Preprint - 92- 33/282
V. E. Tarasov
DISSIPATIVE QUANTUM DYNAMICS AND NONLINEAR SIGMA-MODEL
Moscow 1992 DISSIPATIVE QUANTUM DYNAMICS AND NONLINEAR SIGMA-MODEL.
V, E. Tarasov
Institute for Nuclear Physics, Moscow State University Moscow, 119899, Russia E-mail: tarasovUcompnet. npimsu.su
Moscow 1992 ''№ УДК 539.1.01
Sedov variational principle which is the generalization of the least action principle for the dissipative and irreversible processes and the classical dis- sipative mechanics in the phase space is considered. Quantum dynamics for the dissipativo and irreversible processes is constructed. As an example of the dissipative quantum theory we consider the nonlinear two-dimensional sigiua-inodel The coufonnal luiomaly of the energy momentum tensor trace for closed bosonic string on the affine-metric manifold is investigated. The two-loop metric beta-function for nonlinear dissipative sigma-model was cal culated. The results are compared with the ultraviolet two-loop conterteims for affino-inetric sigma model.
© Institute for Nuclear Physics, Moscow State- University, 1992 Introduction
Vectorial Newtonian mechanics describes the motion of the mechanical systems subjected to forces. The Forces usualy are divided into potential and dissipative forces. The Newtonian approach doesn't restrict the nature of the force [l]. Vari ational Lagrangian and Hamiltonian mechanics describes the systems subjected to the potential forces only [1,2]. The dissipative forces are beyond the sphere of the variational principles [3-6]. For this reason the statistical mechanics does not describe the irreversible and nonequilibrium processes. It is caused by the absence of the Liapunov function [7] in the phase space in the Hamiltonian mechanics ( Poincare-Misra theorem) [8-10], To describe the dissipative and irreversible pro cesses we must introduce the additional postulate in statistical mechanics (for ex ample, the Bogolubov principle of weakening (relaxation) correlation [11] and the hypothesis of the relaxation time hierarchy [12] ) [10,13,14]. Therefore this pro cesses is considered in the sphere of the physical kinetics [13-15]. It is known that the initial point of the quantum mechanics formalism is Hamiltonian mechanics [16|. Therefore the quantum theory describes the physical objects in the potential force fields only. The irreversible and dissipative quantum dynamics is beyond the sphere of quantum mechanics. Sedov L.I. suggests the variational principle [3-6] which is the generalization of the least actional principle for the dissipative and irreversible processes. The holonomic and nonholonomic functionais are used to in clude the dissipative processes in the field of the variational principle. In this paper we consider the Sedov variational principle and the classical (h —» 0) dissipative mechanics in the phase space, Taking into account this Hamiltonian dissipative mechanics we generalize the quantum dynamics for the dissipative and irreversible processes. In the dissipative quantum mechanics we get that the operator of the nonholonomic quantity is nonassociative and consider the properties of this oper ator. We obtain the dissipative analogues of the Schroedingcr equations and the Feynman represantation of the Green's function. The dissipative quantum scheme suggested in this paper allows to formulate the approach to the quantum diesipa- tive field theory. As an example of the dissipative quantum theory we consider the sigma-mode! approach [18] to the quantum string theory [17]. The conformal anomaly of the energy momentum tensor trace [IS] for closed bosonlc string on the curved afTme-metric manifold ( or in dissipative and nondissipative background fields ) is discussed. The two-loop metric ultraviolet renormalization group beta- function [19-20] fur two-dimensional non-linear dissipative bosonic sigma-model is
:\ obtained. The results are compared with the ultraviolet two-loop metric countert- erms for affisie-molric sigma-model suggested in the papers [21,22].
