Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 68

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 68)

Jackiw, Phys. Rev. D 9 , 1687 (1976)H. Kleinert, PATH INTEGRALS3.23 Path Integral Calculation of Effective Action by Loop Expansion315The functional W2 [xcl ] is defined by the path integral over the fluctuationse(i/h̄)h̄2W2 [xcl ]=RDx exp h̄iRno1δxD[xcl ]δx + R[xcl , δx]2onDδx exp h̄i 12 δxAxx [xcl ]δx,(3.715)where D[xcl ] ≡ Axx [xcl ] is the second functional derivative of the action at x =xcl . The subscripts x of Axx denote functional differentiation.

For the anharmonicoscillator:gD[xcl ] ≡ Axx [xcl ] = −∂t2 − ω 2 − x2cl .(3.716)2The functional R collects all unharmonic terms:R [xcl , δx] = A [xcl + δx] − A[xcl ] −Zdt Ax [xcl ](t)δx(t)1−2Zdtdt0 δx(t)Axx [xcl ](t, t0 )δx(t0 ).(3.717)In condensed functional vector notation, we shall write expressions like the last termasZ11(3.718)dtdt0 δx(t)Axx [xcl ](t, t0 )δx(t0 ) → δxAxx [xcl ]δx.22By construction, R is at least cubic in δx. The path integral (3.715) may thus beconsidered as the generating functional Z fl of a fluctuating variable δx(τ ) with apropagatorG[xcl ] = ih̄{Axx [xcl ]}−1 ≡ ih̄D −1 [xcl ],and an interaction R[xcl , ẋ], both depending on j via xcl .

We know from the previoussections, and will immediately see this explicitly, that h̄2 W2 [xcl ] is of order h̄2 . Letus write the full generating functional W [j] in the formW [j] = A[xcl ] + xcl j + h̄∆1 [xcl ],(3.719)where the last term collects one- and two-loop corrections (in higher-order calculations, of course, also higher loops):i∆1 [xcl ] = Tr log D[xcl ] + h̄W2 [xcl ].2(3.720)From (3.719) we find the vacuum expectation value X = hxi as the functionalderivativeX=δxδW [j]= xcl + h̄∆1xcl [xcl ] cl ,δjδj(3.721)implying the correction term X1 :X1 = ∆1xcl [xcl ]δxcl.δj(3.722)3163 External Sources, Correlations, and Perturbation TheoryThe only explicit dependence of W [j] on j comes from the second term in (3.719).In all others, the j-dependence is due to xcl [j].

We may use this fact to express j asa function of xcl . For this we consider W [j] for a moment as a functional of xcl :W [xcl ] = A[xcl ] + xcl j[xcl ] + h̄∆1 [xcl ].(3.723)The combination W [xcl ] − jX gives us the effective action Γ[X] [recall (3.662)]. Wetherefore express xcl in (3.723) as X − h̄X1 − O(h̄2 ) from (3.698), and re-expandeverything around X rather than xcl , yields1Γ[X] = A[X] − h̄AX [X]X1 − h̄X1 j[X] + h̄2 X1 jX [X]X1 + h̄2 X1 D[X]X12+ h̄∆1 [X] − h̄2 ∆1X [X]X1 + O(h̄3 ).(3.724)Since the action is extremal at xcl , we haveAX [X − h̄X1 ] = −j[X] + O(h̄2 ),(3.725)and thusAX [X] = −j[X] + h̄AXX [X]X1 + O(h̄2 ) = −j[X] + h̄D[X]X1 + O(h̄2 ),(3.726)and therefore:1Γ[X] = A[X] + h̄∆1 [X] + h̄2 − X1 D[X]X1 + X1 jX [X]X1 − ∆1X X1 .

(3.727)2From (3.722) we see thatδjX = ∆1xcl [xcl ].δxcl 1(3.728)Replacing xcl → X with an error of order h̄, this impliesδjX = ∆1X [X] + O(h̄).δX(3.729)Inserting this into (3.727), the last two terms in the curly brackets cancel, and theonly remaining h̄2 -terms are−h̄2X D[X]X1 + h̄2 W2 [X] + O(h̄3 ).2 1(3.730)From the classical equation of motion (3.668) one has a further equation for δj/δxcl :δj= −Axx [xcl ] = −D[xcl ].δxcl(3.731)Inserting this into (3.722) and replacing again xcl → X, we findX1 = −D −1 [X]∆1X [X] + O(h̄).(3.732)H. Kleinert, PATH INTEGRALS3.23 Path Integral Calculation of Effective Action by Loop Expansion317We now express ∆1X [X] via (3.720). This yields!iδ∆1X [X] = Tr D −1 [X]D[X] + h̄W2X [X] + O(h̄2 ).2δX(3.733)Inserting this into (3.732) and further into (3.727), we find for the effective actionthe expansion up to the order h̄2 :Γ[X] = A[X] + h̄Γ1 [X] + h̄2 Γ2 [X]h̄= A[X] + i Tr log D[X] + h̄2 W2 [X]2!!2h̄ 1δ1δ−1−1−1+Tr D [X]D[X] D [X] Tr D [X]D[X] .

