic) this reduces to

[£m, Ln] = (m x n) X,m+n . This is the Lie-algebra sdiJfiT2) of volume preserving diffeomorphisms of the 2-torus.

The equations that will be studied here arises from the same linear system as in Example 2.2, but now the gauge potentials are replaced by Moyal algebra valued potentials. The same argument as was described there leads to the equation [18]

{П,г,П,у}д/0!1.1 = I- (3.3) In the limit к —> 0 equation (3.3) becomes (2.6), and so may be regarded as a 'deformation' of Plebanki's first heavenly equation. Central to the constructions in section 2 was the use of Frobenius's theorem to construct the associated twistor space. Owing to the infinite number of derivatives in the definition of the Moyal algebra one cannot use this theorem directly (though it is tempting to think of the elements of the Moyal algebra as some sort of 'deformed' vector field). However, despite the absence of this, equation (3.3) remains intcgrable. One way to study (3.3) for non-zero к is to expand П as a power series in к and equate coefficients1, so with П = J^^Lo ""^n , this becomes (using the definition (3.1)) = Ё (2s + 1 ^ ^

and on using (3.2) this becomes

1 = {Яо,г,fio.vJpoisson , |m/2lm-2j 2s / t\j+j / 9 4 0 = E E Ekrw(7) ™ = 1,2,...,00. «=1 n=0 ^ ' ' Rearranging the last equation yields

m-l

n=l

- E E E (2s + 1 j! ( )

m = 1,2,... ,00. '526

This is much simplier than it might first appear, since the right hand side depends only on ПА , Jb = 1,2,... ,m — 1. Thus it has the generic form

•п0ат = 5т(П0,Пг,...,Пт_1], m = 1,2,... ,oo.

The simplest function, Si, is identically zero. Thus one may build up the solution Я = iteratively, starting with an flo satisfying Plebanski's first heavenly equation, by solving an infinite number of wave equations with sources. Just as one may define a wave equation on a self-dual background by using the Poisson bracket, one may define a 'wave equation' В^ф = 0 associated to a solution fi of (3.3), where В is defined by ВпФ s {П.х, Ф,„}мо«»( - • (3.4) Solutions to this (given an П satisfying (3.3)) may be constructed using the same iterative scheme as above. On writing ф = and equating coefficients, (3.4) becomes the infinite sequence of equations

where T7l m—n,y} Pots Jon — {fln.v^ m—n,r } Poiaton j n=l J 'gff'ri'A (-iу* (2s\ \+{з1-Щn„,„,^Sf- nm.„.x}/»0,„on] H H +!)'• v J J [-{^-Щ^Лд^п^р^. •

One application of this wave equation is the study of the infinitesimal symmetries of (3.3). Under 0 —> SI + SSI (where Sil is infinitesimal), 6П must satisfy the equation

ВпШ = 0..

These ideas have been developed further by Takasaki [19], who has studied the dressing transformations for equation (3.3) (this being defined as a Laurent series with coefficients in the Moyal algebra) and the associated Riemann-Hilbert factorization problem- These results establish the integrability of the equation (3.3). Finally there are a number of further question that need to be addressed. • Are there any other models for which the Moyal algebra may be used? For example, is there a Moyal-Toda equation? This would be of interest because of the property that all finite simple Lie algebras are contained within the Moyal algebra. Thus one could study all 2-dimensional Toda equations simultaneously. A related problem is to understand the links between the ideas developed here are the Toda models constructed using the continuum Lie algebras of Saveliev and Vershik [20]. • Are there any physical applications? Recent work by Vasiliev [21] has show that equation (3.3) is related to a particular sector of a four dimensional physical theory containing field of arbitrary spins. • More generally, these geometrical/twistorial idea have been very successful in describing classical integrable systems. How, if at all, one understand quantum integrability using these idea is unknown. Also, how the Hamiltonian and r-matrix structure of certain integrable model is encoded within the geometry is ill-understood, though there are some preliminary results by Schiff [6]. • It is also possible to obtain supersymmetric integrable systems as reductions of the supersymmetric self-duality equations. The Moyal bracket has a supersymmetric counterpart known '527 as the cosine bracket, so it seems plausible that the results of this paper may have supersymmetric extensions.

Acknowledgements Financial support was provided by the U.K. Science and Engineering Research Council. I would also like to thank K. Takasaki for various comments.

References

[1] A. A. Belavin and V. E. Zakliarov, Phys. Utt. B73 (1978) 53.

[2] R. S. Ward, Phil. Trans. R. Soc. A315 (1985) 451. [3] R. S. Ward, in Twislors in Mathematics and Physics, C.U.P 1990, ed. T. N. Bailey and R. J. Baston. [4] I. J. Mason and G. Sparling, I'hvs. Lett. A137 (1989) 29, J. Осот. Phys. 8 (1992) 243. [5] R. S. Ward, Phys. Lett. A61 (1977) 81. [6] J. Schiff, Self-Dual Yang-Mills and the Hamiltonian Structure of Intcgrablc Systems, IASSNS-HEP-92/34, Oct. 1992 (hep-th 9211070). [7] R. Penrose, Gen. RL-1. Grav. 7 (1976 ) 31. [8] K. Takasaki and T. Takebn, 'SDiflft) Toda equations Hierarchy, Тли Function and Sym- metries', RIMS preprint 790, (hep-th 9112042).

|9j Q. Han Park, Phys. Lett. B238 (1990) 287. [10] E. T. Newman, J. R. Porter and K. P. Tod, Gen. Itel. Grav. 9 (1978) 1129. [11] M. F. Atiyah, in Advances in Mathematics Supplementary Studies, Vol.7 A Academic Press, New York, 1981. [12] E. Brezin, C. Itzykson, J. Zinu-Justin and J. B. Zuber, Phys. Lett. B82 (1979) 442. M. Lusher and K. Pohlmeyer, Nucl. Phys. B137 (1978) 4G.

[13] For review articles see: L.-L. Chau, in Springer Lecture Notes in Physics #189 (Springer-Verlag, 1983). L. Dolan, Phys. Reps. 109 (1984) I. [14] I.A.B.Strachan, Class. Quan. Grav. 10 (1993) 1417.

[15] J. Moyal, Proc. Cam. Phil. Soc. 45 (1949) 99. [16] P. Fletcher, Phys. Lett. B248 (1990) 323. [17] D. B. Fairlie, P. Fletcher and С. K. Zachos, J. Math. Phys. 31 (1990) 1085. [18] I.A.B.Strachan, Phys. Lett. B282 (1992), 63. [19] K. Takasaki, Dressing Operator Approach to Moyal Algebraic Deformation of Self-Dual Gravity, KUCP-0054/92, Dec. 1992. Nonabelian KP hierarchy with Moyal Algebraic Coefficients, KUCP-0062/93, May 1993.

[20] M. V. Saveliev and A. M. Vershik, Com. Math. Phys. 126 (1989) 367, Phys. Lett. A143 (1990) 63. [21] M.A.Vasiliev, Phys. Lett. B257 (1991), 111, Phys. Lett. B285 (1992), 225. '528

RELATIONSHIP IN BARYON MASS SPECTRUM AS A POSSIBLE REASON OF THE FINE STRUCTURE OF NUCLEAR EXCITATION S. I. Sukhoruchkin

Petersburg Nuclear Physics Institute, Gatchina, Russia

1. Introduction The presence of distinguished nuclear excitations (E" and D = E' — E') with multiple values of me and D0 ( me=511 keV is the rest mass of electron, ZJ„-electromagnetic mass splitting of nucleon) as well as the presence of intervals multiple to 9mt= 4.6 MeV in the binding energies of nuclei (Eg) discussed in this work [1] and in accompanying one [2] as ihe manifestation of the common properties of mass/energy spectrum of hadron system [3]. Due to fact that such excitations and long-range correlations appear mainly in nuclei with few- nucleon configurations (above different closed shells or few-holes in shells) ones may conclude that symmetry properties of nucleon interactions reflected in the shell-model [3] are essential to the phenomena. On the other side one of parameters of the long-range correlations in Ев turned to be close to ACg(a)=8 x 9me or near to 1/8 value of Д -excitation of nuclcon [4] and this fact could be matched with the observation done earlier about rational (1:2:3) connection between three values: splitting in decuplet baryon masses (m3), Д-excitation of nucleon (2ДМь) and nonstrange constituent quark mass estimation M, [4,5]. This introduces very usefull mass scale and nuclear effects may serve for control in a search for the connection of such quantitative parameter of hadron mass spectrum with parameters of different interactions. The basic relation between parameters Mq and me in the form mc/Mq = а/2ж ( where Q/2JT is radiation correction of electrodynamics ) in addition to found in [4] rational relation (n=18- 17-13) between Д Мд = тп, = Af,/3 and masses of pion (ra,) and moon (mM) enables one to suggest the distinguished character of relation of the type n=l-13-17 in different structures interconnected by scaling factor а/2* [4]. Nuclear models are based mainly on shell-model picture and close relation of cluster-shell effects of nuclear structure with the symmetry are known [3]. In Table 1 [6] differences ДЕв in the binding energies of the sequences of nuclei with ДZ — 2, AN = 4 and AZ = AN = 2 (a -particle) are shown starting in all three cases from the nuclei with hole in the closed shell 6 (and with N=82,28,20). The proximity of the values AEB( He) to 5 x 9e0 in N=82 shell and АЕв(4а) to 16 x 9e„ in Z=N=20 shell ( where e„ = 2mc ) turn to be within several keV. e Grouping effects in AEB[ IIe) near N=82 and AEB(a) near N=20-28 are clearly seen in AEB- distributions and their reality is beyond of doubt because of strong correlation with the filling in of the well-known shells. Discreteness in the particle masses with interval 16me was established in [4] by an analysis of distribution of 19 values of independent mass splittings in the hadron multiplets and muon and pion masses turned to be close to n=13-17 in these series (Part 2 of Table 1) while mentioned above triple coincidence ДЛ/д, m, and Л/,/3 was represented by real splitting m, (see proximity of them in Table 1). Several cases of close to each other mass intervals found by Takabayasi [7]-ДМг and кт are shown in Part 3 of Table 1 . One may notice that in baryon and in meson masses spin-flip effect is close to the mass difference connected with double change of the strangeness (ку and к in multiplets with J" = 1" and J" = 2+ [8]) so the proximity of ДЛГд to m, in baryons reflects the same effect. In Parts 4 and 5 of Table 1 two found by Wick and Sternheimer [9] relations in the particle masses are shown : proximity of intervals between the masses of very different particles (see also proximity intervals M, to the value of constituent quark mass in [10] ). '529

The SU(6) spin-flavor symmetry method in syStematization of properties of baryons turned to be successful in the framework of Quantum Chromodynamics (QCD) which as all-accepted theory of strong interactions includes the foundation of nuclcon interactions in nuclei where importance of proper understanding of the Д-excitation of nucieon lies in the fact that here two nonstrange quarks of the baryon remain in the same L=0 state (mark ** in Table 2 with notations of SU(6) ). Estimation of the values of nucieon excitations by instanton model [12] is also shown in central column of Table 2. QCD was built by analogy with QED-Quantum Electrodynamics and the hope was expressed by many authors [13,14] to the future unification of all interaction of Standard Model SU{3) 0 SU(2) ® U{\) in which QCD forms the first part and QED (with the parameters 2 о = с /he and mt) forms the last part. But beside the fact that basically very similar vector fields are contained in the description of strong, weak and electromagnetic interactions and along with widely accepted analogy between three families of lcplons and quarks still exists unsettled problem of the origin of their masses [15,16]. This situation permits semi empirical intercomparisori of different parameters and for example Bernstein put into the list of "Great Questions of Physics" [17] the parameter of QED a-1 = 137,04. The value a-1 itself is near to ratio (=137,06) of masses m, + mr to 2ш,. This "corrected" well-known Nambu-relation [18] inspired some semi empirical analysis [5]. hi the review on problems of quantum gravitation—theory with mass scale of Plank mass MR = (IIC/CN)"2 = 1,2210 x 1019 GeV [8]—the hope was expressed that unified theory оГ all interactions will solve among others such a fundamental problem as the origin of electron rest mass [19]. Above mentioned different nuclear structure effects were interconnected in [4] with single scaling factor r>/2x and expected parameters of non-statistical effects in liyperfme-structures were expressed in the form of relations with n=l-2-13-17... by the analogy with the discreteness in particle masses ( SFERC-mcthod [2], values lGnie,m(1 and m„ in the upper part of Table 3). The same ratio (п=1.4-1G-17-18) exist between found in [6] grouping of Д/?в=139,1(1) MeV in heavy nucici (n=17), four-fold value of ДEy(a) near ДДв(4а) in 39 A',see Tables 1 and 3, (n=18) and values of the total binding energies of 15C and l70 (n=13 and 16). The expected and observed intervals of fine- and hyperfmc-structures arc shown in lower part of Table 3. Wc see that in particle masses the numbers n=16 and n=18 arc connected with intervals introduced seinicmpirically by Wick, Sterriheimer and Kropotkin [4,9] Л/,=440 MeV =m=/3 = ЗДМд^З x 18 x 16me and М"=Ш MeV=3(16 x 16 - 1 )rnc and shown in parts 4 and 5 of Table 1. The first of these intervals concides with above discussed value ДЕв = 18 x Kim* in tiie cluster-shell effects in nuclear binding energies [4,6] (Part 1 of'Iablel). This fact is starting point in the SYMMED—SYmmetry Motivated Method of Estimation of Delta-baryon mass or delta-excitation of nuclcon.

2. Relationships in hadron mass spectrum

For the first time structural effects in hadron mass spectrum was discussed by Nambu [18] who pointed out proximity to 6m, of mass interval between nucieon and muon. This is a commonly accepted procedure in the nonrelativistic quark model (NRQM) to introduce the residual interactions between constituent quarks as subtraction from the main part of mass [10,11,20] and so that the Wick's interval (390 MeV) equal to the half of w mason mass is realy close to 1/3 of Д-baryon mass ( тд/3=410 MeV) and could be taken also for the estimated value of constituent quark mass close by the order of magnitude to 3m„. The somewhat larger value then m„ namely Д Л/д=147 MeV serves usually for the estimation of the interaction of quarks and so we may compare masses of particles of the first four multiplets with

integer number of the discussed above parameters ДМд, m* and ты/6 (see Table 4). This type '530 of comparison of mass values based on the empirical observation that combined distribution of differences of masses of all baryon singlets has periodic character [4] and it turned to he consequence of proximity of the masses themselves to the integer number of a period very close to m„ [9]. In Tables 2 and 4 the values mj and mo that are close to m x (140 MeV) with m=8-13 are marked by * and statistical test of the occasional probability of the correlation with certain period in the masses of isosinglet baryons gives the value less 10-3 provided that period is determined as a fractional part of the first value (imitation of m, = тод/8 ). In the NRQM models with residual interaction of quarks by one-gluon exchange (OGE) interval ДM& serves for establishing of the value of matrix clement Я„,=ДМд/3 of OGE [20] and so the discussed above intervals introduced by Sternheimer and Wick turn to be close to 9Л„, and 8Я,„. To consider the existence of several close to each other parameters in empirical representations of hadron masses (n=16-17-18 in Table 5 where n and m-integer numbers) we should start from usual estimate of constituent quark masses from Д-baryon and nucleon masses [11,12] with usual notation (QQQ) and Q[QQ] for the symmetric and antisymmetric configuration of quarks in nonstrange baryons [10]. Then from the assumption that mass value of constituent nonstrange quark Q may be represented by the sum q + [qq] of valence and residual parts we get ^]=П1д/3-ДЛ/д =1/3 (m„ — ДМд) as estimate of nonstrange nonvalence component of quark (792 MeV)/3=264 Mev which turns to he close to part of Wick's interval 2/ЗЛ^'= 2/6m^= 2/3(782 MeV)=261 MeV. Now from mass of Л-hyperon with the notation as S[QQ] the

estimate of constituent strange quark by M,= mA-3[qq]= 1115-528=587 MeV could be made arid it turns to be close to 4ДЛ/д= 4m,= 588,67 MeV= 4 x 18 x 16me. The same procedure for the mass of A*-liyperon gives estimate of charmed quark Mc=m\ — 3[?g]=1757 MeV close to 12ДА/д=1766 MeV=12 x 18 x 16me. The estimate of constituent nonstrange quark Mq= M,-mM= 587-147 MeV=440 MeV is close to JWj=437 MeV from NRQM calculations [12] and to Л'/,=т=/3=440 MeV from mass of heaviest octet baryon This simplified picture contains

two quantitative parameters ДМд=18х 16me and [q]=mu/6=(16 x 16 —l)me which correspond to n=18 and n=16 in the formula (n x 16t0, l)rne. We may turn attention to the fact that intersection of this expression and the sequence of values n' x 9rnc will give besides evident cases ( n=9-18 and n'=16-32) at ДМд and 1/2 ЛМл the single value n=13 (n'=23) exactly at the muon mass .with representation mM=(13 x 16 — 1 )me. We may recollect that the value close to 9mc was found as observed periodic interval in nuclear binding energies structure (see Table 1 [6] ) and simultaneously it is proximate to the value of electromagnetic mass difference of pion [2,4,5]. Analogy in the representation of the muon mass and of the value of the component of

quarks M'q' from w-meson mass (16 x 16 — l)me = mu/6 is evident. The nonrelativistic SU(6) classification of baryon states was initially worked out in NRQM with OGE residual interactions but the parameter ДM& reflects the same spin-flip effect in the framework of nonperturbative QCD with instanton-induced interaction. The latter hasn't some shortcomings of OQE approximation [11,14] and the characteristic parameter of instanton- induced interaction in [12] P = 3vaJ/mJjr3'2 was determined by applying it to the value of Д-excitation P — 2(тд — m\)=580 MeV. In all NRQM models narrow L=1 doublet and L=1 triplet in excited states of nonstrange baryons are expected and the groupings of masses are seen in the experimental values (ordered according SU(6) systematic in Table 2).The splitting of L=1 multiplets was found to be near 145 MeV in accordance with theoretical expectations P/4= 147 McV [12] (difference between seen in Table 2 groupings near M=1665 MeV,S=3/2 and M=1520 MeV,S=l/2 ). It is worth mentioning that equidistance with the same parameter Р/2=2ДМд=290 MeV exists also in sequence of masses themselves 1665 MeV- 1520 MeV-

тд(1230 MeV)- mN(940 MeV). This proximity of AN = 1,L=1 excitation of nucleon 1520- 940=580 MeV to the 4ДЛ/д=588 MeV=4 x 18 x 16me agrees with the assumption that Д- excitation is quantitatively distinguished. The hypothesis about presence in the nuclear date of fine-effects connected with the inner structure of nucleon was expressed by Dcvons [21]. '531

3. Consideration of masses of heavy particles The third family of fundamental fermions contains т-lcpton and b- and t-quarks and recent value rnT=1776,9(5) MeV are definitly higher then estimate 12ДЛ/д= Ш,= 1766 MeV= 12 x 8 x 6me and less then the value m,, + 12m, in [22]. Several mesons have masses close to mT or 4Mq = 3M, . For example mass of K'(3")-meson is related to masses of w-mesons with J = 1" and J = 3" and A"'(l~)-meson by two pairs of equal intervals ДМ= 880 MeV= 2M, (with AJ=2) and ДAf= 109 MeV close to m„ ( connected with Д5=1 ). The proximity оГ the slope of Regge-trajectory of both vector mesons (ш and 1С) to 2Мч = 6ДМд is very interesting by itself and nearness m„+m,, to 2M, leads to the proximity of A"(3")-mass to mT and 4Mq. The comparison of the value mT with doubled value of sum of the real masses of muon and ui-meson shows that difference between these combinations of experimental values accounts only some MeV. Equidistant character of mass values of vector members in bottomium spectum with interval close to 4m„= m(T,2S)-m(T,lS)= m(T,4S)-m(T,2S)=560 MeV was noticed in [22] and in Table 5 the first radial excitations of bb, cc, ss and qq systems are shown for comparison. Closeness of three values to 4m„=520 MeV or tu 4ДМд=590 MeV is seen ( they arc marked by *). The ratios of the values of particle mass themselves ( M\ of J -• 1" states of bb, cc and ss) to Д Мд turns to be close to integer numbers (see bottom line of Table 5). The closeness of ratio m(J/t/>)/[3/4(0, ДЛГ = 1)]= 3096 McV/589 McV=7,009(1) to integer number permits one to make similar estimate with mass of T(lS)-mcson. Value m(T)/16=591,3 MeV turns to be close to 4ДЛ/д=589 MeV. The estimate of b-quark mass from B-meson mass also turns to be close to rational number of 4 ДЛ/д as the result of proximity of the value mB(5278) = 5278,7(2) MeV [8] to 9 x 4ДМд= 5298 MeV. The difficulty in deriving meaningful results from mass values increases in case of the t- quark with estimated value m,= 174(10) GeV (23). If the real value mt will stay iri this mass region (2m,= 348 GeV ) than the ratio (M," = 390 MeV) / 2m, =112 x 10"5 will be of the order of 340 keV / ДМд and near to a/2jr= 116 x 10"s. To make some estimates of the value Mp (Plank mass) comparison of it with ДМд=т, and with the different power (x) of scale factor a/in was made [1] and closeness within two 6 в percent to expression Lxm,x (о/2тг)" = (Mz/3) x (а/2ж)~ was obtained with L-lo.pton ratio m„/me= Mz/M4 (see also [2]). For the value Л/,= m=-/3= 440,44(4) MeV the ratio MZJM,= 207,05 is close to integer L=13 x 16-1=207

The recently observed in L3 experiment object with mass M71=60 GeV [24] forms with discussed above Sternheimer's mass interval Л/, = ЗДЛ/д =0,441 GeV ratio 136=8 x 17 close 1 -1 to parameter o" in the Nambu relation т„/2 = о x me (see general remarks above). Together with Mz, Mq, Мь + Mc=\6Mq and m, (Table 3) the value 2Myl may turn to be part of structure in the mass scale of the order of 100 GeV which might be responsible for other discussed here structures. Taking seriously discussed before main relation of SFERC-method me/M, = aj2x= m„/Mz shown in Table 3 we may extrapolate it in this large-mass region _2 and take 2m( =348 GeV and Mu = me(a/2x) =380 GeV as candidates for the distinguished mass parameter. Author wish to thank S.Sakharov and A.Vasilyev for the help in editing and T.M.Tjukavina for the help in handling of nuclear data. '532

TABLE 1 Comparison of intervals in Ев and in particle masses with n x 2mc j Z,N Quantity Value (MeV) n(eD) n x 2m, Diff. Part.l 6 many Д EB( He) 46,0-41,5(1) 9 x (5-4,5) 46,0-41,4 [6] ,3 4a 57,82 &Еи(Ч1е) 46,009(6) 9x5 45,990 0,019 l33 Cs 55,78 AEB(*He) 45,972(6) 9x5 45,990 -0,018 ,3vLa 57,82 AEB(^HC) 91,981(9) 9 x 10 91,980 0,001 39 K 19,20 AEB(4a) 147,159(2) 18 x 8 147,168 0,007 Part.2 m, 0" M* 139,5679(7) 17 x 8 138,9917 0,5762(7) TTlft 1/2+ mц 105,65839 13 x 8 106,2878 -0,62942 m, 3/2+ MJ- — MJ- 147,8(6) 18 x 8 147,168 0,63(60) ДА/д (шд» - m„)/2 146,85(30) 18 x 8 147,168 -0,32(30) ti m -"-[8] 147,10(10) 18 x 8 -0,07(10) Л/,/3 [m„ - m„)/3 147,26(6) 18 x 8 n 0,09(6) Part.3

«Т 1" — 237,46(14) 29 x 8 237,103 0,36(14)

mw — гПт| 234,5(2) 29x8 к 2+ mji — m; 249,6(14) 31 x 8 253,456 mj — mv 255,5(12) 31 x 8 MT o- mn — m„ 407,9(2) 50 x 8 408,799 0,082 MT o- m4-m„ 410,3(2) 50 x 8 Part.4 к 1- mj 2 390,97(7) 3 x 16 x 8 392,450 -1,48(7) mp — m, 390,82(19) 3 x 16 x 8 (3me=l,53) -1,63 m/(. — 392,01(3) 3 x 16 x 8 -0,44(3) M„ — тп^ 392,12(12) 3 x 16 x 8 -0,33(19) Part.5 M4 1/2+ m„ — тпк° 441,60(3) 3 x i8 x 8 441,50 0,J o- m„ — m„ 441,79(9) 3 x 18 x 8 0,29(9) 1/2+ m=-/3 440,44(4) 3 x 18 x 8 (2me=l,02) -1,06(4) Mi [10] estimation 437 3 x 18 x 8 '533

ТАВЬЕ 2. Baryon states in SU(6) representation and their masses [8].

