Two-Variable Relativistic Tensor Harmonics

Home , Spherical basis

сообщения объединенного института ядерных исследований дубна

Е2 - 12936

M.Gmitro, A.A.Ovchinnikova

TWO-VARIABLE RELATIVISTIC TENSOR HARMONICS

1979 ч

л i

Е2 - 12936

M.Gmitro, A.A.Ovchinnikova

TWO-VARIABLE RELATIVISTIC TENSOR HARMONICS F2 - !2Q4h Гмитро М., Овчинникова А.А. '** *****v Релятивистские тензорные гармоники в случае двух переменных Рассмотрено обобщение релятивистских сферических тензоров на слу чай двух переменных. В сферическом и спиральном базисах а системе от счета Брейта построены лоренцесские коварианты, зависящие от трех пе ременных. Изучены ид свойства: четность, ортонормальность на сфере, вложенной в трехмерное Евклидово пространство. Получено выражение для скалярного произведения в четырехмерном пространстве и условие незави симости релятивистских тензорных гармоник. Переход к нерелятивистским коеариантам является очевидным. Построенные релятивистские тензоры могут быть полезны при изучении реакций с числом частиц в конечном состоянии больше деух, Работа выполнена в Лаборатории теоретической физики ОИЯИ.

Сообщение Объединенного института ядерных исследований, Дубна 1979

F2 • 12Q46 Gmitro «., Ovchi,:r.ikova A.A. *** '*™« Two-Variable Relatrvistic Tensor Harmonics Three bases in the Hilbert space of tensor fields on the unit spheres associated with two independent vectors are discussed: the tensor spherical harmonics and the symmetric and unsymmetric tensor helicity harmonics. Under the conditions which we specify they form complete sets of independent Lorentz covariants which may serve the purpose of the analysis of reactions with several particles in the final state. The investigation has been performed at the Laboratory of Theoretical Physics, JINR.

Communication of the Joint institute for Nuclear Research. Dubno 1979

) 1979 Объединенный институт ядерных исследований Дубна 1. Introduction Recently, a new interest in the intermediate-energy (=к10 MeV) lepton- and meson-induced processes on nucleons and nuclei has arisen in connection with the increasing flow of the high- quality data coming from the SIN, TRIUMF and LAMPF meson faci lities. The Lorentz covariants written in the Cartesian basis have been the traditional theoretical tool to analyse such reac tions. Though manifestly covariant, they frequently do not fit the purposes of the physical investigations, e.g.,already the basic problem of selecting the independent covariants may prove to be practically insoluble in many cases. It is our experience that the tensor harmonics in the sphe rical and helicity bases constitute a convenient, highly flexible framework for the construction of the Lorentz covariants: The most important mathematical properties of the tensor harmonics follow directly from the well-known formulas of the angular-mo mentum algebra, the construction of the independent sets of covariants is straightforward. Orthogonality properties of the tensor bases make the calculations of rates and other physical quantities much easier than with the cumbersome Cartesian tech-

3 niques. Besidea these technical advantages two gratifying proper ties of the new formalism constitute its main merit and should be mentioned. First, the relativistic tensor harmonics allow a natu ral unification of the theoretical treatment of a big class of dif ferent physical processes. Second, the formalism, though fully equivalent to the covariant Cartesian expressions is actually very much similar to the familiar nonrelativistlc multipole-expan- aion formulae. Therefore, the physical results may always be easi ly interpreted by a direct extrapolation to the domain of classi cal nuclear physics. Using the tensor harmonics we need not to perform any "nonrelativistic reduction" which is normally done, e.g., via the Foldy-Wouthuysen transformation. Namely this is the step which frequently makes the treatment of physical processes unwieldy and brings in the approximations which are usually dif ficult to control. The trick hare is indeed in choosing the ap propriate referenoe frame.It is the Breit frame which being fully appropriate physically, gives simultaneously an enormous ampli fication of the formulas. The formalism of the relativistic s-th order tensor spheri cal harmonica has been presented recently by Daumena and llinnaert . For the corresponding analysis performed in the helicity ba- /2/ sis we refer the reader to the paper by Akyeampong ' '. As a matter of fact the method was first introduced by Stech and Schu- lke who have considered, however, only the specific case of nuclear beta-decay. Recently, Delorme ' ' presented the applica tion of the relativistic spherical tensor harmonics in the context of the so-called elementary-particle theory of nuclear currents. The treatment in Refe. is always limited to the one-variable harmonica which correspond to the case of binary reactions. Here we shall present our results concerning the two-vari able Lorentz-covariant tensor harmonics in the spherical and he licity bases. It will be shown below that they provide actually the most general description of the multi-variable tensor fields, which may be needed in the analysis of any reaction of the type atb—• 1»г«- •• и. In Sec. 2 we define the spherical tetrads and build up the second order tensor spherical basis. Section 3 is devoted to the (scalar) spherical harmonics in two variables. There we display the reduction formula which permits an еаву elimination of those