1 Sedov Variational Principle.
The equations of motions of the mechanical systems in H-dinensional configura- lional .space are
U'/'to.n.O + Q. = 0 (П
where T is the kinetic energy, which can be written in the form
7'(
D,.±±-± dt du{ dq* u' = dq'fdt and Q, = Qi{q,v,t) is the sum of external forces. In general case, Q, is the sum of the potential Q? and the dissipative Qf forces. The potential force is the force for which a function V = V(q,u,t):
DiV = ~QK (3)
exist. The dissipalive force Q"f is the force which can not be written in the form (3). Then the Enlcr-Lagrange equations take the form
/A/* + # = 0 (4)
d when- L — L(qyii.t) н T — V is Lagrangian. In the dissipative case { Q j£ 0), the equation '•!} can not be followed from the least actional principle [1,2]:
bS{q)~& Idt L(q,utt) = 0 (5)
The basic variational principle for dissipalive processes is the principle suggest ed by L.I.Sedov [3-6]. It is a generalization of the least action principle. The Sedov variational principle has the form:
6S(g) + 6W(q) = 0 (6)
1 where S(q) is the holonomic functional called action and W(q) is the nonholo nomic functional. The nonhojonomic functional is defined by the nonholonomic equation. Let a variation of the nonholonomic functional be linear in the varia tions Sql and 6ti* that is
i SW = 6Jdt w{q,u)= f dt {w]{q1v)6q + w?{q1u)6u') (7)
where w] and w] are the vector functions in the con figurationa l space. The classical (h —» 0} field theories (in the continium mechanics) for the systems with W ф 0 are considered in [3-6]. Let us consider the Hamiltonian approach to the variational classical mechanics with dissipative forces.
2 Hamiltonian Dissipative Mechanics.
One direct corollary of the Sedov variational principle are the following dissipative equations of motions
d ,dL(q,u) 2i dL(q,u) ,. dq' •
Let us define a canonically conjugate momentum by the equations
p,Sfe>W(^) (9)
and represent this relation in the form u' = v*(q,p). The Hamiltonian is given by
НЧ,Р) = РЛЧ,Р)-ЦЯМЧ,Р)) (io) If we consider the variation of the Hamiltonian, we obtain the dissipative Hamil tonian equations of motion
Sw(qtp) = 6w{q,v(q,p)) = ги'б^' + w*p6pi (12)
Let the coordinates p^,q*}w,t of the (2n+2)-dimensional extended phase space be connected by the equations Sw - а,{ч,р,1)6$ - 6'(g,p, ()Spi = 0 (13)
where ut , 6' are the vector functions in phase space. Let из call the dependence w on the coordinate q and momentum p the holonomic-nonholonomic function (HNF) and denote w = w{q,p) € Ф- H the vector functions satisfy the relation
t)a,(q,p} _дЩд,р)
the coordinate w is the holoiiomic function ( w € F). If these vector functions don't satisfy the relation (14) the object w(q,p) we call the nonholonomic function or the Sedovian ( ш G. F). Let us define the variational Poisson brackets (VPB) for Wajte Ф in the form:
The basic properties of the VPB:
l) Уа,б€Ф [rt,b]= -[6,aj€ F;
2) УаДг,€ Ф : «ViVc€F Л5[а,&,<\] j£ 0; 3) V«A6Ac€f Д5[а,6,с] = 0; where /15[«,A,c] = [л, [fc,c]J + [6, [e,a]]-f [c, [a,6]j. It is easy to verify that this properties for the holonomic functions coincide with the properties of the usual Poissoii brackets [1,2]. Let us consider now the characteristic properties of the physical quantities:
О !;<..;<,) = faV) = o [Ч'>Р,\- ••к
'..') («•,;<.! = "•,' [w,q']= -wt IK/M'ftl £{[«>,PjlPi] [i«WW]* •WW] •1) AS[q',w,p,]:= Щ Ф о
• Ow* i)iv' fi2w p Hi)
ti This object Щ characterizes the deviation from the condition of integrability (14) for the equation (13) and by the Stokes theorem
/ 6w « / Щ dq> Л dpi ф. О Тыл JM * if we take into account VPB the dissipative Hamiltonian equation of motion (11) takes the form J = [/,*-») J = [p„A-H (17)
The total derivative of the physical quantity A = A(q,p) € F with respect to the time t is written in the form