Jp S Oct et members (V ass,MeV [S ]) Singlets

1/2+ 56,0? 1/2 JV(939)" N(940) [12] A(1116)' S(1193) 1/2+ 56, 0? 1/2 N(1440) A(1600) £(1660) 1/2- 70, 1Г 1/2 N(1535) N(1520) [12] A(1670)* £(1620) A(1405)* 3/2- 70, lf 1/2 N(1520) Л(1690)" £(1670) A(1520)' 1/2- 70, 1Г 3/2 N(1650) N(1665) [12] A(1800)* £(1750) 3/2- 70, If 3/2 N(1700) 5/2" 70, 1Г 3/2 N(1675) A(1830)' £(1775) 3/2+ 56, 0J 3/2 Д(1232)" Д(1230) [12] S(1385) E(1530) П(1672)

TABLE 3. The expected from SFERC-method intervals D = n x 16тЛо/2л-)1.

X n=l n=13 n=16 n=17 n=18 n=3 x 18 n=3 x 16 Part 1

-1 Mc + Mb Mx = 2m, 91 GeV approx Part 2

(0) 2(^,-171,) mu (MeV) mr (MeV) ДМд (MeV) M, (MeV) 8166 keV 105,658 139,568 147,1(1) ЗДМд 0 8176=16me 106,288 130,816 138,992 147,168 441,5

(0) Д EB = 106,503 130,893 139,1(1) 147,159 A ,5 17 ( Z) ( c) ( 0) Д Eg Д£в(4а) Part 3 1 9,48 keV 123,3 keV 151,7 keV 161,2 keV 170,7 keV 511,0 keV (1) 123 [2,6] 152 [2,6] 161 [6] 170 [6] 511 [5,6] Part 4 2 11 eV 143 eV 176 eV 187 eV 198 eV .594 eV (2) 2x5,5cV 142 [2] 187 [2] 198 [2] 594 [2,4] '534

TABLE 4 Comparison of hadron masses га,- with integer number (m) of components of rtci if iinnl лкагЬ maf м» m /fi / IR 4/ 1С « „„J А \/Г

J" Symb m ГТ (MeV) MWF 6 (17 x 16 + l)m„ (18 x 16)m„ 1 o- 1Г 1 139,6-146,9 130,30 139,50- 147,17* M, 3 390,8-441,8 390,92' 418,50 441,50" V 4 547,45(19) 521,2 558,0 588,7 2 1" P 6 768,1(5) 781,8' 837,0 883,0 A" 7 892-896 912,1 976,5 1030,2 Ф 8 1019,413(8) 1042,4 1116,0 1177,3 3 1/2+ N 7 938,3-939,6 912,1 976,5 1030,2 A 8 1115,63(5) 1042,4 1116, o- 1177,3 £ 8 1189-1197 1177,3- E 9 1315-1321 1172,7 1255,5 1324,5' 4 3/2+ A 9 1232 1172,7 1255,5 1324,5 £ 10 1383-1387 1303,0 1395,0 1471,7 11 1532-1535 1433,4 1534,5- 1618,8 n 12 1672,4(3) 1563,7 1674,0- 1766,0

TABLE 5. Proximity of values of radial excitations of some; 1"-mesons to 4AM&.

JPG jy2S+l M,(bb),MeV Af,(cc),MeV W,(ss),MeV M,(qq),MeV 3 1— l T(1S) 9460,3 J/4/(IS) 3097 ¥> 1019,41 U> 781,95 з 1— 2 T(2S) 10023,3 tf(2S) 3686,0 V 1680(50) И 1394(17) AN A M= 563,0- 589, Г 660(50) 612(17)" = 1 Л/,/ДЛ/д 10023/ДМд 3097/ДЛГд 1019,4/ДЛ/д =68,01 =21,04 =6,93 '535

References

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MANIFESTATION OF ELECTRODYNAMICS PARAMETERS IN ENERGIES OF NUCLEAR STATES AND IN PARTICLE MASSES S. I. Sukhoruchkin

Petersburg Nuclear Physics Institute, Gatchina, Russia

1. Introduction Conclusion about the existence of distinguished intervals multiple by the value of the electron rest mass mc was drawn earlier from the analysis of correlations in distribution of nucleon separation energies and excitations of a wide range of nuclei [1-3]. Recent data on properties of nuclear states [4] and of particles [5] permit more definite conclusion. Together with [6,7] this work summarizes results of recent investigations [8] and deals mainly with long-range correlations in the data. In low energy nuclear excitations (S") so-called fine-structure of the range of hundreds keV is seen and in Table 1 groupings in spacing D=E,'-EJ distribution are shown for some near-magic nuclei (see also [1,8]). Here N is the number of intervals, N is the mean number of D in the distribution (with averaging interval ЛЕ=5 keV).

Three properties of structural effects connected with mc were found earlier [3] and were associated in single model (SFERC -Scaling Factor Electrodynamics Radiation Correction Method): 1) distinguished character of certain intervals of fine-structure in energies of nuclear states may be expressed by sequence of the integer numbers n=18-17-16-13 and intervals with n=18 and n=17 are rationally connected (1:3 and 1:8) with distinguished intervals coinciding with mass splitting of lepton (mc) and nucleon (Д,=1293 keV)-see Table 1-2; in particle masses structure with n=13-17 is also seen and it corresponds to masses of muon (m„) and pion (m„); 2) manifestation of scaling factor close to a/2Tr-electrodynamics radiation correction between corresponding intervals in different structures was found, say between intervals with n=13 or 17 [3]; In the first part of Table 3 proximity to а/2тг of the dimensionless ratio of the interval of fine-structure e' (in proton and neutron resonances in light nuclei) to the interval e0 — 2ra, found in [3] is shown and analogous ratio between hyperfine-structure intervals (0,01-2,0 keV) and fine-structure intervals (0,01-0,5 MeV) was considered as a consequence of fine-structure. Recent analysis of non-statistical effects in nuclear excitations (see Table 1 and 2,[8]) showed the distinguished character of the parameter D=e0/3=341 keV (connected with AJ=1) and the same parameter turned to be connected with the stability of the residual interactions of nucleons enp=£0/3 on the different nuclear shells. Attention was turned to the proximity to a/2t! of the ratio of the values £„/3 and of Л-excitation of nucleon (also AJ=l,see middle of the third part of Table 3). The general character of the value a/2* is supported by recent finding [7] that it is near to ratio of masses of muon and vector boson (second part of Table

3). 3) Presence of long-range correlations with 2mC=E„ in the binding energies of broad scope of nuclei (and in particle masses) [2,3] formed the third property of SFERC model.

2. Analysis of nuclear data The intervals with n=13-18 in fine- and hyperfine-structures were noticed by several authors [2,3,10] and effect of their stability gives long-range correlations. The last value in the first part of Table 3 gives parameter from additional analysis of stability of intervals with n=13-17 in the fine and hyperfine-structures. The linearity between hyperfine intervals D and values £*(0+) gives estimation of scaling factor between excitations connected with different nucleon configurations. '537

Stable intervals D=£I, and D=£„/2=me were found in nuclei with closing Z,N=20,28,40 shells (see Table 1). An example of correlations found in the binding energies of cluster configurations is shown in Table 4 [3,6] where proximity to each other of the independent values of sums of в separation energies of several nucleons S2p + S4„= ДЕв( Не) and S2j> + Sin— АЕц(а) in the shells N=82,28 and 20 (Z=58-l,28-1,20-1) is seen and their closeness to the integer numbers of the value 9e0 is shown. In case Z=58-1,N=82 two-step interval with the value near 90 MeV has deviations between its components (and between the value itself and 10 x 9s„ ) within small uncertainty of several keV. The same is true for the total effect Д£д(4о) in ld2s shell and analogy in the configurations of nucleons (proton hole in the closed shells) in 13aLa,b5Co and 39 К is evident. So the correlations with parameters mc and Da between values E",D and Ев in nuclei with few-nucleon configurations reflect the general character оГ effects connected with symmetry properties of nuclei [7].

Long-range correlations in Ев with parameter ec may be seen even in the light nucici (see parts 5-7 of Table 4). We turn attention to the fact that analogous to the above discussed in nuclear data long-range correlations with energy spread of several keV (in the values of the order of several hundreds of MeV) have been already observed in particle masses [3,7].

3. Analysis of electromagnetic mass differences

The representations of muon mass тд = 13 x 16m,.—mt and Дra„/m,,=o/2ir found in [3] are shown in part 5 of Table 3. The proximity of well-known particle masses to the integer number of mc was reported in [2] and was connected then with the coincidence of electromagnetic mass difference оГ pion Дrn„ with the integer number me ( 9me within then existed uncertainty about 6 kcV). With the recent data [5] the small difference between Дт, and 9me may be expressed as relative deviation Д(Дто»)/9т, near to a/1ir [7,11] ( see bottom part of Table 3). The improvement of the data does not change correlations observed earlier in particle masses (see Table 5 from [2]) . Comparison of electromagnetic mass differences with integer numbers of m„ is shown in Table 6 and proximity of Дm„, Атк and Amz to 9mc, 8me and 16me is clearly seen. On the right side of Table 6 the expected value Дт= near 13me from Coleman-GIashow relation [12] between mass splitting of octet baryons Am^ + Am= = Дms is shown. In the last lire the results of theoretical estimation of Am, by the model with instanton-induced interaction are shown for comparison and poor reproducibility of values Дт, and Атк was noticed in [13].

It turns out that to the value 9mc (very close to Дт,) are rational not only nuclear intervals in binding energies Д Eg ( Table 4 ) but also some well-known masses of particles ( sec Table 7). The value ЗДМд=М, and other estimations of constituent quark mass iW^'=m„/2 and М^=3тж that are discussed in [7] are also rational to 9me . Mass intervals ЗДМд=тч- m„=440 MeV and тш/2=тр-т,=390 MeV were introduced by Sternheimer and Wick [3,7] and their values are also turn to be rational to 9mc. Proximity of nucieon mass difference to 3m, together with the above discussed correlations of particle masses with 9mc and 3mc leads to the fulfillment of the rule m,=3m«, x A/,- noticed by Frosch [14]. But it is evident that the above mentioned closeness Атк to 8me and Дт= to 13me prevents this rule to be common for all particles. Several authors pointed out the constancy of mass interval in pseudoscalar я~-г)"-т)' masses and proximity of it to the sum m„+2m„. /3,7/. In the upper part of Table 8 the differences from пх16те of the masses of all six particles of pseudoscalar nonet are shown. Closeness of "remainders" m,-n x 16m„ ( middle column of Table 8) to -+me ( for ir~ and rf ) and to •+2mc ( for K° and if ) with constant change of integer number n ( Дп=117-67=67-17=50 ) is clearly seen. Masses of pseudoscalar mesons often are expressed as results of spontaneous chiral symmetry breaking /35/. To continue search for relations between me and other particle masses we may notice that ratio of pion mass to its charge splitting ( m,-m, )/2( Дт,-т, )=17.03 is close to integer and the value 5(тг)=(т»-те)/17= 8,180 MeV close to 16mB=8,176 MeV may be '538 introduced as the parameter of the discreteness. It is obvious that independently very close to it parameter from the proximity of mentioned mass differences between pseudoscalars may be introduced hyperons (near to 3,7 MeV, see part 5 of Table 8) leads to the parameter 5(S)= (1197,43-1115,ЬЗ)/10=81,80(6)/10= 8,180(1) MeV again close to 16mc. Now we continue this type of analysis with baryon masses known even with greater accuracy [5].

4. Analysis of the baryon masses

The similar to mentioned above trend is seen in the deviations of neutral baryon masses from values nxl6mc (part 3 of Table 8): in the sequence of octet baryons n-S°-S° systematic increase of the difference m,-n x 16me may be seen , with neutron mass deviation from nxl6me being minimal. The value of deviation of neutron mass from the nearest level of integer number m, that accounts -161,6(3) keV gives with the electromagnetic mass difference of nucleon Д, ratio D0jДт„=8,003(15) or ratio 9,004(16) between the values of shift of proton mass Дтр=Д,-|-Дтп=1454 kcV and neutron mass Дт„. These three values ( 161-1293-1454 keV ) and sum ( 1454 keV +e0 )= 2476 keV coincide with parameters of non-statistical effects in nticlcar data [8].

The value Дт,,=1455 keV is analogous to the value 3me in the representation of the stable nuclear fine-structure intervals by the neighbor periods 161-170 keV (n=17 and 18 in Table 2). If we compare two independent values Дmn and D0 with periodicity seen in baryon singlet mass spectrum (mentioned by several authors [3| and starting from mass of A-hyperon тл=8т„ ) then the closeness of the ratios Дm„ / m„ and Dc / тд to а/2ж (see Table 3, parts 4 and 6) reflects correspondence between members of n=17 structural intervals in nuclear data and pion mass mentioned in [3,7]. Many authors turned attention to the proximity of masses A- and П-hyperons to the integer numbers of pion mass (see ref. in [3,7]) but as Tables 7-8 show these masses are somewhat less than the values n(17 x 16+l)me , while the component itself is about 65 keV less than real mn (see Table 5 and 8). Wc may further estimate from values in Table 8 corresponding downwards shift from the integer number of m„=( 17 x 16+l)mc (with increasing of the strangeness) taking for the base mass value of omega- hyperon. It gives 12(17 x 16+l)me- mn=fe,-Ami)= 6,132 MeV-4,53(32) MeV=l,6 MeV. The related to Д5=1 value of downwards shift l,60(32)/3= 0,53(11) MeV turns to be close to me. In the table 8 comparison of Дтд» and Дт, is also shown (see the last column in part 1) and proximity of the ratio Дтпк/тпц= 5 s 111 x I0~ to the mc / Л/,=о/2тг=116 x 10~ permits one to recollect the mentioned in [36] proximity to a/ir of the parameter of CP-invariance nonconservation in kaon 2n decay. Besides the need for further analysis of nuclear data some cases of necessity of more accurate data about particle masses may be pointed out. For example masses of rho-mesons and delta- baryons are needed to permit the establishing of the values of downwards shifts and splittings in the symmetric quark configuration. We may compare the shifts of rho-mesons and phi-meson

masses (small Дmp , Дт^=20„, A.S=2, see Table 8,part 2) and turn attention to the fact that ДШф coincides with doubled value of electromagnetic mass splitting of nucleon Da within very small uncertainty (8 keV ). Two additional observations [1,7] connected with nucleon properties is worth mentioning.

Proximity of neutron mass value to the integer number of period 16mc was already discussed and small downwards shift 673 keV (Table 8) may be expressed as sum 511 keV + 162 keV

or as four components with the values 162 keV=£>0/8 and 170 keV=me/3. This number four coincides with the number of fractional parts of electric charge of electron in quarks forming neutron (ddu). The second observation concerns neutron beta-decay parameter дл/ду= -1,2573(28) [5] and '539 its closeness to ratio 2 x 17 /3 x 9 = 34/27=1,2593 where n=17 and n=2 x 9=18 are numbers in the representation of the particle masses.

5. Conclusion Discussed above exact ratio 1:9 in the values of downwards shifts of nuclcon masses agrees with the existence of analogous properties of stable nuclear intervals and it is supported by proximity of the values of these shifts to the distinguished intervals in nuclear levels. This means that properties of nuclear quasi-particles in few-nucleon configurations arc quantitatively correlated with the masses of free particles or in other words that they all are under tough control of the physics of the real vacuum in which different interactions are interconnected [16]. In that sense nuclear physics may play very important role in establishing relations between candidates for the common quantitative parameters. Author wish to thank S.Sakharov and A.Vasilyev for the help in editing of the paper and Z.N.Soroko for the supporting and encouragement.