4 harmonics which can be expressed as linear combinations (with sca lar coefficients) of the harmonlcB which form the basic set. In Sec. 4 the second order tensor spherical harmonics in two vari ables are introduced and their most important properties are listed. In Sec, 5 two different forms of tensor harmonics in the helieity bases are deduced from the two-variable tensor spherical harmonics constructed in the preceding section. Finally, in Sec. 6 we indi cate, using a particular example of the reaction with three par ticles in the final state, how the formalism of the present paper may be applied and indicate some of its merits in comparison with the Cartesian expressions.

2. Tensor Spherical Basis First we have to introduce a set of orthogonal 4-vectors on which to define the projections of the tensor fields. Following Daumens and Hinnaert ' ' we choose three враее-like vectors e" (n = *1,0) on the unit sphere S (e) embedded in the sub- space £ fe) orthogonal to the time-like vector e~ =S^ . The complex vectors e"1 satisfy the following conditions f ^r = (-'r\ ; , (1,

п е; (ерГ^гЛп.. (2)

We иве the Paull metrics (i.e., a^ =a6^ а"& - Q0bo ) and the usual summation convention for repeated Greek indices ( ix - 0,1,2,3). Note that three vectors e" ( *) = *1,0) form the usual 3-dimensional spherical basis. The spherical components of an arbitrary vector Q„ in the basis Just introduced are given from the decomposition CL-Ia*"e£* + aMe"* . (3) This means

00 _ JX> _'»• _ •='•e " (4) а"- «ле™, а™ = °л r •

The construction of the tensor spherical bases of an arbit rary order has bnen discussed in detail by Daumens and Hinnaert '' . We shall deal with the 2nd order tensor basis

5 п,л2 only, where the symbol 1---J denotes the Clebsch-Gordan coeffi cient. The parity and orthogonality properties of basis (5) read as followsj

M^.M'-"'^, (6) ^"•'Г'^'ЧЛЛД". (7)

3. Spherical Harmonics in Two Variables The relativietic spherical harmonica may be introduced by taking the projections of an arbitrary unit 4-vector U^ on the basis #1" • The constructions aa performed,e.g.,in Ref.

holds for the purely space-like vectors (i.e. ue =0 ) only. This condition, however, is not too restrictive for the applica tions, since it can always be met if one works in the rest (Breit) system which is fully appropriate in major phyaical situations.

Indeed, taking the definition of the Breit system аа Q^-(0,iQ0)

l and choosing the spherical basis in such a way that e" *Qh/j-Q > we may always instead of a^ consider the vector

which is orthogonal to е., :

and then define the unit 4-vector U^ = a f\f £г , in this way since the time component of U^ vanish, the spherical har monics defined in (8) on S (e) become identical with /5/ the usual spherical harmonics as defined, e.g., in Ref. . In the simplest oase t = / we have

6 Proceeding to the case of two variables U, and v we construct as usual the objects, which transform according to the irreducible representations of the rotation group!