Note that the equation of motion (17) can be derived from the equation (18) as a particular case. Let us consider the solution of the equation (17) in the form
«' =q'{qo,po,i) p.=Pi(4o,pii,t) (19)
Let us assume that the points of the volume Jo = / &Я0&Р0 in the phase space are initial points at the moment t = to [23]. Then the equations (19) transform the
= volume J0 to the volume J = / S<)6p — f /SqoSpo, where / = яЙлТ f«*lw ~ |*t|^ = [<7',p,jo- The following equation is easily verified
fdJ = /f*d (20)
where Л = £*_( iVt = £"„, ASltftWjpi). The fundamental hypothesis of the statistical mechanics [10,13,23,24] is that the state at the moment t is defined by the distribution function p(q,p,t), called density, which satisfies the normalization condition Jdqdppiwt)*! (21)
The average of the physical quantity A(q,p,t) is defined [23,24] by
7 Using for equation (21) formula (20), we obtain the dissipative analogue of the Liouvillc equation [10,13]:
where д I = I( W-™) _ *(*-"0 д _ n called Liouville operator [10,13,14]. Iu addition to the Poincare-Misra theorem [8-10] (" The Liapunov function [7] of the point of the phase space does not exist in the Hamilionian dynamics.") we obtain the statement: " There exists the Liapunov function of the coordinate and momentum in the dissipative HamUtonian mechanics". Let us define the function rj(q,p,t) = ~tnp(q,p,t) and assume й > 0. The equation (23) shows that dij/dt = П. and the function 17 satisfies the relations dt//dt > 0. It is convenient to introduce the entropy of the distribution [10,13,25,26] defined as follows
9=
The relation dsjdt > 0 is easily verified. In the general case, any function Л«ЫМ) which is the composite function f{q,p,t) = g{p{q,p>t)) and satisfies the relation О {()g{p))f(0p) < 0 (Vt) is the Liapunov function [7], that is [df)/{dt) > 0.
3 Quantum Dissipative Mecanics.
Let us use the usual rule of definition of the quantum physical quantities, which have the classical analogues [16,27,28]: If we consider the operators A»Bi i) [Q\Q'\ = {iJ.J'A = o [Qj,/>,] = !/.*; •г) [w, i\\ = itM? [w,Q'} = -ihw; :\) [[WJ',],^\^[lW,F,lF,\ №Q%Q']?W,Q'\,Q'\ ijij •\) K)',|lV,P,!)ii< [/',,!»'.«'')) «r /1S[Q'.№',/>) = I^0 (25) 8 a where AS[A,B4C) = -l/(ft ) ( \A\BC}] + [B[CA\] + \C[AB]] ) and Q* = Q; />1 = Я; W* = H^; fif = П. Let us require that the usual quantum commuta- tional rules be a part of this rules (25J. To satisfy the commutation rule (25) the operators of the nonholonornic quantities must be nonassociative. It is sufficient to require that the operator W satisfy the following conditions: 1) left and right associativity: ((№А')В>) = {№{А'В>)) and (А*(В*\У)) = ((A'B'JWO 2) left-right nonassociativiiy: ({А'\¥}*)?{ЩКА})) ifi*j and ((А'УГ)В>)$(Ай{\УВ>)) if A* fk B4 where Л and В are P or Q operators. The state in the quantum dissipative mechanics can be represented by the density operator p{t) as usual [19]. This operator is defined by the usual conditions: p^(t) = p(t) and Sp(p{t)) ~ 1. The average of the physical quantity A is defined by < A >~ Sp{A(t)p(t)). The time variations of the operator of physical quantity A(t) = A(Q, P, t) and of the operator of state p(t) ar>2 written in the form Jt = -\[p,b\+ (27) where anticom mutator [, ]+ is the consequence of the hermiticity for the density operator p an. 1 for theoperator П. The solution of the first equation may by written in the form A{t) = S{UtQ)A(to)S%ta) where S(Ut0)~Texpj fdr {H~W)(T) (28) Л Jtg T-exponent is defined as usual [29-31] and we add the following flow chart глр A = 1 + A + \{AA) + Ц{АА)А) + ±{{(AA)A)A) + .... The solution of the equation (27) has the form 9 piO^UitytoMtoWHttb) where V(t,t0) = Texpl- t dr Щт) (29) I Jlo In this way the time evolution of the physical quantity operator is unitary and tin* evolution of the stale operator is nonunitary. It is easy to verify that the pure stale [28} at the moment t =• in { /»7( j( s = < П > {30) Let us define the canonical ( unitary ) transformation [16,27] of an operator An(t) = /1(0 in the form As(tyt0) = SHt,t0)A„(t)S(i1to) The operator 4s(Mo) satisfies the condition As{to,t0) — Ац((&). ID this case ihr equations (26), (27) tak<* the form ^Д,.