TABLE 1. Parameters of nonstatistical effects in spacines D = n x me

Value D D D D D E- E' E- [8] [8] [1] = 0 [8] [81 [8] Nuclei 3M,S 53,58pe "Co A=20-24 A-odd A-odd A-odd Z= 20-2 28-2 40-2 28-1 10-12 Z-all Z-odd Z-odd D,keV 1022 511 1021 511 3576 1022 1293 2586 param. 2 me m. 2 m„ mc 7 me 2 mc A, 2Д, N{N) 43(20) 35(20) 25(10) 27(12) 28(11) 27(12) 18(16] 24 [10]

TABLE 2 Parameters of SFERC-model [3] in expression (nxl6m„(a/2;r)r)xm

X m n=l n=13 n=16 n=17 n=18 3x18 3x16

-1 1 Mz 2m, m m 0 1 16mc n + t m„ - mc ЛМл M, 1 1 9,48 keV 123 keV 152 keV 161 keV 170 keV 511 keV (8£') 123 [8] 152 [8] 161 [8] 170 [8] Table 1 2 246xn=£?'(0+) ЗОЗхп [8] 324 [8] 340 [8] £<. 3 [1] 455 [8] 483 [S] 511 [8] 4 38 492 [8] 606 [8] 644 [8] 682 [8] E'(0+) 8 76 [8] 1213 [8] 1290 [8] [1] 16 492x3=1476 2424 [8] 2576 [8] Table 1 24 2954 [8] 3638 [8] 2 1 11 eV 143 eV 176 eV 187 eV 198 eV 594 eV 1 5,5 eV=4e" 142 [10] 187 [8] 198 [8] (£'/2) 2 22 [8] 286xn=D 395 [8] 1190 [8] 3 33 [8] [1] 528 [8] 594 [8] 8 66-88 [8] 1500 [8] 1586 [8] '540

TABLE 3. Comparison of а/2ж with ratios of energy parameters [1 No Quantity Components of ratios Value xlO5 2 г а/2ж Д^е/^с=(а/2тг-0,328 а /я- )=115,965x10-* 116,141 1 e"/e' 1,35(2) eV / 1,16(1) keV 116(3) e'/Co 1,16(1) keV / £„=1022 keV 114(1) T.D / S£*(0+) 115,9(1)

2 rnu/Mz Мг=91161(31) MeV 115,90(4) 3 mc/Mq M,=m=- /3=3/2(тд-тлг)=3т5 116,02 D Дтд-m/v) 341(2) keV /(1232-939=293 MeV) 116,4(10) ED! 2rn.fi 2224 keV / 2mN =£B/1877,9 MeV 118,4 4 D0 / mA 1293,318 keV / mA 115,927 5 Дтм/гпд [(23x9me=13xl6me-m(:)-mtf]/m(J 112,1 6 Am,/m, (nxl6mE-mc-mrl)/mI=161,7(2) keV/m„ 115,86 7 Д(ДтТ)/9те [9m,.-4593,66(48) keV]/ 9тг 116(10)

TABLE 4. Comparison of intervals AEg = E'B — Eg and E' with the integer number of parameter ee — 2me Z N Quantity Value (keV) n(e„) nxe0 (kev) Ref. iss ь 1 La 57 82 АЕв( Не) 46009(6) 9x5 45990 [6] 133 e Cs 55 78 Д EB( He) 45972(6) 9x5 45990 139 B La 57 82 AEB{2 He) 91981(9) 9 x 10 91980 2 "Co 27 28 AEB( 2q) 73462(3) 9x8 73584 (3,6 3 19 20 ЛЕв(4а) 147,159(2) 18 x 8 147,168 (3,6 ,40 u 4 Ce 58 82 S„(" Ce) 9187(5) 9 9198 И "5Ce 58 84 S„(l42Ce) 7167,1(29) 7 7154 ,40Ce 58 82 AEg(6He) 45981(6) 9x5 45990 [6,81 4 5 tfe 2 2 £np 14321(1) 14 14308 181 8Яе 4 4 &2п2р 28600(5) 28 28616 20 N с 10 10 S2n2p 16320(5) 16 16352 n C 6 6 E'( 0+,T=2) 27595(2) 27 27594 fS| 6 uc 6 8 Sn("C) 8176,48(2) 8 8176,0 10 £ 5 5 Д£'(Д-/=1) 1021,8(2) 1 1022,0 f8 la 7 Ne 10 8 В-(0Г) 3576(1) 3,5 3577,0 [8 187Ve 10 8 4590(8) 4,5 4599,0 TABLE 5. Proximity of particle masses to integer numbers of mz [2]

Differ, т.-/V x me Odd N( mc) keV Even N(r nc ) keV

Year 1963 1992 1963 1992 Charged -120(1) -118,42(3) mesons 7Г+- +90(50) +65,2(7) K+~ -130(140) +21(9) Neutr.meson 7Г" + 100(50) +70,6(7) Positive P +61(10) +78,0(3) baryons £+ -60(200) -238(70) Neutral n -179(10) -161,6(3) baryons A -160(100) + 153(50)

TABLE 6. Comparison of electromagnetic mass differences [5 with N x m5

Particle ж К If D N S = ЛЕ-Д/V

Дт;,MeV 4,594 4,02(3) 4,5(2) 4,8(3) 1,293 8,07(60) 6,4(6) 6,78(9) N 9 8 (9) (9) (3) 8 (13) (13) iV x m, 4,599 4,088 4,599 4,599 1,533 8,176 6,643 6,643 A«L|/ITOR 1,94 2,91 (1,29) 8,70 5,18 (7,41)

TABLE 7.Proximity of masses to integer number of 9m, or m„=(17xl6+l );rce Particle ut V P A n m, (MeV) 105,658 781,95 547,45 938,272 1115,63 1672,43 n' x 9mc 105,777 781,83 547,28 938,194 (1116,0) (1674,0) n' 23 2 x 85 23+3 x 32 119+85 Repr. 13 x 16-1 6(16 x 16-1) ц+ЗЛМА 1 +W2) ( 8m, ) ( 12m, ) Diff. -0,118 +0,13(14) +0,17(19) +0,078 -0,39(5) -1,6(3) '542

TABLE 8. Deviations of particle masses from the integer number of 16m,.

[A SI Simb. rtii (MeV) n 7П,-71 x 16me AS x 0,51 MeV approx.

1 V 105,658 13 -0,629 IT 139,5679(7) 17 +0,5762(7) 1T° 134,9743(8) 17 -4,0174(7) Ие» ) 1 547,45(19) 67 -0,34(39) -0,34(19) 1 K" 497,671(31) 61 -1,064(31) -0,55(3) ("Co) 1 A'+ 493,648(9) 60 +3,080(9) (+3£„ ) n' 957,75(14) 117 + 1,16(14) c+o 2 P° 768,3(8) 94 -0,2(8) w 781,95(14) 96 -2,94(14) (-3e„ ) Ф 1019,413(8) 125 -2,585(8) (-2D. ) 3 0 n 939,5656 115 -0,6726(3) -0,673 1 S° 1192,55(10) 146 -1,14(10) . -0,63(10) 2 3° 1314,9(6) 161 -1,40(60) -0,38(60) 4 0 P 938,2723 115 -1,9660(3) -1,966 2 H" 1321,32(13) 162 -3,19(13) -2,17(13) (-3e„ ) 5 1 Л 1115,63(5) 136 +3,73(5) +4,24(5) 1 1189,37(7) 145 +3,85(7) +4,36(7) I s- 1197,43(6) 146 +3,74(6) +4,25(6) 3 ft- 1672,43(32) 204 +4,53(32) +6,06(32) '543

References

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GEOMETRIC NONADIABATIC PHASES AND SUPERSYMMETRY A. A. Suzko Joint Institute for Nuclear Research, Head Post Office, Box 79, Moscow, Russia 1

Abstract Generalization of the Witten supersymmetric quantum mechanics for gauge equations in two-dimensional space with additional scalar potentials is investigated. Coupling of the supersymmctry and geometric phases; the influence of additional scalar potential under the degeneracy of the ground state and as a result under topological effects arc discussed.

1. Introduction The adiabatic representation method [1], [2] is very useful for the description of many real quantum-mechanical systems where there are "fast" and "slow" degrees of freedom and one should estimate the effect of the slow dynamics on the behavior of the fast ones and vice versa. The adiabatic approach creates a link between nonrelativistic quantum mechanics and field theory due to the appearance of gauge fields in nonrelativistic few- and many-body quantum systems. This is an interesting aspect of the adiabatic representation in view of the discovery by Berry [3] of geometric phases. This approach opens possibilities to predict and explain such effects appearing in quantum systems as geometric phases [3, 4], the Aharonov-Bohm effect [5], the Hall effect [6], superconductivity [7, 8] and nonlinear processes. The conditions for the existence of such phenomena very often occur in the presence of supersymmetry. Important questions arise, i.e. whether it is possible to conserve supersymmetry in a more general case with an additional matrix of scalar potentials and also conditions for topological effects and geometric phases. Is there link between supersymmetry and nontrivial geometric phases? It is necessary to take account of nonadiabatic geometric phases due to the so-called level crossing. The problems of level crossing are not yet understood [9], in spite of the detailed discussion given by Hill and Wheeler [10] in 1953. The problems are closely connected witli the well-known Landau-Zener transitions and with nonintegral geometric phases. The solution of complex multidimensional problems are often based on the dimensional reduction procedure of space M = BuM realized by some partial expansion of the wave function of the original Hamiltonian over the known basis functions. In the geometrical treatment, this corresponds to introducing the Hilbert fibre bundle H = fB (B^Fxdfi(x). When a manifold M is reproduced in the form M x В, В is a base manifold, ц(х) is a positive measure on В and the fibres Tx form a family of Hilbert spaces parametrized by the points of i e B. In the traditional approach of the coupled channel method, this construction corresponds to a fixed fibre Tx spanned over the known basis functions, and unknown coefficients are given on the base В, lower than the initial manifold M. Unlike this, an adiabatic representation, where Hamiltonian H is decomposed into

H = h'®I + h!, (1)

is formulated in terms of the Hilbert fibre bundle, H, with nonfixed fibres Tx. This bundle, constructed from floating fibres Tx is naturally more general than the ordinary one based on a fixed fibre T. Since self-adjoint operators h*(z) act in the Hilbert fibres Fx it is appropriate to

1 Institute of Radiation Physical and Chemical Problems, Byelorussian Academy of Sciences, Minsk 5-15

f call them fibres of operators H: H — Jg (&h (x)dx [11]. The operator h' 0 I acts as h' on the slow "л* variables x e В and as an identity operator on the fast "/" variables y. The searched wave function Ф of the total Ilamiltonian И is given by the expansion

|Ф(х,В) >= |n >< п|Ф >= 22 j Kfcy)F„(x) (2)

over eigenstates Ф„(х; у) of the self-adjoint Hamiltonian h^[x) for each fixed value of x

1 к (х)Ф„(х;у) = £п(х)Фп(х;у). (3)

Due to the self-adjointness of ft'(x),x € 13, the eigenfunctions {Ф„(х;.)} form a complete and orthonormal set

I n >< и |= lfi(y — у')] < n I m >= j Ф1(х-,у)Фт(х-,у)<1у = S„m Vx (4)

which depends on the values x paramctrically, and thus the expansion (2) is valid. I pon substituting this expansion (2) into the Schodinger equation and using relations of the orthonormalization (4) of the eigenstates {Фп(х; •)} of Л-' at fixed x we arrive at the multichannel "slow" system of equations for the expansion coefficients F = {F„} of (2)

- 1/2[V ® f - iA(x)]J/X*) + U(x)F(x) = EF(x), (5)

where operators A(x) and Lf(x) act as operator-values for the vector and scalar matrix components of the gauge field. The elements of A(x) are given by

Л„т(х) =< Фп | t'V | Фт > . (6) The effective scalar potential is

22 = T, < ф»<1; ОИ*)^*; О >= Mx). (7) n n In this case U(x) = diag{£(x)}. In the general case the clfective scalar potential U(x) = diag{£(x)} + U'(x) consists оГ two parts, a diagonal potential matrix (7), whose elements coincide with energetic levels £{T.) of the instantaneous Hamiltonian H^ (3), and an additional potential matrix U'{x), that enters only into the system of Eqs.(5). The properties of this system of equations depend on whether level crossing occurs or not. If Ф„ and Фт arc the wave functions of the continuous spectrum of the operator /i'(x), then A(x) is the integral operator having the kernel Anm{x). The representation of (2) has to be invariant with respcct to cliosing a basis set of functions in the fibre T(x) |Ф(х;.) >= |Ф'(х;,) > U(x). Then, as follows from (2) F(x) is replaced by F'(x): F\x)=U(x)F(x). (8)

It is easy to show that upon such transformation the effective matrices U(1) and A(x) are transformed as gauge scalar and vector fields:

1 1 tf'(x) = U(x)U(x)U-\x), A'M[x) = UAU' - Ш- д„и (9)

and a lengthened derivative in (5) is covariant one:

A, = ® / - iAu{x). (10) '546

Here U(x) is a gauge transformation. The unitary U follows from the condition of completeness of the sets {Ф} and {Ф'} in each fibre Tz И(х)гг4*) = 1. u\x) = u-\x), Vie в. (ii)

Using the orthonormalization and completeness relations (4), we define the frame at a certain fixed point X = x„, \e(y) >= |Ф(х0; у) > . The moving frame |Ф(г,.) > is connected to the fixed one |e > through the unitary bilocal operator U(x,xa) = U(x)

|Ф(х;.) >= |e > U(x, x0) , U(x,x0) =< е|Ф(х;.) >, (12) that accomplishes a translation of the frame from x0 to x. From the definition of the operator A (6) with (12) and (4) we get A,(x) = iU~\x)dJ4(x) (13) and the covariant derivative (10) associated with the gauge field Au can be rewritten in terms of the operator U(x) Д, = c?„ + и~л (х)дМ(х). (14)

Now, by using the unitary gauge transformation lo U(x,x.) = Pexpi J A{x')dx'. (15) obtained from (13) and (12) under the condition (11) we can annihilate A and reduce the system of Eqs.(5) coupled through the kinetic energy operator to a system of ordinary equations coupled through the effective potential matrix for new coefficients F' connected with F by (8). Note, that in the one-dimensional Abelian case path ordering in expression (15) is unnecessary. As is well known from vector analysis, if the curl of the vector potential (6) vanishes at all x = x, В = V x A = 0, we can eliminate the gauge vector potential by a phase transformation.

In the adiabatic representation, vanishing of the matrix tensor R„u = д„А" - д„А" — ig[A", A<*] is equivalent to this requirement, because we have non-Abelian gauge fields. Proceeding with the transformation (8) where U(x) (15) and using (14) we find that the gauge vector potential vanishes. If the vector potential contains singularities, then the vanishing of curvature is not enough for eliminating the vector potential at all x. Note, more interesting effects take

place when Лм„ ф 0. These phenomena are connected with Berry's opening of geometric phases in simple quantum systems [3]. Berry demostrated the existence of magnetic monopole fields in dynamical systems, which arise naturally in a gauge theory framework [12]

6 = ^ t A(x) • dx. — i J J В • dS ф 0. (16)

The relation (16) can distinguish from zero in some cases, for example, when we have the supersymmetry, then В ф 0, or when eigenvalues (potential curves) intersect at some point of R-space, then the vector potential A(x) is singular and closed loop gives a non-zero result. 2. Supersymmetry Let us investigate the first case of distinguish from zero S in the relation (16). We try to construct a model of a SUSY quantum mechanics in twodimensional space with an additional scalar potential И'(г,з/) with respect to the classic problem considered by Aharonov and Casher [13]. They studed the problem spin 1/2 charged particle moving in a plane under the influence of a perpendicular magnetic field. It is shown the total magnetic flux is

B(x,»)cfedys« = 2jr(JV + s), 0 < e < 1 (17) '547

(compere with (16)). If (/V + e) > 1 the Pauli Hamiltonian has exactly N - 1 zero-energy normalizable eigenstates whose spin has the same sign as the flux; all nonzero energy eigenstates are degenerate with respect to spin flip. In fact their proof is based on the supersymmety of the Pauli Hamiltonian

Hp = l/2[-tV-eA]2-=

= 1/2(7^ - te

First, we define the hermitian supercharges Qi = Qi(x,y) (t = 1,2), Q2 = i&aQi [14, 15] for the description of a particle moving in a two-dimensional space

Qx = \/2\rdIW(x,y)%

Qi = l/2h(irJ-3„K'(I,S))-irl(»,+Wx,j))|, (19)

It is a generalization of the definition of supercharges introduced by Witten [16] for the description of a particle moving in a line in one-dimensional space

QX = l/2|

Now, as usual, we introduce the non-hermitian supercharges

Q+ = ^gt-Oa + iO.], Q~ = fy-Q* - «'Gil- (21)

They are represented as block two-by-two matrices

Q+ = " 'W + (*V + = П+,

Q- = ^т.[(-г>г + id,W) + (irv + dtW)) = П-. (22)

They can be written as

+ n = ^r+[nj-inj], n- = i=r..[n- + m;] (23) where coordinate components of П* are defined in corresponding to (19) as follows

П* = ±DX - idyW, П± = ±D„ + idrW. (24)

Here т± = j(

{Q;,Qi} = S.jH' and [Л%0,] = 0, (25) and

Q+a = Q-J = [Я*, Q*] = 0 or П+а = П-3 = [H\ П±] = 0. (26) '548

Using (23) we construct the supersymmetric Hamiltonian

tl -1/^11 ,11 о (П-• П+) - «(П-x П+) j • t27)

In the general case for a Dirac spin -1/2 particle in the extra magnetic field, four-component wave functions have to be introduced. We can make a generalization of relations (18)-(27) in two ways. One of them follows the logic of choosing the supercharges Qi and Q2 (19): in the definition of supercharge

Й" = = • (28)

Another way corresponds to use of definition of supercharge given in [17]

Q+ = -LT+«r.Q

Q" = = (29)

where П* and are represented as block two-by-two matrices combined from the two-by-two matrices generators of the supersymmetry П* and Q* (Q* = ±Д, + 9„W). Then instead of the supersymmetric Hamiltonian (27) in the first case we hawc

if (П+- П") + г'<гз(П+ x П") 0 \

~ 2 О (П- • П+) + г<г3(П- x П+) J

and in the second case

+ + H.-l( (Q -Q-) + 4Q xQ") o\ H ~2\o (Q"-Q+) + »

In addition to the terms of H+ = and H~ = {Q'Q+ of the one-dimensional supersymmetric Hamiltonian in (30) and (31) the field strength tensor

= д,Л" - Э„Л" = (32)

arises, associated with a magnetic field. If the gauge vector potential in ( 24), (28) and in (29) is replaced by

1. Ax -» Ax + dyW; Ay-*Ay- dxW\

2. Au -* + d„W,

respectively, then the tensor (32) Fxy will be represented as

1. Fxy = dx(Av - dxW) - дя(Ах + dyW) = Bxv; (33) 2. Fxv = dx(Ay + d^W) - d„(Ax + dzW) (34)

and H' (30), (31) with a noncentral field is rewritten in the form of the Pauli equation HT with the following exchanges Т+Ж+ -> П+, T~IT~ -> ft", '549

Т+7Г+ -> Q+, т~7Г- -» Q-. Note in the first case this is not a gauge transformation, while in the second case it is a gauge transformation. (We can present supersymmetric Schrodingcr operator with a noncentral forses p in the form of thr auli one if make the following exchanges: dyW -+ Ax, dxW —> — Ay.) At the application of the first approach for 1 /2 -spin charged particles the so-callcd spin-flip effect takes place with the simultaneously changing coordinate dependence of wave functions, when the generators (

X+ = (о- - П+)х- and x- = (

This is the second result of Aharonov and Cipher obtained here for the case whith an additional scalar potential. If Х-аз are eigenstatcs of the H~ Hamiltonian ( 30) then the eigenstates of //+ are

= г[(СТ1(жг - dyW) + <72(x„ + dxW)]x.^. (35)

A zero eigenstate Xo = X-« of 4~ Hamiltonian must be annihilated by (

(<7--П+)х„ = 0.

This relation after multiplication from the left by

{<Гз[дх - i(Ax + dyW)} + (idy + Ay- dxW)}Xo = 0. (3G)

one can easily see that in the adiabatic representation for (5) we arrive at the well-known situation stated by y.Aharonov and A.Casher [13] in the case of zero scalar potential IV. The ground state of the Hamiltonian H (30) is degenerated.

XO(x,y)=f(x,y)Pcxp{-v3J J B(x,y)dxdy}, (37)

where В is defined by ( 33)

Bxy = Fxy - (dx + ia3d,){dxW(x,y) - icr3dyW(x,y))

and the function f(x,y' satisfies

(дх + ш3ду)Дх,у) = 0.

Hence in the presence of W[x,y), the function f(x,y) is an entire function of (x -)- ia3y) as before with W = 0. As a result we have a degeneracy of the ground state modes which are defined by

XO,=(x + ia3yyexp{-

The number N defines the degenerate multiplicity of zero-energy states being governed by the Atiah-Singer index theorem, ft relates to the number of zero modes of a particle moving in a external gauge field with the topological number N defined as the surface integral (33) from

the tensor Bxy (17) J J B(x,y)dxdy = 2x(N + t). If e = 0, the flux, defined by relations (17) and (33) is quantized and one can speak about the Hall effect and nonstandard statistics in nonrelativistic systems (for example, in three-body ones). It is now trivial to see from (37) and (33) that the presence of a scalar potential can lead to an increase and vice versa to a decrease and even to the cancellation of a positive integer N, and as result, removing the degeneracy of the ground State, i.e. to lose conditions for topological effects. '550

3. Geometric phases Now we consider the second case of distingwishing of the relation (16) from zero. Non- diagonal elements of the induced connection operator A realise transitions between states of the parametric so-called instantaneous Hamiltonian and generate nonadiabatic Aharonov- Anandan phases [4]. In the realistic statement of the three-body problem [18] the nondiagonal matrix elements of A have to be taken into account since without them it is impossible to solve the problem correctly. Moreover near the level crossing, the adiabaticity is not valid. When the level crossings or quasicrossings take place, singularities of A„M arise and as a result, so do additional relevant geometric phases. This makes it necessary to introduce nonadiabatic Aharonov-Anandan matrices of geometrical factors in the presence of A singularities

, = exp Anm(R)dR)= (39)

. Л d < Фп I Vv.H'(R) | 4>j > „ < Ф,- | VKtf'(R) I Фт > =expmi~ — ад-uR)—" Then, at n = m we obtain the Berry relations for geometric phases. Now we discuss these problems. Note, that the nonabelian nonadiabatic phases manifest, even though only the radial dependence takes place. It is another aspect of our investigation. In the presence of A singularities nontrivial geometric phases arise in the wave functions Xo and hence х(й, E) at an arbitrary energy (16). These phases are to be taken into account. At the beginning consider some simple examples in the one-dimensional case. For example, if A(R) — i Пу/(Л)/(Я ~ '&)) with a smooth function f(R) having an analytic continuation in the complex plane of R, then S = ^ at f(R) = 2R or /(Я) = (Л + ib,) and the geometric phase is S = ir bj. Generally speaking, R is scaled and is diinensionless. Let us return to our mor® complicated problem related to Eqs.(5), (6). The nondiagonal elements of Л„,т(Л) =< n|iVn|m > realize transitions between different eigenstates of the fast Hamiltonian H1 and the adiabatic assumption is not applied, in particular, near the level crossing. Let us rewrite the matrix elements of the induced vector-potential (6) in another form

J 4 IR\ .< Фп(Я;г) | dH [R)/dR | Фт(Я;г) >

This is a result of the differentiation of an equation for basis functions (3) with respect to й s x and orthonormal relations of Ф. When an incomplete set, Ф, is taken into account, the state- vector is defined in n-dimensional subspace of (n + m)-dimensional, Hilbert space nontrivial gauge fields are induced. Geometric phases appear due to off-diagonal elements of the effective

matrix vector-potential Anm(R) = Ajj(R)Ajk(R) and we obtain the expression analogical to (40)

• V- d < I dH'(R;r)/dR\*j> < Фу | dH'{R\ r)/dR | Фк > =ехр^Яез т^ш щкгт •

In the last case there is an opportunity for three terms to cross at one point. Thon, at i = к we obtain the Berry relations for geometric phases. Note, the nonabelian nonadiabatic phases manifest even though only the radial dependence takes place. The reason is that in the presence of A(R) singularities, the gauge transformation U(R) does not eliminate them, and they occur in the scalar potential analogous to the nonvanishing curl of the vector potential in an N- dimensional space of slow variables. In conclusion it should be noted that conditions for the existence of topological effects and geometric phases arise in the presence of supersymmetry for the gauge Eqs.(5) describing '551 slow dynamics of quantum-mechanical systems. These last ones appear as consequence of the degeneration of zero states. Relationships between gauge vector and scalar potentials are stated. Nonadiabatic extra geometric phases arise due to singularities in the connection operator A in points of level crossing. Supersymmetry for nonrelativistic Schrodinger equations is the particular case of supersymmetry for the gauge equations.