They form a complete orthogonal basis. We would like to keep only independent terms of this infinite-dimensional basie. By indepen dent we mean such terms which cannot be expressed through the re maining onee as linear combinations with scalar (i.e., depending on U , V

(13)

l J цг '< '• which represents actually the relation between the two coupling schemes of the four momenta e,t, s, and t' re-written using the expansion Y„>)Y,'(u)=I«6U) OYi», ' „ '•-< (14) where ц=/2о+7, and (••;( stands for the well-known 93 sym bol. Using (13), any harmonies |(',4') *».| with 4t4>l*i can be expressed through the harmonics J(f,^)''n} with P, • (2 < f, <• ^ , It is easy to see that this proaeas can be repeated as many times as necessary in order to reach the harmonics {«,«,)?».} with t,*ti*t, (15a)

1 ( Ы <>>* - (,5Ь)

The construction just desoribed leaves us with 11<-4 orthogonal terms of the form (12), which are independent in the above sense. They cannot be further reduced in view of the triangular condi tion Т,*Тг -Г.

7 Concerning the case of the spherical harmonica in v vari ables it should be noted, that further generalization of (12) is actually not needed,In physical applications which we have in mind we always deal with the 4-dimensional Minkowski space. In this space there are only four independent vec< re, that is the

time-like vector Qr = (°,-4i) and three space-like vectors, e.g., u v Anv Uf. . Vr, and ^../Y @< flf • other vector can be obtained as their linear combination with scalar coefficients. Therefore, the prescriptions (12) and (15) are sufficient to obtain the multi- variable spherical harmonics as well.

4. Tensor Spherical Harmonics The 2nd order tensor spherical harmonics in two variables are defined as f/г J" n I / V / I Y I 1 V 4 --'-*'•-• I /| | TZ>^--n^l\xs^xs4jT ^- (16) The generalization to the tensor harmonics of an arbitrary higher order is straightforward. Using the properties (6) and (7) of the basis tensor i 2\ and those of the two-variable spherical harmonics (12) we may easi ly see that -n((,W JM (i) the tensors ' (*-,

Now, recalling the definitions of tensor

o4 vr>M^ (V' ° V^U -) and pseudotensor

4 Агх(ч*)-^(-//'°' '%(-« -v)

operators, we can easily see that the tensor harmonics (16) tran sform under the parity operation P like tensors (pseudotensora) if the sum I,' (i ' r<'rz i<> even (odd).

8 , т&'гН J*l

(ill) I (r,ri)r„A are orthonormal on the unit sphere embedded into the space £ (Y Tll)1' JV

= ^JJ' °nn' °'/,' d'/i °W '"••'i' "«!•« °rr' •

(iv) The scalar product of ' (^«-„j in the 4-dimenaional spa ce is

' <<.«0"r i ' ' C<-,'f,)<-' rA

(19) • r;; iiiii-; i VMJO U-j*1 Y;W Y; W where V(....i..) denotes the Racah coefficient, and (v) the number W, of independent tensor harmonics of an arbitrary order s ia

AT-r^ZZ («'«-^^(ггл^м), (20) J i- e-.p-ri r where ?V is the statistical weight of the corresponding basis tensor» E#g» for the 2nd order spherical harmonics (19) with ^0"" we have 11^=2 , »,W-3 .and ^" = 4 .therefore

(zl N s *6 (2.М). (2D

Note that while it seems to be actually Impossible to count the number of independent Lorentz covariants when working with the Cartesian forme, in the spheriOBl basis the result (20) has been obtained in a very natural and elegant way.

9 5. Tensor Harmonica in the Helicity Basis The helicity basis states analogous to those of Jacob and Wick/6' are introduced here in a slightly formal way, therefore the interpretation in terms of conserving quantities may be lost. Nevertheless, the technique has proved to be very helpful when a particular direction may be chosen according to the nature of the physical problem. We keep calling helicities the projections of the (internal) angular momentum vectors (vectors G™ of the basic tetrads) on the direstions ^>y>, and "2,fi_ connected with the scalar spherical harmonics (8)

(22)