(Мц! = 5'(Мо)^^Ь(Мо) (31) jtP*(tJo) = {\ps, (H ~ W)s\ - \\ils, psh (32) This is the dissipalivr analogue of the Schroedinger equations, the operators An(t) and As(t) called Heiseuherg and Schroedinger representations accordingly [16,27]. The solution of the equation (32) has the form Ps(t, to) = 6'J(«, t')ps(t\ to)Us(t, C) where Ul(t,t') = 7Vipy £dr (H-W- j(l)s(T,io) (33) Let us consider some important features of the basis vectors [16]. Account is to be taken of the time dependence of the state operator pn(t) = Co Ра[Фа*1 >н< Фп,(]н and of the wave vectors in the Heisenberg representation [ф,1 >ц. That И; is [q,ti >н ф [я^г >я contrary to usual quantum mechanics. Let us define the basis vectors {[q,t >} [16] at the fixed time point t = tj : 1) QH(t)\q*t>H=\q>t>H4S 2) 3) jdq[q1t>H 5) [0,t >„= Jdq [q,tf >H *//(?, M/) where Ф//(//• It is easy to prove the following statements: " 1. The basis vector unitary transformed is a basis vector ; 2. There exists a unitary transformation for any two basis vectors defined at the non equal time f points." Thus, Schroedinger representation of the basis veclor [q i,(a >s^ S (' — *o)[g,t >н might be considered as the unitary transformation of the basis vector [<7)'o >H= S^{t — ta)[q^t >н~ [q,t,to >• Let us consider now Green*s functions and its Feynman representation [34-36]. If we take into account the equation (27) we can write the dissipative Schroedinger equation for wave vector in the the form «ft^№,Mo >s= (H-W- уП)5(Мо)№Л*о >s The simple example of this equation for the harmonic oscillator with friction is considered in Appendix 2. Account is to be taken of the time dependence of state in Heisenberg representation. Therefore we make the distinctions between following Green's functions *s(«,0 = jdq' G{q,q\t- t') *s(?',*') Фя(9,0 = Jdq' GH{q,q\t-t') Фя(д\*') where Фн(?,() = < q,t\HH',i >s = < 9,<]s[V>,< >H Gs(q, q', t-i')= = < q,t]H Ul(t,t') [,',(' >H 6(1 - f) (34) GH(q-4',t- (') = < q,t]s Uj,(t - (') [q',4 >s 6(t - t') (35) and [q,t>H = [•?,< = '/iirf >н 11 The Green's function satisfies the time-dependent equation 4G («, Let us use tile Fadeev's method [36] and the conditions % < I'.I/IH'J/ - W - —fl)H\q,lt >= (A - in - ~{ld)(qj,Pj) < M/M«.н , <«° ','.Ь'4«»+1-'.)[чМп>5й l : ^< ,"* ,IJlW ^(/l - W - |n)„(/„)[«V» >// The Feynman representation of the Green's functions has the form Gs(q,q',t-t') = = JDq Dp exp^£dr (р£-ЦЧ,р,т) + ш(Ч,р,т) + ^Пы) (36) i» the same way we can formulate the path integration and generating func tional in the quantum field theory [29,31,36-38]. 4 Two-Dimensional Non-Linear Dissipative Sigma Model. Let us consider now the closed bosomc string theory [17j in curved spare [39-41] or more exactly the nonlinear two-dimensional sigma-mode! [19,20,42,43] and the sigma-model approach to the string theory [40,46]. The world sheet swepl out by the siring is described by map X(x) from two-dimensional parametr space N into space-time manifold M, i.e., X(x) : N —* M. The two-dimensional parainelr are j- = (л<т) and the map X(x) is given by space-time coordinates A'*(:r). Let us choose the holonomic and tionholonomic functional in the form l , t S(X) = $[G,*,g) = \f + ~Jd2*y/sXm(e№X) (37) 12 k l,l l i 6W = -ftPx Dikl(X)d(lX y/gg 'dllX6X (38) where дци{х) is