References

[I] Born M., Nachr.Acad.Wiss. Gottingen, 1951, Bd.l, No.6, s.l.

{2] Macek J., J.Phys., B1 (1968) 831; Phys.Rev., A31 (1985) 2162.

[3] Berry M., Proc.R.Soc.Lond., A 392 (1984) 45.

[4] Aharonov Y. and Anandan J., Phys.Rev.Lett., 58 (1987) 1593. • Anandan J., Phys.Lett., A133 (1988) 171.

[5] Mead C.A., Phys.Rev.Lett., 59 (1987) 161; J.Chem.Phys. 49 (1980) 23; Rev.Mod.Phys., 64 (1992) 51.

[6] Semenoff G.W., Phys.Rev.Lett. 56 (1986) 1195; Kohmoto M., Annals Phys. 160 (1985) 343.

[7] Laughlin R.B., Science 242 (1988) 525; Phys.Rev.Lett. 60 (1988) No.25, 2677.

[8] Chen y.H., Wilczek F., Witten E., Halperin B.I., Int.J.Mod.Phys., B3 (1989) No.7, 1001.

[9] Bulgac A. Phys.Rev.Lett. 67 (1991) 965.

[10] Hill D.L. and Wheeler J.A., Phys.Rev., 89 (1953), 1102.

[II] Kuperin Yu.A., et.all. Annals of Phys., 25 (1991) 330.

[12] Wilczek F. and Zee A., Phys.Rev.Lett., 52 (1984) 2111.

[13] Aharonov y. and Casher A., Phys.Rev., A19 (1979) 2461.

[14] Suzko A.A., Russian J.Nucl.Phys., 56 (1993) No.5, 202; Hadronic Journ., 15 (1992) 363- 374.

[15] Suzko A.A., Sov.Part. and Nucl., 24 (1993) No.4, 1133, Energoatomizdat, Moscov (in Russian); Lecture Notes in Physics, " Quantum Inversion Theory and Applications", (Ed. H.V.von Geramb) 1993, Vol. 427, 67-107, Springer-Verlag, Heidelberg.

[16] Witten E., Nucl.Phys., B188 (1981) 513; B202 (1982) 253.

[17] Gainboa J. and Zanelli J., J.Phys., A21 (1988) L.283.

[18] Vinitsky S.I., Kadomtsev M.B., Suzko A.A., Sov.J.Nucl.Phys., 51 (1990), 952; Vinitsky S.t., Markovski B.L., Suzko A.A. Sov.J.Nucl.Phys., 55 (1992) 669. '552

MATRIX OF FINITE TRANSLATIONS IN OSCILLATOR BASIS V. M. Ter-Antonyan1'2, A. N. Sissakian3 1 Yerevan State University, 375049, Yerevan, Armenia. 2 Idjevan College-University, Idjevan, Armenia. 3 J/JVR, Dubna, 141980, Russia.

Abstract It is shown that the probability amplitude of quantum transitions under the action of a sudden homogeneous field coincides with the Charlier function up to a phase factor. The Charlier function is expressed in terms of the matrix of the operator of finite translations with respect to the basis of a linear operator. This formula is used to derive two summation theorems for the Charlier functions.

Instead of Introduction. In October, 1993 it will be a year since the decease of an outstanding scientist, Ya.A. Smorodinsky. Contacts of physicists and mathematicians of the present generation with Yakov Abramovich were highly fruitful and instructive for them as his intellec- tual scope was very wide. Besides, he understood and appreciated everything beautiful, which attracted people to him. This short note is a tribute of the greatest respect of the authors to the charming man with wide-ranging erudition.

Formulation of the Problem. Here we will calculate the probability amplitude of transition of a linear oscillator from an n- to a fc-state under the action of a suddenly switched-on homogeneous field U = — Fx. This problem is interesting for the following reasons: a) is generalizes one of the problems expounded in [1]; b) the amplitude is expressed only through the Charlier function; c) the solution of this problem provides a representation for the Charlier function that accents the symmetry rather than combinatorics,- d) this representation gives new results. When the field is switched on, the oscillator remains linear but with a shifted center of equilibrium, which means that in units the m = h = w = 1 amplitude of transition n —» к is described by the expression oo

Vkn = J Hk{x-F)Hk[x)dx (1) -oo

where Hm are wave functions of steady states of the linear oscillator. When n = 0, the amplitude (1) is calculated by integration by parts as a result of which the probability of transition 0 —» к is determined by the Poisson distribution [1]

wko = (2),

where Ь = F2/2. Our further goal is to compute the amplitude (1) at arbitrary n.

Method of Calculation. The integral (1) is not so simple as it seems at first sight; however, it can easily be calculated by using the operator of finite translation and the technique of second quantization. Writing F6 Hn(x -F) = e~ -Hk(x) '553 we can represent the amplitude (1) as the matrix of the operator of finite translation over the basis of linear oscillator (in rcf. [2] the diagram technique was developed for computing the matrix of finite rotations in the basis of a multidimensional isotropic oscillator). In the Dirac notation we have

Fa Pkn=(n\c~ '\k)

Introduce the creation and annihilation operators a+ = (i - d^/y/2, a = (x + Эг)/уД

which obey the equation + -Fdx = F(a — a)/V^ As [a, a+] = 1, the Hausdorlf-Baker formula holds true, i.e.

L E-FB, _ E-B/2£V5I E-V5D

Now we use the complete set of intermediate oscillator slates and write the amplitude in the form

Ры = е"Ь/3 £(n 1m >(m k) (4)

Using the known formulae

wc can easily prove that

(m к) =

Binomial coefficients in these formulae vanish tor m > к and m > n, respectively, and therefore the sum in (4) is truncated at minimal n or k. The result is a product of an exponential factor and a polynomial.

Transition Probability. As follows from the previous section, the amplitude (1) can be represented in the following final form

= (-l)'Cn(M) (5),

where C„(fc,6) is given by the expression

and Cn(k, b) is the following polynomial

The polynomial (7) is known in mathematics as the Charlier polynomial. It belongs to the class of so-called orthogonal polynomials of a discrete variable [3]. It is fixed if the parameter . and one of the indices are fixed (the second index represents a discrete variable). '554

WE will call Cn(k,b) the Charlier function. Apart from the obvious symmetry in indices n and Jfc, the Charlier functions obey the orthonormalization condition oo

E Cm(k,b)Cn(k, b) — Sm„ (8) fc=0 and the recurrence formula

^/Ь(п+1)Сп+1(к, b) + (k-n- b)C„(k, Ь) + т/ЫСп-х(к, b) = 0 (9)

(The relations (8) and (9) are usually written in terms of the Charlier polynomials.) For the transition probability we have J wkn = {Cn(k,FV 2)) (10) Note that at n = 0 formula (10) reduces to the Poisson distribution, formula (2), for the probability wM. (The model with the generalized Poisson distribution was considered in ref. [4], as well, where the distributions were expressed via the Laguerre polynomials connected with the Charlier polynomials as follows: C„(k,b) = (—b)"n\L^~"(b)). The orthonormalization condition (8) allows us to verify the validity of formula (10). Besides, formulae (8) and (9) can be used to compute the expectation value and dispersion for the distribution (10). It is easy to show that к = Я/2 + n and D = F - It3 = (2n + l)F/2. When n = 0, we arrive at the relation к = D typical of the Poisson distribution. One more interesting property is to be noted: in the limit b = 0 the Charlier function transforms into the Kronecker symbol. Indeed, when n > к (n < k) in the limit 6 = 0 the leading term in (7) is proportional to b~k {b~"), and, as follows from (6), the Charlier function vanishes; whereas at n = к in the same limit the Charlier function equals unity.

Symmetry instead of Combinatorics. Comparison of the relations (3) and (5) leads to the following interesting representation for the Charlier functions:

C„(fc,i) = (-l)t(n|e-^58'|*) (11)

Whereas the definition (6) accents the combinatorics, the representation (11) transfers this accent to the symmetry thus stressing the connection of the Charlier functions with finite translations. Formula (11) implies that the Charlier function coincides (within a phase factor) with the matrix of operator of finite translations in the basis of a linear oscillator. This point of view may be a basis of the theory of Charlier functions. It is not difficult to verify that it immediately gives the above-listed properties of Charlier functions. It is important that this approach is not only a more customary view of old things. It can also be used to derive new results. Indeed, if we start with the obvious identity

e-Fi8iefi3i _ e(-Fi+Fj)9r

and pass in it from operators to matrices in the basis of a linear oscillator, we can easily derive the summation theorem oo

Cn - Vhf) = £с„(т,ь,)С*(т,б2) m=0

that is a generalization of the orthonormalization condition (8) provided 6i = Consider one more identity,

e-F,Bze-F2Sx _ e-(Fi+F2)8r '555

Then, by analogy, we easily deduce one more summation theorem, CO 2 c„ (k,(Vbi+ \AJ) ) = ^ыгакм^км, m=0 that for a particular case, Ьг = b2 — 6, gives eo

mstO

Thus, we see that the representation (11) is in fact not only beautiful but also useful.

Instead of Conclusion. It was a surprise for us to come across Charlier polynomials, one of the five "bricks" of the theory of orthogonal polynomials of the discrete variable. As is known, Ya.A.Smorodinsky was carried away by that theory, thus being a great expert in it. It may happen that our results would evoke a smile as if he were listening a good anecdote. In any case we regret that cannot know the corresponding opinion of Yakov Abramovich himself.

References

[1] L.D.Landau, E.M.Lifshitz. Quantum Mechanics. "Nauka", M., 1989.

[2] G.S.Pogosyan, Ya A. Smorodinsky, V.M.Ter-Antonyan. J.Phys. A14, 769, 1981.

[3] H.Bateman, A.Erdely. Higher Transcendental Functions, vol. 2, McGrow-Hill, New York, 1953.

[4] I.V.Lutsenko, A.N.Sissakian, H.T.Torosian. JINR Commun. 2-13-049, Dubna, 1980. '556

GEOMETRY OF GROUP MANIFOLD AND PROPERTIES OF CHIRAL FIELDS IN VECTOR PARAMETRIZATION OF GROUPS Nguyen Vien Tho and V. I. Kuvshinov Institute of Physics, Academy of Sciences of Belarus 70 Skaryna. Avenue, Minsk 220602, Belarus

Abstract In the vector parametrization of groups the calculation of Cartan forms is considered. The direct connection between the Cartan forms and the composition law of parameters is established which allows to calculate simply the Cartan forms for many groups, not resolving Cartan-Mauer differential equations. For the unitary groups the Cartan forms and Lagrangians of chiral fields are obtained which have the form of rational functional in local vector parameters of group manifold. This form of chiral Lagrangians may be convenient for the investigation of their properties. In particular, for SU(2)-Skyrme model the Lagrangian is obtained which contains only three independent field variables and has the simple form. Using the obtained Lagrangian we consider the scheme of quantization of rotational collective excitations of skyrmions. The spherical top form of quantum liamiltonian is derived without using approximations in the intermediate calculations.

1. Introduction In many different problems of theoretical physics one deals with the nonlinear fields whose interaction is introduced by geometrical method. These fields take values not in the space Л", but in an nonlinear manifold M whose culvature defines their interaction. The simplest example is the п-field model wbore the manifold M is the sphere S2 in three-dimensional space. The manifold M may be a Lie group G (principal chiral fields, Skyrme model) or a coset space GjH (the case of spontaneously broken dynamical symmetries and Goldstone phemomena). In these case the nonlinear fields are identified with local parameters of Lie group or coset space, considered as nonlinear manifolds. It is known that the geometry of group manifold may be considered by the Cartan method [1-3], where the basic role is played by the differential Cartan forms which represent the motion of orthogonal frame in group manifold, considered as an analogue of Riemann manifold. The Cartan forms satisfy the Cartan-Mauer equations which completely define the structure of manifold. In this report it is shown that the Cartan forms may by found simply in the vector parametrization of group manifold. In the method of vector parametrization of group [4,5] the special attention is paid to the find and the exploitation of simple composition law of parameters which corresponds to the group multiplication. In the vector parametrization the group inversion and the similarity transformation take also the simple form: the group inversion correspords to the change of sign of parameters, the similarity transformation - to a linear transformation in the space of parameters. These properties are called, respectively, the naturalness and the linearity [4,5]. Owing them the investigation of groups may be performed only on the level of parameters, not addressing to the matrix expressions of groups or their representations. In vector group parametrization the direct connection between the Cartan forms and the composition law of parameters is established (section 2) which allows to calculate simply the '557

Cartan forms for many (liferent groups, not resolving Cartan-Mauer differential equations. In section 3 the explicit expressions of Cattan forms and chiral Lagrangians for unitary groups are found which have the form of rational functional in local vector parameters of group manifold. In particular, for the SU(2) group the chiral Lagrangians contain three independent field variables and have simple form which is convenient for the investigation of their properties. As an example we consider in section 4 some properties of S(/(2)-Skyrme model, using the form of its Lagrangian in vector parametrization of SU{2).

2. The connection between Cartan forms and composition law of parameters Let the Lie group G be parametrized by the set of parameters Q = {<2n}- Suppose that (a) A composition law of parameters Q =< Q,Q > which is defined by the group multiplication g(Q") = g[Q)g(Q) is given, and Q" continuously depends on Q,Q\ (b) the parametrization satisfies the naturalness, that is s(Q = 0) = /, g(-Q) = g"'(Q) 0) (I - the unit of group). From the view point of geometry of group space of parameters an infinitely small vector dQ which has the origin at the point Q corresponds to the group element 1 9(Q') •• g(Q')a(Q) = a(.Q + dQ), or 5(

hdQ + QdQmdQn + S(Q + dQ) = № + it " \aijk) - =

5(«) + <Ш)+0(Л?2), (2) we obtain J 9(Q') = 9(Q + dQ)g~\Q) = Г + (dff(Q))

g(Q') = g(Q + dQ)g~\Q) = g(Q + dQ)g(-Q) (4) and therefore Q' = (5) In the equation (5) Q' continuously depends on dQ ( by the preposition (a) ) and when dQ — 0 we have Q' =< Q,—Q >= 0 ( by the naturalness, preposition (b)), therefore the parameters Qn are infinitely small and can be represented in the form :

J 1 s Q'n=„= a,lm(Q)dQm + 0(dQ ) = Q'j > + 0(dQ ) (6)

1 where the coefficients anm(Q) depend on the concrete form of the composition formula , Qn ' = onm(Q)dQm is the first order ( in dQ ) term in the expression of Q,. (6). Remember that the tangent space to the Lie group G in the unit element of group is identified with the Lie algebra of this group, g(Q') have the form:

9{Q') = I + iQV)X« + Q(.dQ% (7) where A'n are the generators of the group G. Comparing (3) and (7) we obtain

(dg{Q))g~l{Q) = 'Qn(1)-Xn. (8)

The equation (8) gives a simple method for calculating Cartan forms: in this approach they are just the first order (in dQ) terms (On1') in the expression of the composition < '558

Q + dQ, —Q >. Therefore, for a group, if a parametrization is found which has a composition law of parameter and the naturalness, by this method one can easly determine Cartan forms without resolving Cartan-Mauer differential equations.

3. The expressions of Cartan forms and chiral Lagrangians for unitary groups By using the vector parametrizations of groups and the method devoloped in section 2 the expressions of Cartan forms are calculated for the groups of space-time symmetries [6], supersymmetry [7], and unitary groups /7(2), 57/(2), U(3), SU(3) [8] (see also (9,10)). Because the Cartan forms appear in the formulas of finite local transformations of groups, are basic elements for setting up chiral Lagrangians hence with the obtained expressions of Cartan forms it is possible to find the explicit form of finite local transformations for all mentioned groups [6,7,3,10] (in usual approach they are written in infinitesimal form), to construct Lagrangians of principal chiral fields for unitary groups U(2), SU(2), /7(3), SU(3) [8-10]. We are interested in the form of the expressions of Cartan forms and chiral Lagrangians for unitary groups in vector parametrization. The parametrization of unitary groups in which the composition law of parameters has simple form is parametrization by Calley form [11-13]:

V = U(N) = U 6 U{n)' (9) where N is a antihermitic n x n -matrix N* = —N. The composition law of parameter matrices A is: U{L) = U{,M)U(N),L =< M,N >= 1 — (1 — JV)(1 + MiVr^l - M). (10) In particular, one has [13] : < M + SN,M >~ (1 - M)~lSN{l + M)-1, (U) where 6N is a small addition to the matrix M. It is easy to verify that the parametrization has the naturalness and the linearity. For n = 2 and n = 3 one can choose as vector parameters the components of parameter matrices У in cr-basis and A-basis, respectively [8-10]. We consider here the simplest case: the Sl/( 2) group. Then

U = U(if) = = |~'.1?,<7°, U 6 SU(2) (12)

where oa(a = 1,2,3) are the Pauli matrices, r)a are three real parameters which make up a three dimensional vector. By using (8) and (11) one obtains the following expression for Cartan forms in this case: x (dU)U~ =-2iU^uJi2^3)dlbaa, o,i> = 1,2,3, (13)

/ot(»?l.>ftl43) = >, , -..3 [£ab — 2e„bcVc + (Sadtici + fob ~ $>b£№?<<] , (1 +4') The Lagrangians of principal chiral fields are determined as the left- and right-invariant metrics in Lie algebras [3,14]:

£ = -constSp[(dt/)a-,(^)i/"1]- (14)

For the SU(2) group from (13) and (14) it is easily to find the Lagrangian of principal chiral fields: '559

The expressions of Cartan forms and chiral Lagrangians for the groups U(2), C/(3), St/(3) are obtained in [8-10]. They are rational functionals in local vector parameters of these group manifolds.

4. Vector parameters as independent field variables and collective coordinates of skyrmions Let us consider the Lagrangian of SU(2)-Skyrme model [15,16]:

4 L Ш /Я + £< > = ?1тг(д„ид»иЦ + J-rTr[(aw£/){/t, (16) where V is a S(/(2)-ma.trix, Fr = 186AfeV is the pion decay constant, e is dimensionless parameter. By using the paeametrization (12) for SU(2) group and the expression of Cartan forms (13) one obtains Lagrangian (16) in the form [10]:

IMAM! rm 2 (l + ip)2 ег (l-fi?2)4 K 1 and the corresponding expression of static energy density:

F* (V.y)» 4 6 "" 2 '(1 + Ч2)2 e'' (l-t-тр)4 ' 1 J

(we denotes with [. Л .] the three-dimensional vector products). The Lagrangian (17) contains only three independent fields which make up a vector in three- dimensional isospace. All the informations about interaction are contained in the Lagrangian, not accompagning by neither supplement connection equation. The expression of static energy density (18) manifests an explicit invariance under rotations in isospace. For the Lagrangian (17) the soliton solution is found by using the ansatz:

7to(x) =/g(w(r)/2)~, (19) and the boundary condition (oo) = 0. Let us denote with rfo(£) the soliton solution obtained by ansatz (19). From the invariance of the static energy (18) under rotations of rf0 in isospace one can choose as collective coordinates of rotational skyrmion the parameters of isorotations. Considering the following field configurations:

^(i,t) = 0(j(t)№(x), (20)

(q = {$,} is the vector-parameter of rotation group [4,5]) and substituting them in the La- grangian (17) it is easy to separate the parts which depend on the elements of rotation matrix O, that is on collective coordinates q„,(a = 1,2,3). By using spherical coordinates the La- grangian (17) may be reduced to the form:

I = -M + < ^ / гЧг- {f/' (l+<<72(^))2

-?У r dr (1 + ) 1 560

+1 For elements of some matrix N in spherical coordinates one has

j dflN„ = J dSlRriNikR^ = Nik J dQ^R^

Air 4зг = Nik^-S,k = —TrN, (22) where Иц, are matrix elements in Cartesian coordinates, R is the matrix of coordinate transformation. Because of the equality

Tr(dt0dtO) = -Tr[6(dtO)d(dtO)]

2 = 5аЗьТг(о-аст1) = 2s , where

6d o = . = «,„(4,q) = (23) t s 1 + q' one obtains finally

L = -M-TXf, (24) where A is the following constant:

A = [drr7 + МФг + (25)

From (23) g1 = —4Q,j(g)9i9j, a, = (23')

therefore ^ L = -M + Х<*нШъ, X = "з^-А (26)

Introducing the generalized moment conjugated with the coordinates qi

*< = Щ = 2 (27)

one can write the Hamiltonian

H = ir{q{- L = M + x

= M + r**l, (28)

where (or_1)« are the elements of the inverse matrix of a = [а,у]. It is easy to find

and therefore

H - M + —(1 + f)(lkl + qkq,)vkTt. (29) '561

By using the rule of canonical quantization xi, —» —t^ one obtains for the quantized Hamil- tonian: ^^•M-^d+^Kfe+f")^^- (30)

The second term of (30) is proportional to the square of the infinitesimal vector-operator of 50(3) group [4,5], thus

я„. = м-1/>, + (3i)

For the spherically symmetric ansatz (19) and the field configuration (20) it is evident that the operators of isotopic (/,•) and space-times (J/) rotations are connected by an orthogonal transformation, so T1 ~ S1. Thus one obtains the spherical top form of quantum Hamiltonian of rotational excitations. The utilization of the form (17) of Skyrme model Lagrangian and the vector parametrization for finite rotations simplify considerably the calculation and permit receive the result without using approximations in the intermediate calculations. This approach is also convenient in the investigation of modified variants of Skyrme model with higher order stabilizing terms [18].