(23) hie considered above. The one-variable tensor harmonics in the heli city basis have been discussed in substantial detail by Akye- ampong ' '. We shall follow his definitions and notation where possible. In our case the second order helicity teneor basie may depend on three variables Q-y , «^, and vx . (The dependence on Q\ cornea through C, Just as in the case of the spherical basis (5)). Indeed, we can write '/5 / eN Г К:кЫ W«) = Ц *?Ж№ <"'(*) (*> к к' and consider any combination of the tetrads ^ and ?,. The particular choice of the tensor helicity basis certainly Jepends on the character of physical problem to be aolved. Here we describe two cases, the symmetric Si'1}J and uneynmetric

Vtvjt-k r> teneor harmonics which are constructed using the tensor bases e^'M e^i(v) and e'^Me^'Cu) , res pectively. л To construct the S harmonics we substitute Eqs. (22)-(24) in (16) and perform the standard recouplingi

10 тяг ;:=-^г 11:щ&№хы;м &ш »» л ' (25)

where

are Just the helicity-basis tensor harmonics we are looking for. Using again (22)-(24) and the Clebsch-Gordan expansion of the b -functions we have alternatively

In this laat *хргевв1оп the U harmonics are

f -4=-E k^l r We;W.

II Now we list some uaeful properties of S and U t (i) The harmonics (26) and (28) form the sets of independent (see Sec. 3) Lorentz covariants if just 2J<"f different pairs Jiji (j^i) are taken for each combination of ^Kl^i (r.4rlt^- The reduction formula which allows us to fix these restrictions and thus to deal with the independent covariante only can he easily obtained in the same manner as Eq. (13)• It reads -$- н»)оо\г[г:1<м] ^bM^M*) (29)

(ii) The parity operator acta on them according to

Q { {) {u v }i (30) v^ rx К -*. ) = ' V'-*. ^ ' -

Vt^hr*,a)'^" "&*£(».<.<>). (3D

(ill) The S and U harmonics are orthonormal if integ rated With dD-i - Sm«; сЛ/, cttf; ; e.g.:

- °j/ °мм' Oj •• ,r; °fc,i,; Ofe.t; . (lv) The scalar product is

• °r,r,' £«•,<•; °.|Л' o^ 2 К м, .J 1 * м'«. I

k"***, {33)

V • U,.ir, sJb,-t, eJ bv;; J' TM UO YM4 J. Ц Jj» J n»

12 The scalar product of the U harmonics has 8 similar expres sion, we do not display it.

6. Conclusions In the present paper an extension of the relativistic tensor harmcnics to the case of two variables has been suggested. They appear to be rather ccmpact and very flexible in comparison with the Cartesian forms which «ere usually employed in the applica tions. To give just one example, consider the reactions of the radiative muon capture on atomic nuclei ' ' ':

s A* АЦ,Р{) —+ vr *K*>- fy>P.) <*)

Introducing the momenta Qx- (p, <• р,)д and <£л = (pv -р,)л we can fix the general form of the corresponding weak hadronic mat rix elements in the elegant form which is also easy to manipulate

т (35) т:к-1Г!У(1СМ ^;(Ч°), J/4 Ш ',<•, r where ^оЦ),- are form factors. The structure of this tensor in the Cartesian covariante ie considerably more complicated and for different values of J; and J, must be constructed indivi dually. For example for the reaction ft' * p —*v мг(<0«» one finds /8/ V = -a (ft HF< tf- yx * ы* k»lk> fif?»fj'^fJ + •• • *- Ufa),

/a/ where 68 terms ' containing the Dirac matrices u, and form factors FfiaQ) appear. Applioatione of the tensor harmonics in the helicity basis for the reaction (34) are presently being studied.

