The Lagrangian Euclidean Quantum Field Theory of two interacting vector fields is found, which is equivalent to a special Gibbs system with three body potential

Skrypnik, W. I.

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DIAS—STP 89—33ON THE FUNCTIONAL INTEGRALS ASSOCIATEDTO A SPECIAL GIBBS SYSTEMS WITH THREEBODY POTENTIALS.W. I .SkrypnikAbstract:The Lagrangian Euclidean Quantum Field Theory of twointeracting vector fields is found, which is equivalent to aspecial Gibbs system with three body potential.Institute of Mathematics of the Ukrainian SSR, Repin Street 3,Kiev 4, 252601, USSR.11 Introduction.The Sine—Gordon representation for the grand canonicalpartition function and correlation functions of an r—componentGibbs system with the potential energyU((x, a)) = akac(xk—x.),kj=1plays an important role in modern statistical mechanics. With itshelp the rigorous results for charged systems were obtained [1—6].The analog of the Sine—Gordon representation for the Gibbs systemswith many-body potentials has not been found yet. In this paper wederive this analog for Gibbs systems with two types of three bodypotentialsU((x,a))= U((x,a) ) + a.a.c(x- x) i)k=1,jU((x1a)) = - U((x,a) ) +±j=1Uo((xa)n)} ‘ (1.1 ii)where(x,a) = (x1a;...;xa2U((x1u)) = aac(x.x.)iij=iis the charge of the j—th particle, and the coefficient isintroduced for later convenience.The second expression for the potential energy appears in theheat equation, to which the Smoluchowski equation is reduced bythe substitution= exp { U(X) } ‘1 (Xn)the Smoluchowski equation is the forward Kolmogorov equation forthe stochastic (Ghihman-Ito) equationx(t) = - U((x,a)) + 2 w (t)j,o.where { W3(t) } is the sequence of the independent processes ofwhite noise,The discussed systems are not difficult to treat in the caseof an integrabJ.e smooth potential c(x), since they can be reducedto the Gibbs systems with a complex pair potential with the helpof the transformation3exp - { Jakc(x_ xk) } }3—1 jk—l=(2m) exp { i a(q + c(xj_xk)J=1 jk=1x exp - 1 Q ii2 } dQ , (1.2 i)exp28 U((x,)) ] } == (2n) J exp { i [ q, U((x,u)) ] } xdn j=12x exp { - 1 Q 112 } dQ , (1.2ii)wherenII =If c(x) is an integrabJ.e function the thermodynamic limit ofthe correlation function can be found as the solution of thegeneralized Kirkwood—Saltsbourg equation; these equations do nothelp much when c(x) is not an integrable function, and in thiscase we cannot pass to the thermodynamic limit. The proposedSine-Gordon type representation shows a way of dealing with this4-U)U)U)$51N4-’1-H3-.)4.)3.)4.)9—I;:ci*0U)CY*9-..cz9-4)U)U)C)4-)p4U)rU)>rz9.rjo-‘U)0cn—,.cU)U)r-j4.)--r-ji-I(U)9-r-jU)4.)rit1‘U—U)rH‘(CsU)0c-I(CstY)U)U)4c-i-‘1)9-‘rI>1ot3)‘HEU)QU)‘HU)‘Hr,4.)£)4-)c-i0C)U)-4-”ric‘H‘H(1)(D‘4-i‘HU)4-)‘0(5)-H4-’U)U)04-)C)(CS-H>10U)b4-0U)4-’U)£-HU)0-.iU)-H4)9-i4-’4-)04-’s::-°U)‘HErrj-HEU)_>1c-I4-’U)4-)(CSc-I(Cs-HU)0U)rrj-H*o4-’,QrjU)-H-H4-)(5)ro4-’-H0(Cl(1)0c-Ip4U)U)-P4t34(CsU)tJ(CS4)U)-HU)-H0-HI4-’-ic-I*U)-H0U)9,00:ic-I4.)-H0ri4-’‘—‘C)b+*9-ccLJbirN00(“3‘rbcQje4U)II‘No‘H04-)0II0C).44-)p4(Cs-ci)(CsP4(‘S0cvU)£4-,Ui9-4II-‘0U)CL)--H(I)9---(I)->-Hci.)-H(1.)--HP4H-44)4-)U)—(1)-HU).c::U)‘HC’)c-i0c-I00‘ti(“30(“3N0C)(\J’rbp4U)0C-)009-9-‘rb+Lc.-H*p49.U)0H‘Nc-iII+c-ILC)C-)>:U)00-H4-)C)4-I‘4-)00EU)U)4-)‘H‘H4-4(5)a,-H4.)