References

[1] Cartan E. Geometry of Lie Group and Symmetric Spaces Moscow: Izd. inostr. lit., (1949).

[2] Volkov M. K. and Pervushin V. N. Essential Nonlinear Quantum The- ory,Dynamical Symmetries and Physics of Mesons,Moscow: Atomizdat (1978) ; Volkov M. K. Particles and Nuclei ( Dubna: JINR ) (1973) Vol.4, P.3.

[3] Dubrovin B. A., Novikov S. P. and Fomenko A. T. Modern Geometry, Moscow: Nauka (1979).

[4] Fedorov F.I. Dokl. Acad. Nauk. Byelorus. SSR (1958) Vol.2, P.408; (1961) Vol.5, p.101; p.194; Dokl. Acad. Nauk. SSSR (1962) Vol.143, p.56.

[5] Pedorov P. I. Lorentz Group,( Moscow: Nauka ) (1979).

[6] Kuvshinov V.I., Nguyen Vien Tho Dokl. Acad. Nauk. Byelorus. SSR (1991) Vol.35, P.119; P.597.

[7] Kuvshinov V.I., Nguyen Vien Tho, Fedorov F.I. Dokl. Russ. Acad. Nauk. (1992) Vol.326, P.68.

[8] Kuvshinov V.I., Nguyen Vien Tho Yad. Fiz. (1992) Vol.55, P.2253.

[9] Kuvshinov V.I., Nguyen Vien Tho J. Phys. (1993) Vol26A, P.631.

[10] Kuvshinov V.I., Nguyen Vien Tho Particles and Nuclei ( Dubna: JINR ) (1993), accepted for publication.

[11] Gantmakher P.I. Theory of Matrices Moscow: Nauka (1979). '562

[12] Fedorovykh A.M. . In. Acad. Nauk. BSSR, Ser. Math. Phys. (1983) N.2, P.90.

[13] Bogush A.A., Fedorovykh A.M. and Zhirkov L.F. In "Group The- oretical Methods in Physics" (1983) Vol.1, P.196.

[14] Schwartz A S 1989 Quantum Field Theory and Topology ( Moscow: Nauka ).

[15] Skyrme Т. H. R. Proc. Roy. Soc. (1961) Vol.A260, P.127.

[16] Adkin G., Nappi C., Witten E. Nucl. Phys. (1983) Vol.B228, P.552.

[17] Bogusb A.A. Introduction to Field Theory of Elementary Particles, Minsk, (1981).

[18] Kuvshinov V.I., Nguyen Vien Tho Proceedings of the 8-nd Seminar on Nonlinear Phenom- ena in Complex Systems,Polatsk, Belarus, (1993) accepted for publication. '563

DYNAMICAL POLYNOMIAL ALGEBRAS Ш QUANTUM MECHANICS

Luc Vinet1 and Pascal Letourneau

Centre de recherches mathematiques, Universite de Montreal, C.P. 6128, Suec. centre-ville, Montreal, Quebec, Canada H3C 3J7 1 email: vinet (S (Sere, umontreal. ca

Abstract

The two-dimensional quantum mechanical systems with accidental degeneracies studied by Winternitz, Smorodinsky, ' hlir and Iris are revisited. It is shown that their symmetries are best described by polynomial algebras and the relationship that they bear with certain quasi-exaetly solvable systems is pointed out.

X. Introduction

In a seminal paper [1] entitled "Symmetry groups in classical and quantum mechanics",

Wiiiternitz, Smorodinsky, Uhlir and Fris systematically investigated two-dimensional nonrelativistic systems with accidental degeneracies. They reported the following results.

There exists ь constant of motion which is quadrati' :n the momenta if and only if the potential admits separation of variables in one of the four systems in which the Ilelmholtz equation separates in the plane. This observation has been fundamental in the elaboration of the algebraic interpretation of the separation of variables in differential equations.

In order to exhibit a non-Abelian dynamical algebra, a system needs to possess at least two constants of motion; in other words, the associated Schrodinger equation has to separate in at least two different coordinate'systems.

From these observations, two-di.ncnsional potentials with accidental degeneracies were obtained in Ref. [1]. Let rb i2 denote the Cartesian coordinates of the plane; the polar coordinates

(p,) are defined by x^ = рсо$ф, u = />sin0 (1) and the parabolic coordinates (£+,£_) given by

x,={+£_, (2)

Wc reproduce here the list of these potentials and indicate the two coordinate systems in which they separate.

A. Separation in Cartesian and polar coordinates

K = + + 4 + ^ = +(3) I; 15 P5 VCOS Ф МП Ф/

B. Separation in Cartesian and parabolic coordinates

V = a(*? + 4x'J) + 0x2 + ^ = а(£ - + Й) + - £) + (4) '564

С. Separation in polar and parabolic coordinates

(5)

0. Separation in two mutually perpendicular parabolir systems

In the above expressions, a, /?ь & icd 7 are real constants. Let us point out immediately that only cases A and В are genuinely different. We shall use the notation Pi = —id/dXj, i = 1,2 and ж± = — id/d£±. Take case С for example. Upon expressing the corresponding Schrodinger equation [i(pj + p\) + У\ф = Еф in parabolic r--.- < " des, one finds

1 2 « + T -) + + | + = ^ W Wfi+fl) be rewritten ivi the forr.

ф = -оф. (8)

Under the identifications ir+ —» pt, jr_ —» p;, —> ij, —> xa, a —» —E, E —» —a, one recognizes the Schrodinger equation with potential A. The case D is similarly seen to be equivalent to an off-centered harmonic oscillator. In the following we shall therefore limit our considerations to cases A and B. According to the usual lore, it should be possible to explain the spectrum degeneracies from the higher symmetries generated by the constants of motion C,-, i = 1,2,... Typically, this is done as '"'lows. One works out the commutation relations between the constants С,- to identify the Lie algebra that they realize. From the representation theory of this algebra, we may then determine how the symmetry generators transform the degenerate eigenstates among themselves and we may in particular relate the degrees of degeneracy to the dimensions of the irreducible modules. The problem with the potentials given above is that in general, their constants of motion do not close under commutation to form a Lie algebra. The way out of this difficulty has been [1, 2] to multiply the conserved quantities by functions of diagonal operators in order to obtain as a result, operators whose action on the degenerate eigenstates coincides with that of basis eleme ts of a Lie algebra in some irreducible representation. Clearly, this explanation of the accidental degeneracies is not very satisfactory. To perform the operation that we have just described, one needs to know a priori the action of the constants of motion on the degenerate eigenstates. This action is not, therefore, determined from the representation theory of the symmetry algebra and what is obtained in the end, is merely a correspondence between the energy eigenstates and the basis vectors of a Lie algebra module. With this paper, we purport to supplement the results of reference [1] by describing what is in our opinion, a more satisfactory explanation of the degeneracies of the potentials (3) and (4). Our main point [3] is that we are dealing here with situations where the symmetries are best described by a mathematical structure which is different from that of Lie algebras. We claim that in these cases, it is more natural to use polynomial algebras, that is, finitely generated associative algebras whose defining relations are polynomials of a certain order in the generators. This point of view has also been advocated in Refs. [4]. For the problems at hand, we '565 have control over the representation theory of these polynomial symmetry algebras. It will thus prove possible to determine algebraically how the constants of motion transform the degenerate eigenstates among themselves. We shall indicate in addition that certain quasi-exactly solvable (QES) systems arc related to these same two-dimensional problems. QES Ilamiitonians tiave the property [fij that only a part of their spectrum can be obtained algebraically. The paradigm example is II = —jjp- + V'(i) with the anharmonic potential

V(x) = iwV - 2/?u>V + (2/JV - E)x2 (9) where ui, 0 and E are constants. These connections will be established by exploiting the fact that tkie Hamiltonians of reference [1] separate in more than one coordinate system. Through Pavrl Winternitz, one of his prominent student, Professor Smorodinsky has had an important influence in Montreal and on us in particular. It is our pleasure to testify to that and to dedicate this paper to Professor Smorodinsky's memory. We hope that he would have been interested in its contents.

2. The 2-DimensionaI Anisotropic Oscillator Let us first consider the Iiamiltonian associated to the potential (4). After a trivial redefi- nition of the parameters, it can be cast in the form

Я = II, + Ih (10) with

"> = + + (и)

•h = jW + ^V*»-*)2] (12) and ш, ct, 0 real parameters. It describes an anisotropic oscillator in two-dimensions with a 2 : 1 ratio of the frequencies. It is ccntcrcd around 0 in the 2-direction and then; is in addition a singular 1 /x\ potential term. We shall make use of the annihilation and creation operators

«1 = + «1 = ~ 4»i), (13)

a3 = у/ъЬ [(i2 -/3) + ip2], «1 = v^J [(x2 - 0) - £рг] for which the non-vanishing commutators arc

[«„a!]=l, [a,,<4] = 2. (U)

The normalization is related to the anisotropy. In terms of these, the operators Hi and arc expressed as

i/Л = la i(aJ-l)(a alr2 + i (15) to a 1 + 4 1 + I -H = 4a + 1. (16) Ш2 2

The eigenvalues E of H can be obtained by diagonalizing H, and H2:

Hi |£,) =£,|£,)> Ih\E7) = Ei\E3), Е = ЕЛ+Ег. (17) '566

This may be accomplished with the help of algebraic techniques. Upon adjoining to Hi, the operators

Bt = («V-j^-iM".+«lra

= №)' (18) one obtains the commutation relations

[H, Bf] = ±2[Bt.BT) = --H (19) ш and identifies sti(l, 1) as its dynamical algebra. From there, standard representation theory gives for the spectrum of Hi:

Ei = + 1 ± w, n, 6 N. (20)

The series associated to the minus sign is present if and only if 0 < |o| < 2. The operator H2 is part of the oscillator algebra:

[Я2,«а] = -2ша2, [Я„а*) = 2и<4, [а2,<4] = 2 (21)

from where follows that

Et = (2n2 + 1)ш, ni 6 N. (22) The spectrum E = Ei + Ег of H is thus given by

Eij = ^2N ± ^ ui, JV € N+ (23)

where

N = m+n2 + l. (24) It exhibits at the level EN a ЛГ-fold accidental degeneracy. We shall now show how these degeneracies can be explained. We shall determine to this end the invariance algebra of H and work out the representations of this algebra that are relevant to the present problem. This algebra will turn out to be quadratic. First, one needs to obtain the constants of motion. This is easily done by combining the generators of the dynamical algebras of Hi and H?. Let

D = Hi - (25) C+ = wB+oj, (26) С. = шВГ4 = C|, (27)

it is checked that these operators commute with Hi

[Я,Д] = [Я,С±]=0. (28)

The structure of the symmetry algebra of H is characterized by the commutation relations of the operators (23). A straightforward computation yields

[£>,C±] = ±4u>C±, (29)

[C+,C_] = | D* + DH-\H*+(2-^U\ (30) '567

The invariance algebra of the anisotropic oscillator is thus the associative algebra generated by H, D, C± with (25) as defining relations. Since the commutator between C+ and C_ (sec (30)) is a polynomial expression of second order in the generators, the algebra is said to be quadratic. As will be seen, the dynamics of our problem is completely determined by a class of representations of this algebra. It is possible to exhibit a "Casimir operator". Indeed

3 l 2 К = £(C+C_ + C-C+) + ~D + -HD - D is shown to commute with all the generators. Moreover, since H is central, we might equivalently take К = К - ~H [H1 - (4 - osVs] . (32) 8 It is verified that A' = 0 in our realization. Consider a basis of state vectors \E, d), which are the simultaneous eigenvectors of H and D-. ff\E,d) = E\E,d), D\E,d) = d\E,d). (33) In the coordinate realization, this amounts to separating the variables in Cartesian coordinates since both H\ and Hi are diagonal. We shall now determine the action of C+ and C_ on these vectors. From the commutation relation (29) and the fact that C+ = Cl, we have

C±\E,d) = td^\E,d + \u), (34)

C.\E,d) = ld\E,d-Aw). (35)

Clearly, C+ and C_ transform the degenerate states among themselves. The coefficients It arc obtained as follows. From the commutation relation (30) and the definition of К we have

C+C_ = + D - 2u> + w|o|) (if + D - 2w - x (H - D+ 2ы) +-K. (3G) ОW CJ

Since I< = 0 in our realization, the action of C+C_ on the vectors |/i, d) is readily obtained. 2 Comparing with C+C_|E,rf) = l \E,d), we find

~[d-d^)(d-d.)(d-d) (37)

where di = -£ + w(2± H), d=E + 2w. (38) The spectrum of D is now determined from the following two requirements. First,, the representations should be finite-dimensional as there are only finitely many degenerate energy 2 eigenstatcs. Second, for these representations to be unitary, l d must be non-negative. The first condition implies that there must exist 2 eigenvalues of D, dmm and dm»x that are such that

C-\E,dmia) = 0, <

C+|fi, = 0. (40)

In view of (33), the admissible eigenvalues d of D will thus run from Jm;„ to d,„nx in steps of 4cj with the proviso that is kept non-negative. A first series meeting these criteria is

d = d+, d+ + 4w,..., d - 4

If we denote by N the number of states in this series, we have

d-4u-d+ = 4w(A'~ 1) (42) '568 which from (38), is immediately seen to give the following energy formula

Я£= + (43)

If |Q| < 2, a second series is present:

= d- + 4cj,...,d-4u. (44)

Again, if there are N eigenstates of D, with the eigenvalues (44), they are seen to have the energy

E-n = (2N - (45)

Thus, from working out the representations of the quadratic symmetry algebra of the anisotropic harmonic oscillator, we have completely resolved its dynamics and most naturally accounted for its accidental degeneracies. The representations of the invariance algebra are labelled by a single integer N which actually gives their dimensions. It should therefore prove possible to establish a correspondence between these representations and those of su(2). In fact, this mapping is realized by letting d = ^4m + If1) u> and Ei = (ij + 2 ± ) ш and by using j, m and 6 = ± instead of E± and d to identify the basis states. Then for

Lf = 1 C_ (46) + D + 2u + 6u>\a\)

1 № = C+ (47) y^Z + B- 2u» + &,|«l)

Lf = l(D-fJ^), (48)

we find that

L(i}\3,m,S) = V(j±m)(yT"i + l)|j,m±l,f) L{?\i,m,6) = m |j,m,6). (49)

This is nothing else than the old connection [2] between su(2) and the anisotropic oscillator. We clearly see that it can be established only once the structure of the invariance polynomial algebra has been identified and its representations constructed. This is why we claim that the above quadratic algebra is really the fundamental structure which is appropriate for describing the symmetries of the anisotropic oscillator. As already mentioned, the diagonalization of

2 D = - P\) + 5Л? + ~ 1)~5 - 2ш (Х2 - pf (50)

is associated to the separation of variables in Cartesian coordinates. Let L denote the angular momentum

L = ххрг - х2рг. (51) It is shown in Ref. [1], that the separation in parabolic coordinates correspond to the diagonalization of the generalized Runge-Lenz vector

J Л = piL + Lpi + 2w x,(x2 - 20) + ^(1 - (52) '569 which can be written as follows in terms of II, D and C±:

R = у/2ш(С+ + С.) - 20(11 + D). (53)

This operator is tridiagonal in the bases {|/J,c/J), n = 0,1,..., jV — 1} with

di = d± + 4 nu). (54)

From (33) wc have in fact,

R\E,d±) = v^ |E,d*+l) + t^dU)) ~ ШЕ +

Tiie eigenvalues u>\ of R are given by the zeros of the characteristic polynomial of TV11' order associated to the matrix (Я)±„ = (E,d±\R{E,d*). Let \E, A) be the corresponding eigenstatra,

Н|Ё, Л) = шА\Е, Л), (56) a 3-term recursion relation for the overlap functions (E,A\E,d*) is easily obtained in our algebraic framework. Take

(E,A\E,d±) = (Е,А\Е^)РНЬ). (57)

Clearly = 1. From (55) and (56), the coefficients Я*(Л) are seen to satisfy

^Л^(Л) = V^ + t^Pti(A)) - 2/3(/J + ^)^(Л). (58)

If we set ^

P*(A) = (2u/)-" (Й'-f) ^ (5!)) and use (38) as well as E= (2N ± Ifl) ш, we find that the factors Q„(A) obey the relation

Qi+M = [£ + /*(<«+ 2 ±|«l) QHA)

-4um (n ± ^ (N - ».)

They thus define an orthogonal polynomial set. (This is true for the functions

+ + (61)

-4/3^(41 + a + i(a2 -i)(C - {:')] (62)

If we set ш, Л) = = (63) '570 the eigenvalue equations

Нфвл = ЕФЕ,Л, ЛФЯ,Л=и\фвл (64) are seen to amount to the following separated equations for the functions ид({±):

+ = (65) with Ve(£b) = + - - + (20V - ml- (66) Both equations (65) are then recognized as the one-dimensional Schrddinger equation with the quasi-exactly solvable oscillator potential (9). lf one considers the equation

jt' + VBU)] «.({) = «.({) (67)

with E = (2N± |o|/2)w, the eigenvalues and eigenfunctions of N levels can thus be determined algebraically as follows from the above connection. The energies e are given by the eigenvalues

of Л. For the eigenfunctions, one starts from the wave functions фЕ^(хi,x2) = (xi,x3\E,d%) of the 2-d anisotropic oscillator in Cartesian coordinates which are easily obtained. The overlaps (57) for which we have a recursion relation, allow to write /V-1 (*„«a|A)»£

uA(0. That (67) is only quasi-exactly solvable is due to the fact that the potential V depends on E. Therefore, the only eigenfunctions u(£) that will be obtainable algebraically are those that correspond to the N degenerate eigenstates with energy E of the 2-dimensional system. Usually, the fact that a system is quasi-exactly solvable is imputed to the presence of a hidden symmetry algebra generated by first order differential operators. Consider for example the one-variable realization of su(2):

Г+ = (jV-l)*-**^, (69)

= _I(JV_ + (70)

= I' ™ 0 |ToT±] = ±r±) [T+,r-]=:2T . (72)

For N integer, it entails a AMimensional irreducible representation with (1 ,z,...,zlv"') as basis vectors. Hamillonians of the form H= Y2 CaiT'T* + 22 С°Т* (73) a,b=0,± a=0,± clearly leave this finite-dimensional representation space invariant and their spectra on this space can be obtained through a finite matrix eigenvalue problem. If one sets z = and takes H = —2T°T~ - NT~ - (1 + |a|)7- - 40uTa - 2wT+ (74) one recovers the Hamiltonian Я = ^тг2 -f VE(0 through the following conjugation:

Я = е-Я<Л a = — f)uf? — ^(1 + |a|)ln{. (75) '571

3. The 2-Dimensional Singular Oscillator We will now discuss (more briefly) the Hamiltonian corresponding to the potential (3). With a slight change of the parameters, we have

Я = Я, + H2 (76)

with

+ + 8 ' x J = 1.2. (77)

The operators H\ and H2 are here of the same form and Hi coincides with (11). These two one- dimensional Hamiltonians therefore both have su(l,l) as dynamical algebra with the ladder operators given by

Bt = (78)

В- = да. (79)

The commutation relations are given as in (19) by

[Hl,Bf] = ±2uH,JBf, [B+.B-] = —S^Hj, [Of, Bf] = 0. (80)

Once again, the constants of motion are easily expressed in terms of the symmetry generators. It is in fact immediate to check that the operators

, t D = Hi-H2, C+ = B+B;, C_ = (C +) (81)

all commute with Я:

[С±,Я] = [£>,Я] = 0. (82) Straightforward computations yield for the commutators

[D,C±] = ±4u>С±, (83)

3 г |C+,C_] = -\d ~ £Й(8-а;-^ + 2Я ) + 4Я(а1-а?). (84)

The invariance algebra of Я is thus the associative algebra generated by the 3 elements D, C+ and C_ with (69) as defining relations. From (84), we see that we have a cubic symmetry algebra. Note that if ai = Q2, we get the same algebra as found by Iliggs [6] in his analysis of the Coulomb problem on the 2-sphere. In this case also, we can construct the physically relevant representations. Working those out yields the following formula for the energy spectrum

t,N - IN + i(±|<*i| ± |oj|)| w (85)

and explains the degeneracies. The minus signs in (85) are admissible only for |oy| < 2. Furthermore, connections between exactly and quasi-exactly solvable one-dimensional systems and the singular oscillator (76), can also be established by exploiting the fact that the two-dimensional Schrodinger equation Нф = Еф, separates in coordinate systems other than the Cartesian one. Consider first separation in polar coordinates. After the change of coordinate

P2 = -e-«, (86) 572 the radial equation becomes

(87) while the angular equation reads

We recognize in equations (87) and (88), Schrodinger equations with respectively a Morse and a Poschl-Tcller potential. These are as is well-known, two exactly solvable problems [7]. The fact that the equation Нф = Еф separates in Cartesian as well as in polar coordinates implies that it also separates in the elliptic coordinates (u,v) defined by

Xj = rchu cos v, x2 — rshu sin v. (89)

In this case, the separated equations are found to be

1 ^ u. 1 2 4 ,4 . Л 2 4 „ Л X A ft _ ш l/(u) = KU(u) (90) o2du Ti2 + п2 r'sh'u + I\ 2rw V - Er J sh'u + —sh= u ch- Tu-

These two quasi-exactly solvable problems [5] are thus related to the 2-dimensional singular oscillator. Our reexamination of a classic paper of Winternitz, Smorodinsky, Uhiir and Fris has led us to the following two observations which we would like to stress in concluding. First, polynomial algebras can prove best suited to describe the symmetries of certain quantum mechanical problems. Second, quasi-cxactly solvable systems can be obtained through dimensional reduction by exploiting the fact that superintegrable systems generally separate in more than one coordinate system.