13 The authors are indebted to J.A.SmorodinskiJ and R.A.Eram- zhyan for discussions and reading the manuscript.

References

1. Daumens M. and Minnaert P. J.Math.Phye., 1976, 17, p. 2085. 2. Akyeampong D.A. J.Math.Phye., 1979, 20, p. 505. 3. Stech B. and Schulke L. Z.Phys., 1964, 179, p. 314. 4. Delorme J. in Mesons in Nuclei, vol. 1, ed. M.Rho and D.Wilkinson, (North Holland, Amsterdam 1979, pp. 107-147.). 5. Wigner E. Group Theory (Academic Press, Hew York, 1959); VarSalovid D.A., Moskal'ov A.M. and Khersonskij V.K, Quantum Theory of Angular Momenta (Nauka, Leningrad, 1975) (in Russian). 6. Jacob M. and Wick G.C. Ann.Phys., N.Y. 1959, 7, p. 404. 7. Gmitro M. and Ovchinnikova A,A., JINR Reports E2-12639 and E2-12640, Dubna, 1979. B. Hwang W.-Y.P. and Primakoff H. Phys.Rev., 1978, 18C, p. 414. 9. Note that the dependent covariants always may appear in the Cartesian basis. Comparing with Eq. (21) we can see that al ready in the simplest case of reaction (34) on a proton Oi'J) - i) there are four superfluous covariants. The cor responding coupling equations, when constructed with the help of the symbol manipulating program SCHOONSCHIP appeared to contain 26 terms each.

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PI,2-7642 Proceedings of the International School of Young Physicists on Hlgb Energy Physics. Gomel, 1873. 10.53 Dl,2-8405 Proceedings of the IV International Symposium on High Energy and Ele mentary Particle Physics. Varna, 1974. 3.*» PI ,2-8528 The International School-Seminar of Young Scientists. Actual Problems of Elementary Particle Physics. So chi, 1874. 4.70 D6-8846 XIV Symposium on Nuclear Spectro scopy and Nuclear Theory. Dubua, 1975. 3.70 E2-«086 Proceedings of the 1975 JINR-CERN School of Physics. Alushta, 1975. 15.00 D13-9184 Proceedings of the International Mee ting on Proportional and Drift Cham bers. Duma, 1975. 10.00 Dl ,2-8224 Proceedings of the IV International Seminar on Sigh Energy Physics Problems. Dubna, 1975. 4.80 Proceedings of the VI European Con ference on Controlled Fusion and Plasma Physics. Moscow, 1973, v.n. 16.00 D13-92B7 Proceedings of the VIII Internatio nal Symposium on Nuclear Electro nics. Dubna, 1975. 5.5 D7-9734 Proceedings of the Interaational School-Seminar on Reactions of Hea vy Ions with Nuclei and Synthesis of New Elements. Dubna, 1975. 4.86 D-9920 Proceedings of the International Conference on Selected Topics in Nuclear Structure. Dubna, 1978. 15.00 Dl-9788 Nonlocal, Nonlinear and Nonrenor- dingnullssnls of eth Fielt IVtd » TheoriesIatemattona. Proceel Sym- posuur . Alusata"" , 1978. D9-10500 Proceedings of the Second Symposl- • urn on Collective Methods of Acce leration. Dubna, 1978. 11.00 D2-10533 Proceedings of the X International School on High Energy Physics for Young Scientists. Baku, 1076. D13-11182 Proceedings of the IX International Symposium oo Nuclear Electronics. Varna, 1977. D10,11-11264 Proceedings of the International Meeting on Programming and Mathe matical Methods for Solving the Phy sical Problems. Dubna, 1977. 13.00 D17-11490 Proceedings of the International Symposium on Selected Problems of Statistical Mechanics. Dubna, 1977. D6-11574 Proceedings of the XV Symposium on Nuclear Spectroscopy and Nuc lear Theory. Dubna, 1978. D3-11787 Proceedings of the III International School on Neutron Physics. Alush- ta, 1978. 12.00 D13-118D7 Proceedings of the III International Meeting on Proportional and Drift Chambers. Dubna, 1978. Proceedings of the VI All-Union Conference on Charged Particle Accelerators. Dubna, 1978. 2 volumes. 25.00 Dl ,2-12036 Proceedings of the V International Seminar on High Energy Physics Problems. Dubna, 1978. 15.00 Dl,2-12450 Proceedings of the XII Interna tional School on High Energy Physics for Young Scientists. Bulgaria, Primorsko, 1978. Proceedings of the V International Symposium on Nonlocal Field Theories. Alusht.-,, 1979. 9.50 Proceedings of the XI International School on High Energy Physics and Relativistlc Nuclear Physics for Young Scientists. Gomel, 1977. 14.00