the two-dimensional metric tensor; o/ is the inverse of the siting tension; Ф(Х) is the dilaton field. The Lagrangian and Sedovian define a closed bosonic string propagating in the presence of dissipative and nondissipative background fields or in curved affine-metiic space-tim?. The classical equation of motion for this model is an equation of the geodesic two-dimensional 6ow on the affine-metric manifold (the two-dimensional analogue of the geodesic line ), where Diki{X) is a connection defect tensor [47,48]. Let us choose a parametrization for two-dimensional metric tensor guv in the form [49] д^х^Ых» = c(x)(n2(z)(dT)2 - [da + m{x)drf) (39) In this case the densities of the Hamiltonian without the dilaton field term, Sedo vian and Omega are rewritten in the form k lk 1 h = - jG*'(X)IUn, + mUkX' - ~Gki{X)X X' (40) г ,: 1 k w = jfAf'lUn, + Л к1Х -Х' ) ; О = 2nD {X)Uk (41) k k } k k where D {X) ~ D }{X)G* {X) ; X' н {dX )f{ds) ; П* is the canonical momen tum ; Д are tensorial integral operators, which can be written in the conditional form of indefinite multiple integral SXk : Л*1 = 2j ЙЛ'*Дк'(А') Д£, « -ijbX'D^X) (42) Unfortunately we have no correct mathematical definition of this operators. This difficulty can be removed by expressing the nonholonomic functional as a power series in a covariaut field £k(x) which is'the tangent vector to the geodesic line containing X£ and Xk = X*, + fk{Xa,£). The background field expansions of the Д - operators are written in the form Д*' = 2В?'(ХоК' + 0(«г) Д|, = -2Дм(ХоК' + ОК2) (43) ' The covariant background field method [48,50,51] in the phase space is defined by the usual expansion of the coordinates Xk{x) only. Note that the model defined by (37) ( and (38) ) in the conformal gauge n = l,m = 0 called two-dimensional . 13 nonlinear ( dissipative ) sigma-model. U'i us define the generating functional for connected Green functions [29,31] in the form W[Jtff) = -IA in J DXDU expj-f^i (Zx(X,n,g) + Z2{X,J)) (-И) where Z,(X, П, ) e 'U4A'* - Л + ш + ^5} + ^Я|2|Ф(Л-} (45) Я ат 11 and Z-i[X,,/) is the source term discussed in [53-60]. We derive the invariant back ground field expansion of Z\, ?*i and define a new generating functional W{Xotg,J) by «,Д( W(Xa,g,J)+W(Xa)) = = J IKDW W(X0,g,J) = -IA hjD( «pi( АЩХо.О) - + J(CA(O)Wei(G-1 +Д1)"1 ) (46) where .-I(.V) = -^[<7"' + Air'./** - "•Jf* + ,H»fc(.V))(A'' - mA"' + m^Jf)) + + § I G + Д3|„Г*Х'' (47) and A'* = №)/{Нт) and О'(Л') = №й„(Х). The elTectiv; action Л(А') is rewritten ill the form Л(А) = 5(6',Ф,*?) + S{D,g) , where A'lO.s) = SdP.g) + S2(W,y) + 53(/>,tf) t 5, = - у ^ jAk,^A' v^S'"'4A'' = -W(X) (48) •Ч'2 = /l/'J ^/«W^A'^""^' <49) II 53 = / ^rv^VWA,.** + B{X)) U, = \G-' + Д.Г'и - [С + Л2]*, = 4Д\С^Г«' + 0«3) (50) B{X) s ic-'fxMO-1 + Л%1Ок{Х)0'{Х) (51) *•* = ft',**) = (-Ье-'.ьтс-1) к»" = (к",к",к") = (-n-V.mn-V.-mW) (52) Note that the effective action is conformally invariant, because baa no c(i) dependence . Account is to be takei; of the parametrizatioo of the two-dimensional tensors K"" and i" are connected with the parametrization of two-dimensional met ic tensor g"" , i.e. к = к{д) and k = k{g). The energy-momentum tensor is defined as usually [18,61] by r(t) = 4r^.