Acknowledgements We thank J. LcTourncux, V. Spiridonov, P. Winternitz and A. Zhedanov for discussions. One of us (P.L.) is the recipient of a NSERC postgraduate scholarship. The work of Luc Vinet is supported through grants from NSERC (Canada) and FCAR (Quebec).

References

[1] Winternitz, P., Smorodinsky, Ya. A., Uhlir, M. and Fris, I., Symmetry groups in classical and quantum mechanics, Soviet J. Nucl. Phys. 4 (1967), pp. 444-450.

[2] Demkov, Yu., The definition of the'symmetry group of a quantum system. The anisotropic oscillator, Soviet Phys. JETP 17 (1963), pp. 1349-1351.

[3] Letourneau, P. and Vinet, L., Quadratic algebras in quantum mechanics, in "Symmetries in Science VII: Spectrum Generating Algebras an^ Dynamic Symmetries in Physics", B. Gruber Ed., Plenum Press (New York), to appear. '573

[4] Gal'bcrt, О. F., Granovskii, Ya. 1. and Zhedanov, A.S., Dynamical symmetry of anisotropic singular oscillator, Phys. Lett. A153 (1991), pp. 177-180; Granovskii, Ya. I., Zhedanov, A. S. and Lutzenko, 1. M., Quadratic algebra as a "hidden" symmetry of the llartmann potential, J. Phys. A24 (1991), pp. 3887-3897.

[5] Turbiner, A. V., Quasi-Exactly solvable problems and sl(2) algebra, Comm. Math. Phys. 118 (1988), pp. 467-474. Shifman, M. A., New findings in quantum mechanics (partial algebraization of the spectral problem), Int. J. Mod. Phys. A4, (1989), pp. 28S7-2952; Ushveridze, A. G. Quasi-exactly solvable models in quantum mechanics, Springer-Verlag (New York), to appear.

[6] Higgs, P. W., Dynamical symmetries in a spherical geometry I, J. Phys. A12, (1979), pp. 309-323; Zhedanov, A.S., The "Higgs algebra" as a quantum deformation o/.su(2), Mod. Phys. Lett, A7, (1992), pp. 507-512.

[7] Alhassid, Y., Giirsey, F. and Iachello, F., Group theory approach to scattering, Ann. Phys. 148, (1983), pp. 346-380. '574

ON QUASITRIANGULAR HOPP ALGEBRAS RELATED TO THE BOREL SUBALGEBRA OF ah A. A. Vladimirov

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia E-mail: [email protected]

Abstract

Explicit isomorphism is established between quasitriangular Hopf algebras studied recently in papers [5] and [6].

In my recent papers [1, 2] a regular method has been proposed for constructing quasitriangular Hopf algebras (actually, quantum doubles) out of invertible matrix solutions Я of the quantum Yang-Baxter equation

Я12Я13Я23 = Я23Я13Я12. (1)

As an illustration, the Я-matrix of Jordanian type [3, 4)

(2)

has been shown (2, 5] to produce, among others, a quasitriangular Hopf algebra with generators {u, A} obeying the relations

[и, Л] = 2sinb A ,

Д(«) = ek ® v + v ® e~h , Д(A) = h ® 1 + 1 ® h, (3)

S±»(A) = -A, S±l(y) = -t)T2sinhA.

Its universal Ti-matrix is found [5] to be

= exp {sinh(?®t+T® fe)(sinh ' ® V-"в15inh • (4)

After submitting the paper [5] I have learned from O.Ogievetsky that in paper [6], using quite different approach (which allows classification of all Hopf structures on the Lie algebra [1, A] = 2/i), he obtained a quasitriangular Hopf algebra {т,<т} defined by

[r,a]=2(l-e°),

Д(г) = т®е° + 1®т, Д(<т) = tr ® 1 + 1 ® ff , (5)

-1 - S(T) =-re"", S (r) = —те~° + 2(1 — е "), S*» = -trf

with universal 7£-matrix a a Ti = exp(— ® т) exp(—T ® -). (6) '575

Since the Hopf algebra (3), due to a reparametrization

v=^x => [x, h]=2h, (7) evidently belongs to the class considered by Ogievetsky, it is only natural to look for an isomorphism between the Hopf algebras (3) and (5). Such an isomorphism really exists and can be fixed by

r = -e"S, a = —2h. (8)

Relations (5) are readily obtained fro;n (3) and (8). As a by-product, we come to an alternative form of the 7J-matrix (4):

7? = ехр(Л ® e~hv) exp(—e_At> ® ft) • (9)

In [2, 5] (sec also [7]), the following quasitriangular Hopf algebra {b,g,v,k} (the quantum double of {v,h}) was also considered:

[g, b] = [fi, b] = 2 sinh g , [jr, t>] = (ft, u] = —2 sinh h,

(6, d] = 2(cosh g)V + 2(cosh h)b, [g, ft] = 0 , Д(Ь) = es®6 + 6®c-s, Д(и) = ел ® и + u ® e_A , (10)

Д(а) = А® 1 + 1®S, Д(Л) = ft® 1 + 1 ® ft, S^ig) = -5, 5±1(A) = -ft, 5^(4) = -b ±2 sinh g, S±l(v) = =F 2sinh ft. The (anti)duality relations between {b,g} and {u,h) are:

< l,b >=< l,g >=< v, 1 >=< Л, 1 >= 0,

< 1,1 >=< ft,b>=< v,g >= 1, (11) =-l, < h,g >= 0. Universal 72-matrix looks like [5]

8 1 Л 7l = cxp( . f, '! ® ..(sinh3®v + b®sinhft)\ . (12) lsmh(s® 1 + 1 ® ky J v '

In terms of {r, 0} and their antiduals {/i, i/} the same quantum double is characterized by the relations |т, o) = [r, U) = 2(1 - e"), [„, a) = [ft, v) = 2(1 - e"),

[г,м] = 2(м-т), M = 0,

Д(т) = т e e* + 1 ® Т , Д(IT) = A ® 1 + 1 ® cr, A((i)=A«e4l®/i, A(i/) = i/®l + l®f, (13) S(r) = -re"' , 5"'(r) = —re-" + 2(1 - e~"),

S(M) = -PT- , S-'Ы = -/«В"* + 2(1 - О ,

S±>(

< 1, V >=< 1, V > = < T, 1 >=< (7, 1 >= 0 , '576

<1,1>=1, < г, I/ >= 2, < <7, >= -2, (14) < Г, jl >=< tr,tr >= О and universal 72-matrix [6]

ft = exp(|®r)exp(-/i®|). (15)

Apparent similarity of (15) and (6), as well as (12) and (4), is due to 'selfduality' [5, 6] of both {r,o-}- and {t>, A}-algebras. Hopf algebras (10) and (13) are isomorphic by

p = e_sf>, v = —2g, T = -e~hv, a = -2h. (16)

In particular, this produces an alternative representation for % (12):

71 = cxp{g ® e~hv) ехр(е~3Ь ® h). (17)

Surprisingly, it appears problematic to derive (12) directly from (17) by the mere use of Baker- Campbell-Hausdorff formula.

Acknowledgments The present work was supported by the Heisenberg-Landau program, 1993. I am very grateful to Prof. J.Wess for organizing my visit to Munich where this research was carried out. I appreciate stimulating discussions with O.Ogievetsky, A.Kempf, M.Pillin and R-Engeldinger.

References

[1] A.A. Vladimirov, Mod. Phys. Lett. A 8 (1993) 1315.

[2j A.A. Vladimirov, Z. Phys. С 58 (1993) 659.

[31 V.V. Lyubashenko, Usp.Mat.Nauk 41 Vol.5 (1986) 185; Engl, trans).: Euss. Math. Surveys 41 (1988) 153.

[4] E.E. Demidov, Yu.I.Manin, E.E.Mukhin and D.V. Zhdanovich, Progr. 'fheor. Phys. Suppl. 102 (1990) 203.

[5] A.A. Vladimirov, Mod. Phys. Lett. A 8 (1993) 2573.

{6} O. Ogievetsky, Max-Planck-Institut preprint MPI-Ph/92-99.

[7] C.Burdik and P.Hellinger, Charles Univ. preprint PRA-HEP-93/2. '577

GEOMETRIC MOTION ON THE CONFORMAL GROUP AND ITS SYMMETRY SCATTERING R. F. Wehrhahn and A. 0. Barut*

Theoretical Nuclear Physics, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. * Department of Physics and Astrophysics, University of Colorado, Bouider, Colorado 80309-390, U.S.A.

In [l]the possibility to study scattering problems of systems with internal structure using a group theoretical method called "symmetry scattering" for spaces of rank higher than one has been considered. In this paper we pursue this approach and study in detail the symmetry scattering for the Riemannian symmetric space associated with the conformal group, X = SU{2,2)/S{U{2) ® (/(2)) 2 SO(4,2)/SO(4) ® SO(2). This space has already emerged several tl'nes in physics[2,3]. The space X is a non-compact Riemannian symmetric space of dimension eight and rank two. Hence, in th:1 eighth dimensional space, the maximal flat subspace is two dimensional. X is generated by the orbits of К = S{U(2) ® U(2)) in SU(2,2). Symmetry scattering for this space is obtained from the asymptotic properties of the joint eigenfunctions defined on X of the SU(2,2)-invariant differential operators. Thes>e invariant differential operators are generated by the two Casimir operators C2 and Сл of second and fourth order, respectively. For physical applications we consider only the operator C2, which is identical to the Laplace Beltrami operator on X. This is a second order differential operator on Hvo variables x and у parametrizing Л, and on the six angies parametrizing В defined below. The eigenfunctions of Сг hence depend on eight variables, two of them with unbounded domains. As usual, in the symmetry scattering approach, the Casimir operator C2 can be written as a sum of an operator depending only on the unbounded variables x and y, the radial part, and an operator acting on the remaining six variables, the transvers part. The eigenfunctions of C% then separate into the product of a function depending only on x and у and an eigenfunction of the transvers part. As an initial step in the SU(2,2)-symmetry scattering, we consider here the case where the cigenfunctions do not depend on the six angle parameters. Already for this situation a very rich class of scattering problems is obtained in the theory. Thus, we deal only with the radial part Cr of Gi and hence with the scattering properties of systems described by the zonal spherical functions. The applications of symmetry scattering that we consider here result from identifying or relating the operator CT with the Hamiltonian of some physical systems. Symmetry scattering then provides the scattering properties of such systems. After setting up the formalism and obtaining explicit expressions for all required quantities like the radial Casimir CT and the scattering operator, we discuss in detail the application of symmetry scattering to describe one dimensional potential scattering of a system, with an internal degree of freedom. Potentials describing confined systems as well as purely repulsive potentials appear. We begin recalling the decomposition of the Laplace Beltrami operator defined on a Rieman- nian symmetric spacc into a radial and a transversal part. For this purpose we first introduce some notation and the polar or Cartan decomposition of a semisimple Li" Group. Let G be a semisimple Lie group of non-compact type with Q its Lie aigebra over IR. Let Q — V + 1С be a fixed Cartan decomposition, ДсРЬе any maximal abelian subalgebra of T and denote by M the centralizer of A in K. An clement H £ A is called regular if oc(ll) / 0 for all a € S, the set of restricted roots. A! denotes the set of regular elements in A. '578

Fix a Weyl chamber A+, we call a toot positive if its restriction to A+ is positive. We have A+ = [H 6 Д|о(Н) > 0} where a runs over all simple roots. Let К = cK, A = ел, A+ = ед+, A+ its closure, A' = e*' and M = e*1. We have the following polar or Cartan decomposition [4]:

Theorem (i) G = KA+K (it) x = А'Л+ • г where £ is the identity coset in G/K. Let G' = К А'К and X' = G' • I we have, (Hi) X' = (K/M) x (A+ • t) in the sense that (kM,a) -* ка-£ is a dilfeomorphisin of K/M x A+ onto X'. Thus В = K/Af can be viewed as a boundary of X = G/K.

Corresponding to the above coordinates we now state the decomposition of the Laplace Beltrami operator on X into a radial and a transvers part [5].

Theorem The Laplace Beltrami operator C2 on G/K has the following form:

a ['C2f\(ke' •£) = С A + £ ma coth(a)/4„ f(ke • f) »6£+

+ £ sinh-2(a(a)) (лф-К/) (ie"),

where к G К, a 6 A+, and С a is the Laplace Beltrami operator on At. with again I being the

identity coset in G/K. Furthermore A„ € A is defined by < Aa, И >-- a(H) for all H € A.

In order to apply the above decomposition to the Laplace Beltrami operator of SU(2,2)/5(t/(2)®f/(2)) wc note that this is a rank two space. The two dimensional root-space + A2, is spanned by the roots a and 0. The set У, of positive, restricted roots consists of the four roots 2a, 2/9, a + 0 and a - в having multiplicities 1, 1,2 and 2, respectively. For the scalar products between the roots we hc.ve < a,a >=< 0,0 >= 1 and < a,0 >= j. And in order to obtain the Laplace-Beltrami operator as a differential operator we take the following realization in the two dimensional Euclidean plane е-4 С G/K:

a = x

л/з H = 2 29i 2 dy'

Since < A0, H >= a(H), it follows that Q Aa = Hi = —ox and similarly The Laplace Beltrami operator on A • I becomes, '579

Further we have: ^ m , 2a = 2x, A — 2— 2a 2a ox д 3 m2ff, = x + \Ду, A2i3 = 2^ + л/3^

о j. я 3x±V?iy .Q± = з а л/з а mai0 = 2, Q±/3 = - , " 2 9x ~2~dy' Taking this into account we obtain for Lhf radial part of the Laplace Beltrami operator of

SU[2,2)/S(U(2) ® f/(2)), Cr,

5У 3 3 32 г = , , г 4 3x2 2 ЗхЭу 4ауг

Л /а Q \ 2 coth(2x)^ + 2coth(x + \Ду) ( — + \/3— ) +

coth

The eigenvalue equation for the radial Casimir operator is thus a second order partial differential equation. This equation can immediately be interpreted either as the Schrodinger equation describing two particles interacting through a two dimensional potential, or as the Schrodinger equation for a particle in a two dimensional space. Symmetry scattering provides in both cases the scattering properties of such systems. However, since these potentials are not immediately related to known physical systems in the above parametrization, we may consider various coordinate transformations, or linear relations between the variables x and y, or equivalently, choose a fixed asymptotic direction for the scattering on the (x,y)-plane and thus obtain from the eigenvalue equation an ordinary second order differential equation that can be interpreted as the Schrodinger equation of a one dimensional system having one internal parameter which is scattered by a local potential. Before going into the details of this application, we first recall the main features of the symmetry scattering method which will permit us to obtain the scattering properties for various applications. The symmetry scattering operator for a Kiernannian symmetric space G/K is given by [6]

SX(A) = where \'s a representation of K, s" is a Weyl reflection sending a € -4+ into —a. The functions

Cfd(A) and Cf.(A) are generalized Harish-Chandra c-functions [7-8]. The. argument of C*.(A) is related to the eigenvalues of the radial part of the Laplace Beltrami operator through [9]

Стф\ = Афх, where Л = -(< A, A > + < p,p >)

with, P = 5 t>€£+ The valuta of A are not completely arbitrary, in fact they are determined by the series of representations of SU(2,2). However, this constraint only applies to the imaginary part of A which determines whether a scattering is possible. The vector 3fte(A) determines the asymptotic direction. It can take any arbitrary value on the plane A giving rise to different scattering problems. '580

As mentioned in the introduction, here we confine ourselves only to the zonal spherical functions, hence, x corresponds to the trivial representation of K. The scattering operator becomes

S(A)=c(A)c(-A)"\ with [7,8,10]

> TT 2~ '°° Г(< »A,Qq >)

^ »€SJ + 1+ < iA,°° >5) + mj°+ < iA'a° >W here Co is a constant, is the set of indivisible positive roots, <, > is the scalar product induced by the Cartan-Killing form, and o0 = • For SU(2,2)/S(U(2) ® U(2)), has four elements a, /3, a + 0 and a — 0 having multiplicities ma = mp = 0, m2o = тгр = 1, rn„+e = 2 and rnD_0 = 2. Using A = A„a + XB0 with Aa, Xp e С and < a,a >= 1 it follows

< A,q> _ . Xff < X,p> _ A„ — < a,a > ° ~2~' +

< X,a + 0> _ A0 + < A,a-0> _ Xa - Xp ~ 2 a" ~ 2 Further, with p = X 22 т"а = 3a + P oeE+ the eigenvalues of CT become,

A = —(< A, A > + ) = -(,Xl+Xl + XaXp + lZ).

If we take these results into account and use the duplication formula for the Г function,

2у/тг2~*'Г(2г) = Г(г)Г(г + i),

we obtain for Harish-Chandra's с function, the explicit form

fi, r(H^i) r(^) l

00 О Г (I + Г (I + i^i) № - W

where CQ is some constant. As shown in [6], the c-function corresponds to the Jost function of a scattering problem described by the Hamiltonian represented by the radial part of the Laplace Beltrami operator in which the numbers A„, Xp are related to the momentum or to internal parameters of the system. Let us now apply the SU(2,2) symmetry scattering to describe the scattering of a one dimensional system with an internal degree of freedom. First we recall that the systems that can

be treated are described by the eigenfunction of the radial Casimir CT. Secondly, the symmetry scattering operator gives the asymptotic behaviour of these functions when translated along a given geodesies, here determined by a ray in the two dimensional flat space A. The general idea of symmetry scattering is to interpret the scattering process geometrically due to the symmetry of the system, i.e., to the non-commutativity of certain of the generators of the underlying Lie algebra. Since these commutators are related to the sectional curvatures of the space, we can attribute symmetry scattering to the curvatures of the subspaces transvers to the geodesies which we choose to follow for the asymptotic translation of the system. Because each asymptotic geodesies will determine a different sequence of transvers manifolds, each having its SSI own curvature, wc see that with each geodesies, a different one dimensional scattering problem can be associated. Now as mentioned above the geodesies can be related to rays in the flat space A [1], hence, by fixing a unit vector in A we have chosen the interaction that the one dimensional system will experience. As a first application of this approach we consider asymptotic directions parallel to a. Along such an asymptotic direction there is no change in the ^-component, hence, Ар = 0 which also implies in the realization considered here, that O^Ll yQ js constant. The eigenvalue equation for the radial Casimir operator becomes:

a2 — + 2[(coth(2i) + coth(x + i I + coth(x - y0)] Фд = ЛФд dx where Л = А „а + \pfi yields A = —(A2 + A J + А„Ад + 13). After a similarity transformation, this eigenvalue equation results

4cosh(2x) 2 2 -8 ФД = ~(A + Э)ФЛ Эх ' sinh(x + yo)sinh(x - y0) sinh (2x)

where ФА = КФЛ. The function F = e-U"^, with

Q(x) = 2(coth(2x) + соth(x + yo) + cothfr — ya)) becomes

F(x) = [sinh(2x) + 2sinh(x)cosh(y0)]"».

The above equation corresponds to the one dimensional Schrodinger equation with potential

'1 cosh(2x) 1_ _g 2 sinh(x + y0) sinh(x - y0) sinh (2x) and momentum 2 2 A +A +A„A/, + 4. The term —8 has been included to the potential to adjust its value at infinity equal to zero. Since the asymptotic direction is parallel to o, the real part of A„ is related in a natural manner to the momentum while Xp acts as an internal parameter of the system. Note that since the real part of A = A„Q + АД/3 is arbitrary, Ap and ya arc independent quantities. Using the general expression for the Jost function of the the SU(2,2)-symmetry scattering,

j с(А<,,Уо) = со + R^ + II^J) (A2-A2)' we obtain the phase shift:

f Г(2^г) Г , 1 г( arg ^ - { + rfi + s^i) (^1) j ' with

A„= ^ where E = it2 is the energy.

In figure 1, we have plotted for several values of the parameter y0 the resulting scattering potential. We find, that the parameter y0 can be associated wilh the "size" of the system. At '582

л Ь

Figure 1: The scattering potential for the SU(2,2)/S(U(2) ® C(2)) symmetry in the cartesian

coordinates realization for fixed value of the variable y0. Figure la corresponds to the value ya = 0. Figure lb corresponds to the value yo = 0.4. Figure lc the point-dotted line corresponds to the value ya = 1.0. Figure Id is a synthesis of figures lb and lc. The axes are in arbitrary units.

the value of x = yo the potentials have an infinite repulsion. Fbr values bigger than yo the potentials drop exponentially while for values smaller than yo the potential is negative having

two infinite traps, one at zero, the other immediately left from ya. This potential may be used for example, to describe a nucleus consisting of two equally charged particles. The phase shift above would then describe the scattering of spinless particles by mesons or by alpha particles. Using other symmetries more traps in the inner part of the system would be available opening the application to more complicated systems.

References

[1] E. F. Wehrhahn and R.D. Levine, Europhys. Letters 16 (8), pp. 705-710 (1991).

[2] A. O. Barut and R. Raczka, Theory of group representations and applications, (Second Edition, World Scientific, Singapore, 1986).

[3] R. Wilson (editor), Proceedings of the San Antonio conference on No-i-compact groups and applications, (Plenum press. 1993).

[4] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, p. 402. (Academic Press, 1978).