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сообщения объединенного института ядерных исследований дубна

Е2 - 12936

M.Gmitro, A.A.Ovchinnikova

TWO-VARIABLE RELATIVISTIC TENSOR HARMONICS

1979 ч

л i

Е2 - 12936

M.Gmitro, A.A.Ovchinnikova

Сообщение Объединенного института ядерных исследований, Дубна 1979

Communication of the Joint institute for Nuclear Research. Dubno 1979

п е; (ерГ^гЛп.. (2)

00 _ JX> _'»• _ •='•e " (4) а"- «ле™, а™ = °л r •

The construction of the tensor spherical bases of an arbit rary order has bnen discussed in detail by Daumens and Hinnaert '' . We shall deal with the 2nd order tensor basis

5 п,л2 only, where the symbol 1---J denotes the Clebsch-Gordan coeffi cient. The parity and orthogonality properties of basis (5) read as followsj

M^.M'-"'^, (6) ^"•'Г'^'ЧЛЛД". (7)

Indeed, taking the definition of the Breit system аа Q^-(0,iQ0)

l and choosing the spherical basis in such a way that e" *Qh/j-Q > we may always instead of a^ consider the vector

which is orthogonal to е., :

6 Proceeding to the case of two variables U, and v we construct as usual the objects, which transform according to the irreducible representations of the rotation group!

(13)

1 ( Ы <>>* - (,5Ь)

Now, recalling the definitions of tensor

o4 vr>M^ (V' ° V^U -) and pseudotensor

4 Агх(ч*)-^(-//'°' '%(-« -v)

operators, we can easily see that the tensor harmonics (16) tran sform under the parity operation P like tensors (pseudotensora) if the sum I,' (i ' r<'rz i<> even (odd).

8 , т&'гН J*l

(ill) I (r,ri)r„A are orthonormal on the unit sphere embedded into the space £ (Y Tll)1' JV

= ^JJ' °nn' °'/,' d'/i °W '"••'i' "«!•« °rr' •

(iv) The scalar product of ' (^«-„j in the 4-dimenaional spa ce is

' <<.«0"r i ' ' C<-,'f,)<-' rA

(19) • r;; iiiii-; i VMJO U-j*1 Y;W Y; W where V(....i..) denotes the Racah coefficient, and (v) the number W, of independent tensor harmonics of an arbitrary order s ia

(zl N s *6 (2.М). (2D

(22)

10 тяг ;:=-^г 11:щ&№хы;м &ш »» л ' (25)

where

are Just the helicity-basis tensor harmonics we are looking for. Using again (22)-(24) and the Clebsch-Gordan expansion of the b -functions we have alternatively

In this laat *хргевв1оп the U harmonics are

f -4=-E k^l r We;W.

(ii) The parity operator acta on them according to

Q { {) {u v }i (30) v^ rx К -*. ) = ' V'-*. ^ ' -

Vt^hr*,a)'^" "&*£(».<.<>). (3D

(ill) The S and U harmonics are orthonormal if integ rated With dD-i - Sm«; сЛ/, cttf; ; e.g.:

- °j/ °мм' Oj •• ,r; °fc,i,; Ofe.t; . (lv) The scalar product is

• °r,r,' £«•,<•; °.|Л' o^ 2 К м, .J 1 * м'«. I

k"***, {33)

V • U,.ir, sJb,-t, eJ bv;; J' TM UO YM4 J. Ц Jj» J n»

12 The scalar product of the U harmonics has 8 similar expres sion, we do not display it.

s A* АЦ,Р{) —+ vr *K*>- fy>P.) <*)

/a/ where 68 terms ' containing the Dirac matrices u, and form factors FfiaQ) appear. Applioatione of the tensor harmonics in the helicity basis for the reaction (34) are presently being studied.

13 The authors are indebted to J.A.SmorodinskiJ and R.A.Eram- zhyan for discussions and reading the manuscript.

References

Reoelved by Publishing Department on November 20 1979.

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