-HU)0p4U)‘H0rdt’CsThe most remarkable and unpleasant feature of the Lagrangiansis that they are degenerate in q, i.e. there are no quadraticterms in in them.2. The main equations. Grand partition function.To derive the introduced Lagrangians we start from thefollowing identitiesc(x.-xk) jk + ajak(+ ) c(x_xk) =jk=1= a.a(l+iq.)(l+iq) c(xj_xk) + aaqqc(x_x) -- i a q c(O) + c(O) ](2 1 a)6•rl+-C%i>1or1oX4-,09.U)+U)U)C)9-Fri+xrtiI—01b-9.IFri+Xcr---b.-—.4-,o-<1-U)ooN‘-I(C)9-e-IIC’40xbr-I.—+br-I+-e-Ibxr-jcyb+tyiII0..—-—--r-HC’Jn-+____v__1baXbU)0b__;,-4Ie4+bbbb—U)-U)rj•11be-I4-’II0—xI(I)oU)e-IU)+V-IIU)sa,0U)U)U)8(•){}dx(*thp)HVVuot.uesdeuopJoD—eutsn.GAJ8M();odeqI]:IUJ’IIUt’ILIKi)()K)(O( 1)XL)(1)K)(D’(Lz)K)Vo)u=tU:ozu(S)U{U(,D(J)x))fl_}dxeUI=V2V-uotun;uor.edpu.zeq.&zeprsUO3SniMON{°o+(x)th+(x)thl1—}dxex{(th’x)/}(*thp)(thp)rl-}dxeuqo8MsoqreAUbu6ui.puos9UUteDUtAOD9WS9t..TUtMussne6.upuedeputSTM—UeUOdWODOMpU8SDSJTJ1.Ut()9DUtJAQD9LLtMSpitutssn5wopuIE1DS.UpUadapUtOM.euep(*thp)rI‘(thp)rIsensewteUO.DUn 1(o)(Dv)03‘(o)D=°o=°3(T)ndaMptreSa.OUap(‘b)(t)asoatp.uçL((x),(x)) = zexp a(x,) - ai*(x)+i(x)+aJCiI2}(1.3) follows from (2.4) if we change the variables in thefunctional integral, making the complex translation [5-6).3. Correlation functions.The correlation functions are defined by the following expressionm r= ( flzxA(x) ) flz(n5 !‘ xnOF exp -U((x,u.) ,(x,a’) ) dX_jm n(a’) Anwhere xA(x) is the characteristic function of compact set A, andXn = (2.3) yields the representationPA((X?aj)m)=SA’ $ p(d) J (d) exp { LA(I*) } x (3.1)p1zjxA(x)exP .(x1,)-a. lII2}9U)0X[‘J000CD>(I)0CD i-CDHCDCDCDfTh_CD-CDCD>00r1H-CDX0‘1.0CDCD01_i.-0CD>CDCD-CD-6MiH-CDCDC)CDH-CDCD*mI>0Cl)c1--H-CDCDt3II->z41--;-S0-*Mi(QC)>0CD))-CD>H->0II.-fl-s>1>+>Mi-S-eCD5*0>-sH-*0rtCD>-Cl)>0CD4;-1J--S-e-e-s*CDITH--SCT)*0CDEJc-H-CDqCD0CD>-SMiQH-m-s-SCD>*MiH>0-SjH(I),XCDH(-i)0CDxHCDip)CDz. cost u. (xi)}ex{_ aI(x)+1a.lC , r=2k.In spite of the fact that the introduced Lagrangians aredegenerate the rigorous approach can be developed. It demands thatwe integrate out first the field q to find the effectiveLagrangian L(). It can be easily found as a bounded functioninthe case of integrable potential.It is worth remarking that the field div p(x) has a shortrange covariance whenc(x) = (2n) 2 exp { i(k,x)} [ kfl (k2+ m) ]dkIf we prove with the help of a cluster expansion that theeffective Lagrangian depends on div q(x) then the problem of thethermodynamic limit is solved in the case (ii).REFERENCES1.J.R.Fröhlich, Y.M.Park, Commun.Math.Phys., 59, 235—266(1978),2 D.Brydges, Commun.Math.Phys., 58, 311-350(1978).3.J.Jmbrie, Commun.Math.Phys., 87, 515—565(1983).4.D.Brydges, Commun.Math.Phys., 73, 197-246(1980).5.T.Kennedy, Commun.Math.Phys., 92,,269-299(1983).6.J.R.Fontaine, Commun.Math.Phys., 103, 241-527(1986).7.W.I.Skrypnik, Teor.Mat.Fiz,, 69, 128—141(1986)8,D,Ruelle: Statistical Mechanics.Rigorous results. W.A.Benjamininc.,New—York, Amsterdam, 1969.11

On the Functional Integrals Associated to a Special Gibbs Systems with Three Body Potentials

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On the Functional Integrals Associated to a Special Gibbs Systems with Three Body Potentials

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