*.5) (53) in the general case , the objects in S{D,g) are the nonholonomic objects. We use the covariant background field method [21,22]. Therefore this object is the power series in quantum fields £*(*). We note that the background field expansion of the nonholonomic functionals and the two-dimensional metric variation of it are not commutative operations, i.e. (iW(X))l(6gT) = 0 and (6W(Xa,())/(%"*) & 0. Therefore the vacuum expactation value of the energy-momentum tensor [61] < T"" >= Nexp(jW(J)) j Df T--(x) exp~A{X) (54) can not be written in the form — $}(6W(J))l(6g„) because we consider the non holonomic functionals as the background field power series. (In the opposite case, we must prove the Gaussian momentum integration formulas for the tensorial in definite integral operator Д in functional integral.) Let us define the two-metrical generating functional in the form W(g'-,a'\XeJ) = = -Л Ы JD(,exp~(S(G,b,g»u) + S(D,a<") + j(РхМхЦк(х)) 15 The usual functional \V(g,Xo) is derived by W{gtXo) = W{gya1XQ):}=a. The vacuum expectation value (54) can be written in the form <7'„, >= (-—т^Щд.а.Хв)],^ (55) It is easy to derive the ronformal anomaly of the trace of the energy - momentum tensor (18] in the form < 7'; > = т/и вГХЖК + j"'21!»)/?* (56) where ft, = fit, + ...; P = ^ - -&,(«" + -) $1 = i'kt + 2ft'VtV,* ; $* = /?* + «'У^ФУ'Ф m *<; = i^ot'"" ; v*v, = a«v; + (pwi + £>jt„)v; (57)' The law of the change for the energy momentum tensor has the form к If we take into account the background field expansion J21,22S5D] of (*^Х *= CtiXttvQfiitXl) the vacuum expectation value of this law can be written in the form < V"7„„ > = < Ц^± > ИХ* (59) Let us choose the following solution of the classical equation of motion Х$(х) s= const, The equation (59) is rewritten in the usu;' form [40] < VT;1„ >= 0. It is easy to derive that tfie central charge of the Vtr&aoro algebra [62] is proportional to the dilaton Li -function as usualy [40]. The sufficient condition for the validity of this relation [40] is ff1 ~ const and /i£J = 0. Where jflG is the metric beta •function of the two-dimensional nonlinear dissipative sigma-model. In the two- loop metric beta-function calculation we use affine-metric background field method [42,1*1], introduce an auxilliary mass term [44,66], the dimensional regularization t —* n = 2 — 2e [0:3-65] and the minimal subtraction with the general prescription for contraction for the two-dimensional к**" tensor к^ц^^ = z(n) where z(n) = 16 l+zi£+0(e2) and 17^ is two-dimensional Minkowski metric. Different prescriptions may correspond to different renormalization shemes and thus their results should be related through redefenition of the couplings Gu and fkt by analogy to Riemannian two-dimensional non-linear sigma-model with the Wess-Zummo term [71]. It is known that propagator of the quantum fields £k{z) is not standard. Therefore we introduce an m-bein ei(X) 1 where m is dimension of the manifold M and derV^ £a(x) = eJ^C-r), where V*ef = 0. After this modification the kinetic terms become V^V„{" , where This covariant derivative for the affine-metric manifold involves the Schouten- 1 Vranceanu connection [67-69] Лаьс = Лаьс + 2Q(,j)i е ае*ее'ь , (where Q^ is the torsion tensor of the affine-metric manifold ) which is Ricci rotation coefficient on the Riemannian manifold [70]. Note in addition to [42,44] we take into account the diagrams whose external background field lines involve the Schouten-Vranceanu conection- This diagrams must not cancel [45] in contrary to the usual non-linear sigma-model [50]. It caused by the relation Л(а/б/с) = КЩ ^eM, where rY,,j is nonmetricity tensor of affine-metric manifold. The two-loop metric beta-function G Wfl of the dissipative sigma-model is /? = /3jfM -f 0p + /Sf > ere (У%м is the metric beta-function [19,20,50] of the affine-metric sigma-model defined m [21,22] , i.e. the part of the metric beta-function from the action S{G, $,g) only, where two- dimensional metric is the Minkowski metric; /?