(3.734)2 2δX2δXWe now calculate W2 [X] to lowest order in h̄. The remainder R[X; x] in (3.717) hasthe expansionR[X; δx] =11AXXX [X]δx δx δx + AXXXX [X]δx δx δx δx + . . . . (3.735)3!4!Being interested only in the h̄2 -corrections, we have simply replaced xcl by X. Inorder to obtain W2 [X], we have to calculate all connected vacuum diagrams for theinteraction terms in R[X; δx] with a δx(t)-propagatorG[X] = ih̄{AXX [X]}−1 ≡ ih̄D −1 [X].Since every contraction brings in a factor h̄, we can truncate the expansion (3.735)after δx4 .

Thus, the only contributions to ih̄W2 [X] come from the connected vacuum diagrams18+ 121+ 18,(3.736)where a line stands now for G[X], a four-vertex for(i/h̄)AXXXX [X] = (i/h̄)DXX [X],(3.737)(i/h̄)AXXX [X] = (i/h̄)DX [X].(3.738)and a three-vertex forOnly the first two diagrams are one-particle irreducible. As a pleasant result, thethird diagram which is one-particle reducible cancels with the last term in (3.734).To see this we write that term more explicitly ash̄2 −1DAD −1 AD −1 ,8 X1 X2 X1 X2 X3 X3 X30 X30 X10 X20 X10 X20(3.739)3183 External Sources, Correlations, and Perturbation Theorywhich corresponds precisely to the third diagram in Γ2 [X], except for an oppositesign.

Note that the diagram has a multiplicity 9.Thus, at the end, only the one-particle irreducible vacuum diagrams contributeto the h̄2 -correction to Γ[X]:3 −11−1−1−1−1iΓ2 [X] = i D12AX1 X2 X3 X4 D34+ i 2 AX1 X2 X3 DXDXDXAX1 X2 X3 .1 X102 X203 X304!4!(3.740)Their diagrammatic 2representation isih̄ Γ2 [X] = 18h̄+ 121(3.741)The one-particle irreducible nature of the diagrams is found to all orders in h̄.3.23.5Finite-Temperature Two-Loop Effective ActionAt finite temperature, and in D dimensions, the expansion proceeds with the imaginary-timeversions of the X-dependent Green functions (3.687) and (3.688)GL (τ1 , τ2 ) =cosh(ωL |τ1 − τ2 | − h̄βωL /2)h̄,2M ωLsinh(h̄βωL /2)(3.742)GT (τ1 , τ2 ) =h̄cosh(ωT |τ1 − τ2 | − h̄βωT /2),2M ωTsinh(h̄βωT /2)(3.743)andwhere we have omitted the argument X in√ ωL (X) and ωT (X).

Treating here the general rotationallysymmetric potential V (x) = v(x), x = x2 , the two frequencies are1 001 0v (X), ωT2 (X) ≡v (X).(3.744)MMXWe also decompose the vertex functions into longitudinal and transverse parts. The three-pointvertex is a sum 00∂ 3 v(X)v (X) v 0 (X)L 000T,(3.745)= Pijkv (X) + Pijk−∂Xi ∂Xj ∂XkXX22ωL(X) ≡with the symmetric tensorsXi Xj XkX3The four-point vertex readsLPijk≡andTPijk≡ δijXkXjXiL+ δik+ δjk− 3Pijk.XXX 00000∂ 4 v(X)v (X) v 0 (X)LT v (X)S,= Pijklv (4) (X) + Pijkl+ Pijkl−∂Xi ∂Xj ∂Xk ∂XlXX2X3(3.746)(3.747)with the symmetric tensorsLPijkl=Xi Xj Xk Xl,X4TPijkl= δij(3.748)Xk XlXj XlXj XkXi XlXi XkXi XkL+δik+δil+δjk+δjl+δkl−6Pijkl,22222XXXXXX2SLTPijkl= δij δkl + δik δjl + δil δjk − 3Pijkl− 3Pijkl.(3.749)(3.750)H. Kleinert, PATH INTEGRALS3.23 Path Integral Calculation of Effective Action by Loop Expansion319The tensors obey the following relations:Xi LLP = Pjk,X ijkXi TTP = Pjk,X ijkXl TXj TXk TTPjl +Pjk , PijL Pikl=P , P T P L = 0,XXX kl ij iklLTTTL TTL TT= Pijkl, PhijPhkl= PijT Pkl+ PikPjl + PilL Pjk+ PjkPil + PjlL Pik,L TTLT LL LLT TL= Pij Pkl , Phij Phkl = Pij Pkl , Pij Pijkl = Pkl , Pij Pijkl = (D−1)Pkl,LLTPijL Pikl= Pjkl, PijT Pikl=LLPhijPhklLTPhijPhklTTPijL Pijkl= Pkl,LPijT Pijkl= 0,STSTLPijL Pijkl= −2Pkl, PijT Pijkl= (D + 1)Pkl− 2(D − 1)Pkl.(3.751)(3.752)(3.753)(3.754)(3.755)(3.756)Instead of the effective action, the diagrammatic expansion (3.741) yields now the free energy(i/h̄)Γ[X] → −βF (X).(3.757)Using the above formulas we obtain immediately the mean field contribution to the free energyZ h̄β M 2−βFMF = −dτẊ + v(X) ,(3.758)20and the one-loop contribution [from the trace-log term in Eq.