[5] S. Helgason, Groups and Geometric Analysis, Integral Geometry, Invariant Differential Operators, and Spherical Functions, p. 310. (Academic Press, 1984). 583

[6] R. F. Wchrhahn, Phys. Rev. Lett. 05, 1294 (1990). R. F. Wehrhahn, Yu. F. Smirnov and А. M. Sliirokov, J. Math. Phys. 32 (12) p. 325-1 (1991).

[7] S. llclgason, Ann. оГ Math. 98, 451-480 (1973). Adv. Math. 22, 187-219 (1976);

[8] Harish-Chandra, Am. J. Math. 80, 241-310 (1958); Am. J. Math. 80, 553 613 (1958); Acta Math. 116, 1-111 (1966).

[9] S. llclgason, Groups a net Geometric Analysis, Integral Geometry, Invariant Differential Operators, and Spherical Functions, p. 427. (Academic I'ress, 1984).

[10] G. Gidikin and F.l. Karpelevic, Dokl. Akad. Nauk SSSR 145, 252-255 (1961). '584

SPHERICAL FUNCTIONS FOR THE QUANTUM GROUP su ,(2) 1 P. Winternitz

Centre dc recherches mathcmaliques, Universit6 de Montreal C. P. 6128, Succ. A, Montreal (Quebec) Canada H3C 337 and G. Rideau

Laboratoire de Physique Thcoriquc et Mathematique, Universite Paris VII Tour Centrale, 2 place Jussieu, 73251 Paris Cedex 05, France

Abstract The representation theory of the quantum group su,(2) is used to introduce g-analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The method amounts to an extension of variable separation from Laplace equations to certain differential-dilation equations.

1. Introduction One of the present authors (P. W.) had the privilege of working for his Ph. D. degree under the guidance of Professor Yakov Abramovich Smorodinsky in Dubna, starting 30 years ago. The first article he was given to read was one by N. Ya. Vilenkin and Ya. A. Smorodinsky on invariant expansions of relativistic scattering amplitudes [1]. This paper introduced a variety of different types of "spherical functions" as basis functions for irreducible representations of the Lorentz group, realized an 0(3,1) hyperboloids. This turned out to be a pathbreaking article that lead to many interesting developments in the representation theory of noncompact groups and their physical applications. In addition to the original idea of two-variable expansion of scattering amplitudes [1-5], this includes a group theoretical approach to the separation of variables in partial differential equations [2,6-9]. Other scientific programs influenced by Ref. 1 are those of systematically classifying subgroups of Lie groups [2,10,11], or of generating completely integrable Hamiltonian sys ems in various spaces [2,12-14]. A much more recent development is the current interest in quantum groups [15-19] With it carre the realization that the so called qr-hypergeometric series and other ^-special functions have a similar relation to quantum groups [20-24] as the classical special functions have to Lie groups [1,25,20]. The pmpose of this presentation and an accompanying article [27] is to use the theory of

irreducible representations of the quantum group su4(2) to construct g-spherical functions on an ordinary (commutative) sphere S2, i.e. the homogeneous space SU(2)fO(2). The motivation for this study is two-fold. First оГ all, if quantum groups are to play a role in physics, then the corresponding g-special functions should occur in physics as wave functions, or in sor .c similar guise. Secondly, a systematic use of quantum group representation theory should provide methods for introducing new special functions and obtaining new properties of known functions

'Invited talk presented by P. Winternitz at International Workshop "Symmetry Methods in Physics" in memory of Ya. A. Smorodinsky (Dubna, Russia, July 1993). '585

We hope that this article demonstrates how strong the influence of one aspect of Ya. Л. Smorodinsky's work is on research being conducted now, almost 30 years later.

2. The Quantum Algebra su,(2) and its Representations 2.1 Realization of the Quantum Algebra by Differential-Dilation Operators

The algebra su7(2) is a deformation of the Lie algebra лц(2) and is characterized by the commutation relations

„ill, _ „-2//3 [//з,Я+] = //+, [//3,//-] = -//_, [Я+,//_ 1 = * J., , (1) where q is some real number. For q —> 1 Eq. (1) reduce to the usual su(2) commutation relations

[//3,Я+] = //+, [//3,//-]=-//_, [//+,//-] = 2Я3. (2)

Finite-dimensional irreducible representations of su,(2) are characterized by an integer or half- integer number J. Basis functions for these representations can be denoted \JMq) and satisfy

H3\JMq) = M\JMq)

H+\JMq)=ainill\JM+\q) (3)

H. I JMq) = ai4 IJM - 1 q) vitli 1/2 qJ-M+1 _ q-J+M-1 qJ+M _ (j-J-Wl «л/. Vz7r> «F7/—J • (4) For q = 1 Eqs. (3) and (4) reduce to standard su{2) formulas with

"лм = «л/ = ((^ + my -M+ l)]"2. (5)

Thus, in Eq. (3) we have chosen a basis of eigenfunctions of the operator //3, corresponding to a iionduformcd {/(1) subalgebra of .su,(2). We have = = 0, hence there is a highest and lowest weight N = ±J and the representations are finite-dimensional. We shall now pursue an analogy with the construction of spherical functions HJM(0, for sn(2). These functions can be viewed as being defined 011 a sphere S2 defined by the relations:

xo = ^ sill 0 cos ф, IJ„ — i sinOsin ф, ?o=jcosfl. (6)

Using a stereographic projection S2 —• R2

we can see the spherical functions as being defined on the real plane (x,y) with

х = рсояф, у — рз\пф, p = col 5 (8)

0 < 0 < ir, 0 <

We shall also use the complex variable

z = x + iy = pc*. (9) 586

We now need to construct operators H3, Я+, and Я_, acting on functions }{в,ф) on S2, or equivalently, on functions /(г,г) on the plane (г,г). The operators should satisfy the relations (2). An "inspired guess" yields the following relations

Ha = -zd, + zdt - N

U+~ z q-q~> 4 ' * Ч-Г1 (10)

„гО.+N _ n~iB,~N 1 JB, _ - //_ = г2 q'B,-N/i + ^a.+N/iil _ г q-9" whcre N is an integer or half integer parameter. The operators q'a' and qi0' act like dilations:

q'a-S(z,z)^f(qz,z)

qlB'f(z,z) = f(z,qz) (11) and it is easy to verify that H.t and H3 satisfy the commutation relations (1). For q = 1 relations (10) reduce to su(2) expressions

2 H3 = —zdt + zdi — N, Я+ = -d, - z% + Nz, 11. = z dt + dM + Nz. (12)

For.q = I we sec that Я±, as well as H3 are first order differential operators. For q ф 1 the operators become nonlocal: in addition to derivatives, they involve dilations of the independent variables.

S.S Basis Functions for Irreducible Representations We will now look for a realization of the basis functions

\JM ?) = Фк(М) (13) satisfying Eq. (3) with //„ as in Eq. (10). The Casimir operator of su,(2), commuting with Ям, p. = 3, ±, is

iFrom Eq. (3) we deduce that the basiasis functions of an irreducible rrepresentatioc n satisfy

. 2

For q —» 1 Eq. (15) reduces to the standard angular momentum relation

J c*i,N = J[J + I)V MN. (I6)

We are after explicit expressions for the basis functions (13) that for q = 1, N = 0 reduce to su(2) spherical functions (and for N ф 0 to Jacobi polynomials). To do this we put:

NM/ imN *i= K«,9- *QJ4Wi,M,W > (17)

IJ = zz = cot'l ^ '587

To get a finite-dimensional representation (of dimension 2 J + 1) we request the existence of a highest and lowest weight, i.e.

=0, H.VtJNq = 0 (18) and by analogy with the q = 1 case, wc put

Rjn4 = const. (19)

The first of relations (18) implies a functional relation for QJ.,(T)), namely

Qj^Di 1 -I- v) = Qj,(v){ 1 + 4-"v)- (20) Its solution in terms of Exton's q-binomial function [28] is

QjAi) = *o{J;-;q*-,-vq~") (21) which for q -t 1 reduces to Qji(q) = (1 -f rj)~J. For q = 1 the expression Ritmiv) are polynomials related to the Jacobi polynomials. For general q they are also polynomials, satisfying certain relations, following from Eq. (3). They arc

! 2M J (om+1,,) 1«M.и.лиОг) = ^гргЬО + я'^Мы^п) + (1+ 4- vW MN,m (22)

M+N M N 1M J Rif.,^0) = —-h 0 + - 9~ - (l + q r,)R MthW (23)

The normalization constant Ni,Nq in Eq. (17) was chosen to be

where [e,q]! denotes a ^-factorial and fa, g] a ^-number [28,29]

P=1

J Relations (22) and (23) are nonlocal, in that the functions R MNq are evaluated at г/ and 2 rom 1 We can eliminate R\f/ifq{q v) f these two "difference-dilation' equations and obtain a recursion relation, namely

a («M+I.,) ^M+1,jv„C?) + Ям-UN,яМ = {[M + JV, 7] - [M - N,q]v}Ri,N,iv). (26)

3. The g-Vilenkin-Wigner Functions and

J R MN,{V) = \J ~ N,q)\\J - M,q)\x

x V (-')V (27) v MV -M- M'V -N- k,q]\[M + N + fc,?]! '588

and thus the basis [unctions (17) are completely determined. With an appropriate choice of

the normalization constant CJN4 in Eq. (24) we can rewrite them as

*mjv,(M) = 4=([2J + l,

1, = гг = |з|, 4= cos 0. (29)

For q = 1 these functions reduce to the functions PjbN(cosO) extensively studied by Vilenkin J [25] and directly related to the Wigner rotation matrices [30] d. MN[S). They are also related to Jacobi polynomials

1/2 (J - M)l{J + M)\ I'mn(0 = 2~M (i)~N+M x t {J - N)\(J + N) 1

x(l _ ^)(-W+M)/2(] + £)(JV+M)/Jpfi>.»)^) (30)

к = J — M, p = M -N, q = M + N.

The functions Pjhp/{£) are usually introduced as rotation matrices, but they can just as well arise as basis functions for representations of su(2). It is this second role, that of basis functions for irreducible representations, that has been generalized to the quantum group su,(2). Most properties of the ordinary (q = 1) Vilenkin-Wigner functions have their (/-analogues. We shall just list some of them and refer to a related article [27] for proofs Recursion formula

i([M + JV,7]T'/2 —[M— N,qWn)PJMm =

1 2 = -([./ — M,q][J + M + 1, ?]) ' ^Y/t+|i/y1,+ (31)

+([./- M + l,q][J + M^yfpi^.

Generating function, defined as

= ([J (32)

has the form

J-N-1 J+N-1 J] (bjq-J+N+1+2> + iq-1'1) J] H",-"+,+2'-*y/2)- (33) p=0 p=0 For 9=1 this simplifies to the well-known generating function

J+N cos J&M = FH = KJ_N)JJ + NWn ( \ + 5) x '589

/ в 0\J~N x(u)cos- + tsin-J (34)

Symmetry relations PJmn№ = (35)

flWf) = (36)

J 2 QЧМЧЯJAW + ) ' ) ,P /с, ,,7\ =1 CJ,4—п. r„i l^O QMn) where

<*L+ifi,4 = 9(t+1)Jo-i/2,.i for J half-odd integer

_ \+g 1 02(O) '^"l + r'v^Wo)'

(02(u) and 03(u) are ordinary theta functions). It is now quite natural to introduce 7-spherical harmonics in the same manner as ordinary harmonics, namely

1 M Ф) = [J_ M|g]!^o,(cos 0)t~ *. (38)

Pj,(cos 0) = P^( cos 0) = <"Qj,( l)nia,(q). (39)

Notice that Pjq(cos в) for q ф 1 is not a. polynomial in cos 0 in view of the properties of Qj4(q). Finally we mention the relations between the ^-Vilenkin-Wigner functions and other q- functions in the literature. These relations are best written in terms of the polynomials ЯА(л,ч(г/) of Eq. (27). For instance, in terms of the basic hypergeometric series [28,29]

[e,fc;?] = [a,9][a+ \,q)---[a + k- 1,9] we have, for M + N > 0

= + ~lN-J-,M + N+l-,q- -V). (41)

Using the "little

= [^ff,,]!^'4^'^'"'' ' (42)

(for J - M,J-N).

4. Conclusions The full power of Lie group theory in its application to special functions only becomes apparent, when afplied to partial differential equations and combined with the separation of variables. One way of viewing the results presented above is that we have extended the Lie algebraic treatment of variable separation to ^-special functions and to quantum groups. 5!Ю

The separation occurs in differential-difference equations of type (14) and (15), rather than in Laplace-Beltrami equations. In the future we plan to apply similar techniques to other quantum groups and hence to other types of (/-special functions.

Acknowledgements The research of P. VV. was partially supported by grants from NSERC of Canada and FCAR

References

[1] N. Ya. Vilenkin and Y. A. Smorodinsky, Zh. Eksp. Tear. Fiz. 46 (1964) 1793.

[2] I'. Winternitz and 1. Kris, Yad. Fiz. 1 (1965) 889.

[3] P. WinLernitz, Y. A. Srnorodinsky, and M. B. Sheftel, Yad. Fiz. 7 (1968) 1325.

[I] M. Datimens and P. Wintcrnitz, Phys. Rev. D21 (1980) 1919.

[5) J. Bystricky, P. LaKrance, F. Lehar, F. Pcrrot, and P. Winternitz, Phys. Rev. D32 (1985) 575.

|6] Iv G. Kalnins, W. Miller Jr., and P. Winternitz, SIAM J. Appl. 30 (1976) 630.

(71 W. Miller Jr., J. Patera, and P. Wintcrnitz, J. Math. Phys. 22 (1981) 251.

[8| W. Miller Jr., Symmetry and Separation of Variables (Addison Wesley, New York, 1977).

[9) E. G. Kalniii:!, Separation oj Variables for Hiemannian Symmetric Spaces of Constant Curvature (Longmans, Essex, England, 1986).

[10] J. Patera, P. Winternitz, and II. Zassenhaus, J. Math. Phys. 15 (1971) 1378 and 1932; 16 (1975) 1597 and 1615; 17 (1976) 717.

[II] J. Patera, R. T. Sharp, P. Winternitz, and 11. 'Zassenhaus, J. Math. Phys. 17 (1976) 977 and 986; 18 (1977) 2259.

[12] 1. Fris, V. Mandrosov, J. Smorodinsky, M. Uhlir, and P. Winternitz, Phys. Lett. 16 (1965) 354.

[13] A. Makarov, J. Smorodinsky, Kh. Valiev, and P. Winternitz, Nuovo Cim. A52 (1967) 1061

[И] M. A. del Olmo, M. A. Rodriguez, and P. Winternitz, J. Math. Phys. 34 (1993) 5118.

[15] V. G. Drinfeld, in Proc. Int. Congress Math. Vol 1 (Amer. Math. Soc., Providence, RI, 1986).

[16] M. Jimbo, Lett. Math. Phys. 10 (1985) 63; 11 (1986) 247.

[17] S. L. Woronwicz, Commun. Math. Phys. Ill (1987) 613.

[18] Yu. 1. Manin, Quantum Groups and Noncommutative Geometry (Centre de recherches mathematiques, Montreal, 1988). '591

[19] L. D. Faddeev, N. Y. Reshetikhin, and L. A. Taktajan, Leningrad. Math. J. 1 (1990) 193.

[20] R. Floreanini and L. Vinet, Lett. Math. Phys. 22 (1991) 45; J. Phys. A Math. Gen. 23 (1990) L1019; J. Math. Phys. 33 (1992) 1358.

[21] E. G. Kalnins, H. L. Manocha, and W. Miller Jr., J. Math. Phys. 33, (1992) 2365.

[22] L. L. Vaksman and Ya. S. Soibelman, Fund. Anal. PriL 22 (1988) 1.

[23] N. M. Atakishiyev and S. K. Suslov, Tear. Mat. Fiz. 65 (1990 ) 64.

[24] S. K. Suslov, Russian Math. Surveys 44 (1989) 227.

[25] N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (Amer. Math. Soc., Providence, RI, 1968).

[26] W. Miller Jr., Lie Theory and Special Functions (Academic Press, New York, 1986).

[27] G. Rideau and P.Winteruitz, J. Math. Phys. 34 (1993) 6030.

[28] H. Exton, q-Hypergeometric Functions and Applications (Ellb Harwood, Chichester, 1983).

[29] G. Gasper and M Rahman, Basic Hypergeometric Series (Cambridge Univ. Press, Cam- bridge, 1990).

[30] E. P. Wigner, Group lheory and its Applications to the Quantum Mechanics of Atomic Spcctra (Academic Press, New York, 1959).

[31] Т. H. Koomwinder, Proc. Nedcrl. Akad. Wetensch. A92 (1989) 97; SIAM J. Math. Anal. 22 (1991) 295. '592

INFLUENCE OF COLOUR SYMMETRY ON STRING TENSION V. I. Yukalov, E. P. Yukalova and A. A. Shanenko

Laboratory of Theoretical Physics Joint Institute for Nuclear Research P.O.Box 79, Dubna, Russia

Abstract The ratio of the fundamental string tension in the SU(3) pure (quarkless) gauge theory to that in the SV(2) gluonic system is estimated within a statistical model approach to the description of the QCD thermodynamics. The obtained estimation is in agreement with the latticc evaluation of the ratio.

One of the most important problems of the nonperturbative QCD is to clarify the dependence of string tension on the numbers of colour and flavour degrees of freedom of the theory. In this report the ratio of the fundamental string tension in the SU(3) gluonic system to the tension of the fundamental string in the SU(2) quarkless gauge theory is explored in framework of a nonperturbative statistical model of the deconfinement proposed and substantiated in papers [1-3]. The main peculiarity of this model is the consideration of the nonstratilicd mixture of quark-gluon plasma and hadrons. To put it in another way, we do not ignore the possibility of simultaneous generation of colour particles and colourless ones in hadron-hadron collisions. As follows from experiments, such a generation is impossible if hadrons collide in vacuum where there are no neighbouring particles. But in hadron medium the conditions of particle existence differ from those in vacuum. So, investigating the thermodynamic behaviour of strong interacting matter one should not give up simultaneous production of colour particles and colourless clusters in hadron-hadron collisions. Just considering the nonstratified mixture of quark-gluon plasma and hadrons we have obtained a good description of the deconfinement phase transition which agrees with the lattice predictions in the both pure gauge cases (SU(2) and SU(3)) and for the full QCD variant [1-3]. The model Hamiltonian of pure gauge system has the following form d l{ = 11J s) + *) * +

2 + (V-V + + Un]) Ф„;(х,s)dx- n j J J -CV, (1) where

y U3 = Ap' ,

Uni = n\Ap-> - A(p - р3Г + -

T С = -j^- V" + j-2-A(p - Ряу-> + ^(p - Ps)\ (2)

In expression (1) ф3(х,з) is the field operator of unbound gluons, фп](х1.ч) stands for the field

operator of n - gluon glueballs of the j - th sort; the both operators фд(х, s) and t/>„ j(x, s) are '593 of the boson type. In formulae (1) and (2) p denotes the full gluon density of the system which obeys the expression p=Pt+52apn" nj here p, is the density of unbound gluons, pnj is the density of nj - glueballs. Besides, г V denotes the system volume and Ф2 = J Ф22(.r)d r, where Фгг( ) stands for an effective potential of the interaction of two - gluon glueballs. Details of the choice of glueball specifications as well as reasons for such a form of the Hamiltonian can be found in Refs.l

-3. At last, A, 7 and Ф2 are free parameters of the approach. The best agreement of our description of the deconfinement phase transition with the lattice results turns out to be at 3 2 7 = 0.62, = 175 Mev, Ф2 = 5 • 10" Mcv~ for the SU(2) gluonic system, and at ,/(3 гИ) 3 2 7 = 0.62, Л " = 225 Mev, Ф2 = 2- Ю" Mev~ for the SU(3) gluon theory [1-3]. In these cases the deconfinement is a second order phase transition in the SU(2) theory and first order one in the SU(3) system. Using values of the parameter A it is possible to estimate the ratio gStl(3) osu(i) ' where a denotes the fundamental string tension. In order to do this let us consider Us , the energy of the interaction of an unbound gluon with surroundings, in gluon plasma within the relativistic potential model [4,5]. Here we have +00 U, = p9 f s(r)^(r)dr . (4) Jo In formula (3) з{т) is a smoothing function [6] which takes into account gluon correlations. For the potential of gluon - gluon interaction the following approximation [7]

Ф Oadj r (5) is valid in the range of distances (яг 1 fm) corresponding to the values of unbound gluon density near the deconfinement phase transition, o„d, being the tension of the adjoined string. Substituting relation (5) into formula (4) we derive r+oo ид = <та^ря / s(r)rrf r . (6) Jo At the present time the exact calculation of the integral in formula (6) is impossible because s(r) is not known. But one can roughly estimate the integral with the help of the approximation

s(r) = 1 ( r < r„ ) ,

s(r) = 0 (r > r0) ,

1 3 where r0 stands for the screening radius in gluon plasma and r„ ~ pj ' .Under this approximation the evaluation of the integral results in

/•+00 - JT s(r)rdr = -j^. j: Рз As soon as the obtained estimation is sufficiently rough we should suppose the following approximation u, = (7) P3 '594 for the energy of the interaction of an unbound gluon with gluon plasma in framework of the potential model. In expression (7), I) and a are constants which are independent of a colour group. Thus, comparing (7) with (2) one can conclude that 0 = 7 and /lj(/(„) = DaadjSU^ . Now, taking into consideration the relation [7]

a <*SV{n) = 2nl °4SU(n) , we can get the relation

^suft) _ 32 /W(3) _ 2 43 ^ <^sup) 27 Asu{i\

The lattice estimate of this ratio is 1.9 ± 0.7 for the dcconfincinent temperature 0jec = •ПО Mtv{SV{2)) and 0iec = 225 Mcv(SU(3)) [8]. Note, that these values of 0j„ arc usually accepted for pure gauge theories [8,9]. Our numerical investigation of the thermodynamics of the system with Hamiltonian (1) is also in agreement with these dcconfinement temperatures.