£ К = 1,2 is the part of metric beta-function from Sn{D,g) in equations (48), (49). The full expression of the two-loop ultraviolet counterterms is very complicated , but it is easy to see the following ultraviolet finiteness conditions. The one and two loop counterterms for two-dimensional non-linear dissipative sigma-model vanish if the correlation be tween the affine connection and the metric structures on the manifold N is given by the ultraviolet finiteness condition [45]: VkGi} = Niik = Niijk) ; %Nk)ii = ^(kNi)jp ; %/ЫЛ) = ^Г*/(Л>/0я (60) It is easy to see that the ultraviolet finiteness conditions have not the zi dependence. Note that the affine-metric beta-function is zero in all loops if the affine-metric manifold with the nonmetrisity tensor A'yi and torsion tensor Q'u is defined [45] R-kiji = Rkiji - 2V[y/Q4j//] - iQ'ii/Qkn/ii = o (61) 17 VkGij = A'yfc - Щ.})к = 0 (62) It is easy to see that this affine-inctrjc manifold is not flat space. I would like to thank Belokurov V.V. and Stelte K.S. for conversations and valuable discussions and Tficoretical High Energy Physics Department of Nuclear Physics Institute of Moscow State University for their support during the work . Appendix 1 The equation of the geodesic line in the afTine-metric manifold [48,21,22] h»-. the form I»' + ( U + о* )«V = о where ['kl] is Christoffel symbol for the metric a^(g) . It is well known that (his equation can not be derived from the least action principle when the connection de fect is other then null. Note the Riemannian geodesic line can be derived from this 3 variational principle if the Lagrangian defined by L(q,u,t) = 7'{q,u) — ~j,atJ(q)u'u k l l . Let us choose Sedovian in the form $w{q% u) = —a,„(^) D^(q) u u 6q . Then the geodesic line equatiun can be derived from the Sedov's variational principle ['15}. Appendix 2 Let us choose Hamiltonian and Sedovian in the form ft =z - (- ; UW = finp bq This objetls define the one dimensional harmonic osciJIator with friction. Lei us мне the background field method and expand the objects in Taylor series of Q — 4 ~~ Voi where qo is the solution of the classical equation of motion in the coordinate space. Let us choose <•/« - 0. In this case the dissipative Schroedinger equation takes the form d t ,, , г Л & 18 The stationary state Ф(£,£) = u(£)exp — j^Et is defined by the equation u"(t)-aiu'{() + {e-eH() = 0 where Let us consider the function u(£) in the form 4=0 Z where э is the solution of the equation s2 + as — 1 = 0 and n < oo . 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[68] Vranceanu G., Rend. Ace. Line, 1926, v.4, n.6, p.508-511. [69] Vranceanu G,, AtLi del Congresso hiternaz. del Math, di Bologna., 1923, p.6. [70] Chandrasekhar S., The Mathematical Theory of Black Holes..Oxford, 1983. [711 Metsacv R.R., Tseytlin A.A., Nucl. Phys.B.,1987, v.293,p.385-419. Received October, 1992 22 Василий Евгеневич Тарасов ДИССИПАТИВНАЯ КВАНТОВАЯ ДИНАМИКА И НЕЛИНЕЙНАЯ СИГМА-МОДЕЛЬ Рассматриваются вариационный принцип Седова обобщающий принцип наименьшего действия на диссипативные и необратимые процессы и клас сическая диссипатпвная механика в фазовом постанстве. Строится кван товая динамика для дпесипативных и необатимых процессов. В качестве П]>пмера дпееппатпвной квантовой теории рассматривается двумерная нелинейная сигма-модель. Исследуется конформная аномалия тензора энергии импульса замкнутых бозонных струи на аффинно-метрическом многообразии. Вычислена лвухпетлевая метрическая бета-функция не линейной дисеппатпвноп сигма-мода ли. Результаты сравниваются с уль трафиолетовыми двухпеглевьшялонтр';лгнам.и аффинно-метрической сигма- модели. Препринт НИИЯФ МГУ - 92 - 33/282 Работа поступила в ОНТИ 28.10.92г. Подписано в печать 28.10.92г. Печать офсетная. Бумага для множительных аппаратов. Формат 60x84/16. Уч.-изд. п. 1|38.у(:л. п. л. 1,5. Заказ N 5242 .Тираж ЮО экз. Бесплатно Отпечатано в лаборатории офсетной печати и множительной техники Отдела научно технической информации НИИЯФ МГУ 119899. Москва, ГСП