(3.734)]:−βF1−loop = − log [2 sinh(h̄βωL /2)] − (D − 1) log [2 sinh(h̄βωT /2)] .The first of the two-loop diagrams in (3.741) yields the contribution to the free energy 00v (X) v 0 (X)2(4)22−β∆1 F2−loop = −β GL (τ, τ )v (X) + (D − 1) GT (τ, τ )−X2X3 000000v (X) 2v (X) 2v (X)+ 2(D−1)GL (τ, τ )GT (τ, τ ).−+XX2X3From the second diagram we obtain the contributionZ h̄βZ h̄β12−β∆2 F2−loop = 2dτ1dτ2 GL3 (τ1 , τ2 ) [v 000 (X)]h̄ 002 00v (X) v 0 (X)−.+ 3(D − 1)GL (τ1 , τ2 )GT2 (τ1 , τ2 )XX2(3.759)(3.760)(3.761)The explicit evaluation yieldsh̄2 β=−(2M )212(4)(X)(3.762)2 coth (h̄βωL /2)vωLD2 − 1v 00 (X) v 0 (X)2+coth(h̄βω/2)−TωT2X2X3 000v (X) 2v 00 (X) 2v 0 (X)2(D − 1)coth(h̄βωL /2) coth(h̄βωT /2)−++.ωL ωTXX2X3−β∆1 F2−loopand112h̄2 β0002 1[v (X)]+−β∆2 F2−loop =ωL (2M ωL )33 sinh2 (h̄βωL /2) 00216h̄2 β(D − 1) 1v (X) v 0 (X)−(3.763)+2ωT + ωL 2M ωL (2M ωT )2XX2ωT1ωTsinh[h̄β(2ωT −ωL)/2]× coth2 (h̄βωT /2) ++.ωL sinh2 (h̄βωT /2) 2ωT −ωL sinh(h̄βωL /2) sinh2 (h̄βωT /2)3203 External Sources, Correlations, and Perturbation TheoryIn the limit of zero temperature, the effective potential in the free energy becomes(h̄ωL1 (4)h̄ωTh̄2Veff (X) = v(X) ++ (D − 1)+(X)2vT →0228(2M )2 ωL)D2 − 1 v 00 (X) v 0 (X)2(D − 1) v 000 (X) 2v 00 (X) 2v 0 (X)+−−++ωT2X2X3ωL ωTXX2X3()2 00h̄213(D − 1) 1v (X) v 0 (X)0002−+ O(h̄3 ).[v(X)]+−46(2M )3 3ωL2ωT + ωL ωL ωT2XX2(3.764)For the one-dimensional potentialV (x) =M 2 2 g3 3 g4 4ω x + x + x ,23!4!(3.765)the effective potential becomes, up to two loops,Veff (X) =1g41M 2 2ω X +g3 X 3 +g4 X 4 + log (2 sinh h̄βω/2)+h̄22β8(2M ω)2 tanh2 (h̄βω/2)1h̄2 (g3 + g4 X)2 1+ O(h̄3 ) ,(3.766)+−6ω (2M ω)33 sinh2 (h̄βω/2)whose T → 0 limit isVeff (X)=T →0M 2 2 g3 3 g4 4 h̄ωg4ω X + X + X ++ h̄223!4!28(2M ω)2−h̄2 (g3 + g4 X)2+ O(h̄3 ) .18ω (2M ω)3(3.767)If the potential is a polynomial in X, the effective potential at zero temperature can be solvedmore efficiently than here and to much higher loop orders with the help of recursion relations.

Характеристики

Список файлов книги