References

[1[ A.A.Shancnko, E.P.Yukalova and V.I.Yukalov, Phys. At. Nucl. 56 (1993) 372.

[2] A.A.Shanenko, E.P.Yukalova and V.I.Yukalov, Hadr.J. 16 (1993) 1.

[3] A.A.Shancnko, E.P.Yukalova and V.I.Yukalov, Physica A197 (1993) 629.

[4] J.Carlson et al., Phys. Rev. D27 (1983) 233.

[5] N.lsgur and S.Godfrey, Phys. Rev. D32 (1985) 189.

[6] V.I.Yukalov, Nuovo Cim. A103 (1990) 1577.

[7] Yu.A.Simonov, Yad. Fiz. 54 (1991) 192.

[8] B.Pctersson, Bielefeld Univ. Preprint Bl-TP 92/58,1992.

[9] J.Engels, J.Finberg, K.Redlich, H.Satz and M.Weber, Z. Phys. C42 (1989) 341. '595

FRAGMENTS OF REMINISCENCES AND EXACTLY SOLVABLE NONRELATIVISTIC QUANTUM MODELS Boris N. Zakhariev Laboratory of the Theoretical Physics, Joint Institute lor Nuclear Research Dubna, 141980, Russia; e-mail: [email protected]

When I was still a student of the Moscow University, my friend V.B.Belyaev offered me to go to the Landau seminar at the Institute of Physical Problems and ask A.B.Migdal to be our diploma chief. So we did, but it happened that during the whole seminar break Migdal discussed something with Landau and the Migdal's aspirant suggested us to choose professor Ya.A.Smorodinsky who was free at that moment instead of A.B.. I am sure now that it was a happy idea. Our diploma were devoted to the double beta-decay. In his review on the defence of the diploma Ya.A. wrote that we "have shown the interest in physics". It is remarkable that this aspect of our work was not less important for Ya.A. than the scientific results themselves. It remains a guide for me for the whole scientific life not to lose the interest and helped me to search for the beautiful effects in physics and theoretical results with surprizes. • After diploma Ya.A. invited us to work at Dubna where he just became a chief of a sector at the Theoretical laboratory of the just organized JINR. It seemed to us then that it would be better to become post-graduates of MGU instead of going "so far" from Moscow, but fortunately Ya.A. didn't agree with us. So, it turned out that we began our carricr due to Ya.A. in contact with two famous scientific schools of Landau and Bogolyubov. Unfortunately, there were not the best relations between these schools (with losses for both of them) and all the more wa.'. important the part of Ya. A. as the bridge between them. And we are glad that we were involved in this ts far as possible. We were unaware that just at that time Landau was busy with his Nobel prize investigations, lie had a peculiar method of organizing his seminar. He had preliminary discussions with reporters before seminars. It was very useful for them. After that Landau was especially good oriented in the subject and could help his collaborators to understand it. But during the seminar he was used to interrupt the reporter, asked him to jump from one part of the presented paper to another without taking care of those who couldn't follow him. Of course, it had also a positive aspect of mobilizing the auditorium. And although it was difficult for us to understand something at these seminars, I made different profits of them. For example, I thought: why these wise men cannot explain their interesting ideas to the whole auditorium (it seemed to us that only first rows can follow the discussions). Later I decided that in my reports and papers I should take into account the weakest listener and reader. You can check now particularly in the references I give here whether I succeeded in this. From time to time Landau was quite severe to the reporters. For example, after an error of a speaker Landau could, as a punishment, stand up and go out followed with his colleagues neglecting the emotions of the unhappy reporter. These actions were not loo pleasant but nevertheless they stimulated his school to work harder. Ya.A. was also sometimes a victim of hard treatment of his teachcr. May be, this was the cause that Ya.A. even outside the Landau school was often not very confident during his talks and after an arbitrary question he used to rub off the last formulae from the blackboard as if there was an error. He also used to go out from the first row during seminars (to check the self-confidence of the speaker ...?). Fortunately it hindered the seminars not too strong. But his own learners Ya.A. treated much more delicate and we greatly appreciate this. '596

Very important for us was the suggestion of Ya.A. to pass the theoretical minimum examinations by the Landau school. It is remarkable that when we met Landau one day somewhere on the stairs at the university and said him about our wish, he immediately gave us the program of his first examination without asking us who we are and other formalities what was absolutely unusual at that time of prosperity of the bureaucracy. Such openness can serve as a brilliant example also now. Every wish to work must be welcome without any obstacles. Another remarkable feature of this thcor. minimum was the very definitely restricted program. You could be sure that no unexpected questions would appear at the examination. So, you should not fear and waste the nervous energy as it was with me at the university and only care about understanding the given parts of books from the series by Landau and Lifshitz. Besides, the fact that all the theor. minimum textbooks wore written by the same authors in the same style made them more and more habitual when you pass from one book to another during the examinations. So, I got more fundamental knowledge of the subjects with less expense in comparison with the usual university courses. Landau considered it to be not clever to waste the creativity where the problem can be solved by standard methods. Landau even turned my smart friend out of the examination not once when he calculated integrals originally. Although it happened so that I had no common publications witv Ya.A, all my investigations were more or less stimulated by him. He sent us to Ya.B.Zeldovich who posed for us the mu-molecular problems wonderfully clear in contrast with the problems of Ya.A, which were instructive, but as a rule too difficult for us and often even for Ya.A. himself (so, Ya.A. suggested us to solve the problem of violation of parity before it was done by Nobel prize people). Under the guide of Zeldovich and with the help of A.D.Sakharov together with S.S.Gerstein we succeeded in getting important results in mu-physics. These collaborations gave me the valuable experience and qualitative feeling in multichannel quantum mechanics, which later allowed us with V.D.Efros, V.P.Zhigunov et al [1] to generalize the method of hyperspherical harmonics to the scattering problems after stimulating information from Ya.A. that nobody knows how to do it. My idea was to subtract from the many-body function the almost known asymptotics of free motion of fragments which prevent the hyperspherical expansion. The most interesting compact middle part of the wave function ("quantum kitchen") where the features of the nuclear reaction are determined is then suitable for the expansion. And the asymptotics appear in equations through the source terms with constant partial amplitudes of the waves of reaction products (that was not dangerous for the solvability of the problem). The same idea allowed us also to overcome the difficulties in generalizing the Feshbach unified theory of nuclear reactions to the collisions with rearrangement of particles and break-up processes. So, the theory became really universal. 1 worked then already about ten years in the sector of Ya.A. I don't think that anybody else at that time could tolerate so long a subordinate who has no common papers with his chief even in spite of my comparatively big (in my scale) success. Finally I understood that it was time to make Ya.A. free of this ballast. In spite of some internal discomfort (may be for both sides) our relations didn't become worth. I think it was a positive precedent at our laboratory (or even wider) that science must not suffer from nuances of relations of individuals and we and our work only benefited from our smooth divorce. In our first book "Methods of Close Coupling of Channels in Quantum Scattering Theory" we summarized our experience in the direct quantum problem and our understanding how to assemble wave functions of simple blocks as in the children toy "Constructor" (popular in Russia). It appears in Energoatomizdat where Ya.A. was ai; active member of the editorial board. Fortunately, accidentally Ya.A. didn't mention the process of acceptance of this book by the publishing house (as he said me later). Ya.A. was the first who pushed us in the direction of quantum inverse problem (QIP) that became later my main business. But he himself and his pupils were elastically scattered this time from QIP. Ya.A. after that didn't like QIP for ever and for me it also remained rather long '597 time the tiling in itself. I got another push in this direction from L.D.Faddeev by his three- dimensional QIP. One of my students (S.A.Nijaegulov to whom 1 am greatly thankful) plunged then bravely into the QIP. Helpful for us were also the papers of American mathematicians Case and Kac on the discrete QIP. Instructive was also my practice of the finite dilferencc approximations when as in the microscope one can see the simple although model coupling of wave functions and potentials in the neighboring points. I included all our understanding of QIP into our second book "Potentials and Quantum Scattering. Direct and Inverse Problems". It was evidently necessary to publish such a book at that time becausc the first hook by Agranovich and Marchenko about QIP was written more than 20 years ago and there appeared periodically different reviews on the subject. But this time Ya.A. sent our suggestion to write the book to the predisposed referee who prevented the publication. His only objection was: "There is no need to publish now any book on the QIP". As the proof of the mistake of this decision, Chadan and Sabatier soon published their well known book (about QIP) which wc soon translated into Russian. So we let them pass ahead and we published our own book later (nevertheless, in Energoatomizdat [2]). it wasn't competing with their book because of minimal overlap and was met with positive reviews. Ya.A. himself praised it, although he didn't like QIP. Of course, the delay of the publication lias also positive consequences: we have w-'M.en our book more carefully. It was also useful to discuss with Ya.A. all my recent results on QIP. There was a strong violation of symmetry in the development of the two halves of quantum inech.4iiics: its direct and inverse problems. QIP was for a long time as its invisible (for most physicists) side. QIP was developed with a big delay mainly by mathematicians who were satisfied by the proof of the existence theorems. It became clear that wc should ourselves develop the physical intuition. In particular, by analyzing the "inverse problem pictures" we can get a deeper insight into the laws of the microworld and acquire the ability to make the qualitative predictions without computers and formulae. Thus appeared the idea to write a new book of the type of "Picture Book of Quantum Mechanics" by S.Brandt and II.D.Dahmen, but about QIP: "Lectures on Quantum Intuition" [3]. QIP allows another approach in quantum science: the spectral and scattering management, the construction of quantum systems with desired features. So, wc can shift up and down the chosen energy levels of discrete bound states even inside the continuum without shifting other levels. We can make the discrcLc spectrum denser or sparser in the given placc. And what is very important, wc can understand qualitatively what a particular shape of the potential perturbation is needed to do this (see the simple corresponding pictures in [2]). It is the motion management in vertical direction. But there arc other fundamental spectral parameters norniing constants which, besides the energy level positions, compose the coinpletc set of spectral control levers. We can change the desired norming constants and push the chosen bound states to the left or to the right (sec. the pictures for the motion of eigenstates in horizontal direction and corresponding potential perturbations with their simple explanations in [2]). We can destroy and creatc the bound stales. Wc have understood this graphically just before the conference [4]. It is interesting that the independent parameters (levels and the corresponding norming factors) are connected nevertheless in some sense. In the process of going to the limit of small or large norming constants, the bound state is gradually removed (pulled out) from the original potential or pressed into the infinite vertical potential wall and so disappears. The notion about the possibility of resonance management appeared to be closely connected with the analogous carrier of the chosen quasibound states [3]. We have understood visually how to split the continuous spectrum and create the forbidden zone at the chosen energy height (and which shape of the periodical perturbation performs this). It is simply connected with the variation of norming constants of eigenstates of the Sturm-Liouville problem on the period. '598

We can manage the reflection of waves and even construct the absolutely transparent systems with no reflection at any energy. We have drawn ccrtain pictures of such multichannel potential matrices and simply understood the new mechanism of suppression of the reflection when the reflection from potential barriers is destroyed by the decay waves from other channels [5]. It is instructive to compare the usual quantum mechanics and that with discrete variables. The exotic behavior of waves on the lattices allows a better understanding of interchannel motion or mixing of configurations. For example, there arc many common features and wonderful peculiarities in the algorithms of spectral management in the discrete quantum mechanics (see [2] 1992). We can here better understand the influence of the nonlocality of potentials with examples of its minimal manifestation (coupling of neighboring points). We get also here the concept of the "Schroedinger equations of order higher than 2" which are important for the interchannel motion. It is time to explain all these things to students, although it is still not known to most specialists. In this respect our Laboratory is now a unique place where this information is collected in a kind of the textbook. Ya.A. has done very much to spread the quantum culture. It is important because the situation with information in the world is far from excellent. Of 5.5 billions of people on the Earth hardly one tenth can use the achievement of the contemporary culture. The others get only its poor fragments, as if the wealthy minority fenccs itself by the iron curtain out of the other humanity (nonclever egoism ...?). Although it is the efTect of the self-consistent influence of all the people, but the distribution of the responsibility for the situation is proportional to the possibilities. So, the most responsible are those who has more possibilities. And the partial realization of the influence of Ya.A. and the continuation of his noble deeds is that after example of his enlightenment activity we have organized already two school-seminars "Secrets of Quantum and Mathematical Intuition" in our laboratory to bridge the gap between the now achievement in our science and the level of understanding the quantum world among the teachers of physics. 1 used these schools to spread my last book (by copying its LaTex variant) before its publication. I have written here about some peculiarities of my first teacher in science to give an outline of a real individual. We like him with all his merits and weak sides. In conclusion it is worth to say that there is the social law of momentum conservation. The waves which Ya.A. excited in our brains and soils don't disappear. They are spreading wider and wider even if we don't mention it. This conference is also an obvious manifestation of his influence: Ya.A. continues to organize us. When we communicate we transfer part of us one another. A significant part of me is that of Ya.A. So he is a virtual coauthor of this report. This is one of the ways of our unlimited continuation in space and lime.

References

[1] Zhigunov V.P., Zakhariev B.N. "Methods of Close Coupling of Channels in Quantum Scattering Theory", Energoatomizdat M.1974. [2] Zakhariev B.N., Suzko A.A., "Direct and Inverse Problems", Springer-Verlag, Heidelberg, 1990. Zakhariev B.N., Sov.J.Part.& Nucl.23 N5 (1992) 603; ibid 21, N4 (1990). [3] Zakhariev B.N., "Lessons on Quantum Intuition", to be published (Energoatomizdat,, os- cow, 1994); Preprint JINR P4-93-427, Dubna 1993. [4] Zakharev B.N., Chabanov V.M. Preprint JINR P4-93-179, Dubna 1993

[5] Zakharev B.N., Chabanov V.M. Preprint JINR P4-93-111, Dubna 1993; Phys.Lett.B319, 13, 1993. '599

QUANTUM DEFORMATIONS FOR THE DIAGONAL R-MATRICES В. M. Zupnik

Research Institute of Applied Physics, Tashkent State University, Vuzgorodok, Tashkent 700095, Uzbekistan

Abstract We consider two different types of deformations for the linear group GL(n) which correspond to using of a general diagonal It-matrix. Relations between braided and quantum deformed algebras and their coactions on a quantum plane arc discussed. Wc show that tensor-grading-prcserving differential calculi can be constructed on braided groups , quantum groups and quantum planes for the case of the diagonal R-nialrix.

Deformations of the linear group GL(n), vector spaces and commutative algebras are inten- sively discussed in the theory of quantum groups [1-4]. More simple (G, /)-graded deformations of Lie algebras are well known [5,6] and recently their R-matrix generalizations (braided algebras) have been considered [7,8]. We shall discuss the simplest multiparamctric deformations of the commutative algebra and linear group generating by a diagonal До-matrix [9]. The general diagonal unitary R-matrix has the following form: (fio)'L = (1) where o{ji) are complex deformation parameters

q(ji) = 1/ФЛ . 4{H) = 1 (2)

This R-matrix is a partial case X = 1 of the multiparametric nondiagonal R-malrix [10]. Let us define a quantum plane [2] as the formal-series algebra with generators x' (i = 1,2... n) «• x> = (/?o)L ** xm = (3) One can treat the deformed groups as different cotransformations on the quantum plane. We shall study two types of GL[n) deformations using Д/з-matrix (I): braided group DGL(n) and quantum group GLp(n). Symbol D in our notations corresponds to the diagonal structure of До-matrix. One can use the natural tensor ID-grading on the quantum plane and corresponding differential calculus. Here is a commutative semigroup, which can be defined as the set of covariant and contravariant multiindices with the multiplication rule * '©•(fei)' 'Ю-'©-'(SO м

The order of upper or low indices is unessential for the tensor grading so we can use some type of symmetrization of indices, for instance, D-symmetrization. Note, that we must distinguish the multiindices with equal number indices of different sort , e.g. (123) Ф (345). Nevertheless one can consider the equivalence relation E in the set /p:the multiindices are equivalent if they have some identical subset of indices and also an arbitrary number of upper and low indices coinciding in pairs , for example шм»и»нда) (5) '600

Let us denote Io = IofE the corresponding factorset which has also a multiplication rule for classes of multiindices . Consider the generators dx%, 3; of the special differential calculus preserving ID-grading [9]

dxi xk = [ifc] xk dx' dx' dxk = -[it] dxk dx1 di xk = tf + [fei] xk di (6)

д{ dk = [ik]8k д{ i ak dx = [ifcjdi' ak

Here the notation [ifc] = q(ik) is used. Let us introduce the following notations for /о-graded algebras: Mo(n) = Mo(x) is the algebra with generators x\ An(n) is the external algebra with generators x',dx* and Di(Mp) is a tangent vector space of first-order differential operators on Мд(п). D\(Mo) can be treated as the infinite-dimensional Lie D-algebra with the generalized D-commutator of basic elements [9] Dm =xHr)gk = x<>... xipdk (7)

Consider a simple braided D-deformations of the linear group DGL(n) and corresponding

Lie algebra dgl(n) [5,6,9]. DGL(n) is an unital algebra with noncommuting generators L'k

44, = [ij}Uk}[km][mi]L'mL\ = [^Jij, l\ (8) where a special notation for the commutation factor is used. Mo(n) and DGL(n) are examples of D-commutative algebras. A structure of Hopf D-algebra on DGL(n) is consistent with

70-grading Д(4) = Ц 4 = [ij']MM4 4i s(Li) = (L-*)\, = (9)

where a special D-symmetrical tensor product is used [9]. We can choose the exponential parametrization for the braided matrices

4 = (exPA)i = «5j-Mi + ... (10)

= Vm(iWf )i , (M?)[ = S'6? (11)

Here Afj" are generators of the fundamental representation of dgl(n) and X'm(Mf) is an element of the corresponding noncommutative envelope [9]. Note , that the matrix elements of d^/(n)-generators are not deformed. Cotransformation of DGL(n) on the quantum plane Мд(п) conserves /p-grading

x" = L\ z* = [ifcjx* 4 (12) k dx" = L'k dx (13) 3 = &»,(£->)? (14)

It is convenient to use the equivalence relations for the grading of matrices 4 ~ (£2)i ~ ~ (in^)i (is) One can impose an additional restriction on the deformation parameters [tit] = q(ik), which results in disappearance of deformations for dgl(n)

[i*][*j] = [01 => Ш = 1 (16) '601

Note , that dgl(2) ~ gl{2) without any additional restriction. Using the Eq(16) one can define the coaction of commutative matrices L\ on the noncommutative space Mp(n)

L\ I?m = Ll L\

!.'k s' = [i(][ife]i' ®, L[ (17)

Now we shall consider a quantum deformation GLp{ri) of the linear group for the diagonal R-matrix (1). F(GLD(TI)) is a Hopf algebra with generators a),

!lDA\A2 = Л2Л,ЛЦ => <4 n'm = [ij][femR 4 (18)

Stress , that multiplication and comultiplication in F(GLt>(n)) conserve some tensor grading , but an antipode map change tensor grading

V») = = (is) 4 = 4 The /^-differential calculus on Mo(n) (6) is covariant with respect to the coaction of the quantum group GLp(n)

x1' = a\ ® xk , dx" = <4 ® dxk

д[ = дк<а(а^)} = (а-Ч*в>дк (20)

where the symmetrical tensor product is used. Stress , that different deformations of the linear group are closely connected as different cotransformations of the quantum plane . bet us discuss the manifestations of this connection and examples of the combined using of GLo(n) and DGL(n) structures : A) Consider a differential calculus on the quantum group GLo(n)

da't aL = [y'l da\ 'K <4 = -[ij][mk)dal d4 (21)

4 = \mi}lkj)(a->ym da),

Right-invariant differential forms p'k on GLo(n) [3] can be treated as 1-forms with dgl(n)- structure

pi = daKa"1)!

A PL = - [$r] PL Pi (22)

B) £>(7L(n)-matriccs generate "left orbit" on the quantum CiD(n)-matrices

4 - Si = L) 4 L\ aj = [tr][rf]a; Lj

4 Sg, = \ij}[mk}Pm a' (23) 51 "L = I'jlMVm ai 4 =5; (a->)i

Note , that DGL(n)-matrices can be constructed in terms of quantum matrices. C) Quantum Gin-matrices can define "two-sided quantum orbit" on DGL(n)

4 - «j Ll (о"1)? (24) '602

-1 where the elements aj^n )™ commute with L'M. This construction was used in Ref[ll] for deformation of gauge fields . Deformations of the special linear group SL(N) for the diagonal До-matrix can be obtained by a factorization of quantum G£(n)-matrices [12,13]

4 = 4(a,)l/n (25) where DQ is the quantum determinant and A\ are generators of 5£D(n)-group . Note , that a quantum deformation vanishes for SLO(2) group , however we have nontrivial one-parametric deformation for the case of SLD[3) quantum group. Let q(n),q(i3),q(3I) are independent d-; irmation parameters of GLD{3) (18). Consider a quantum determinant of GZ/o(3)-matrix a'k

DQ = EJAJAG + [JILPILAJAJA? + PIJPQAJAJAJ- [2i]aja?a= - [32|a|a^ - [2i][3i][32]ajala? (26)

The quantum determinant is not central element in GLQ(3). Eq(25) determines elements of the SLo(n)-matrix a\ which satisfy the commutation relations (18) with the following deformation parameter

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