Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 63
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For thepurpose of generating correlation functions, this constant is irrelevant. We haveseen in Fig. 3.7 that W [0], which is equal to −F/kB T , consists of the sum of allconnected vacuum diagrams contained in Z[0].3.22.2Correlation Functions versus Connected CorrelationFunctionsUsing the logarithmic relation (3.556) between W [j] and Z[j] we can now derivegeneral relations between the n-point functions and their connected parts. For theone-point function we findG(1) (τ ) = Z −1 [j]δδZ[j] =W [j] = G(1)c (τ ).δj(τ )δj(τ )(3.577)This equation implies that the one-point function representing the ground stateexpectation value of the path x(τ ) is always connected:hx(τ )i ≡ G(1) (τ ) = G(1)c (τ ) = X.(3.578)Consider now the two-point function, which decomposes as follows:δδZ[j]δj(τ1 ) δj(τ2 )!)(δδ−1= Z [j]W [j] Z[j]δj(τ1 )δj(τ2 )G(2) (τ1 , τ2 ) = Z −1 [j]no= Z −1 [j] Wj(τ1 )j(τ2 ) + Wj(τ1 ) Wj(τ2 ) Z[j](1)(1)= G(2)c (τ1 , τ2 ) + Gc (τ1 ) Gc (τ2 ) .(3.579)In addition to the connected diagrams with two ends there are two connected diagrams ending in a single line.
These are absent in a x4 -theory at j = 0 because ofthe symmetry of the potential, which makes all odd correlation functions vanish. Inthat case, the two-point function is automatically connected.2923 External Sources, Correlations, and Perturbation TheoryFor the three-point function we findδδδZ[j]δj(τ1 ) δj(τ2 ) δj(τ3 )#)("δδδ−1= Z [j]W [j] Z[j]δj(τ1 ) δj(τ2 )δj(τ3 )ioδ nh= Z −1 [j]Wj(τ3 )j(τ2 ) + Wj(τ2 ) Wj(τ3 ) Z[j]δj(τ1 )G(3) (τ1 , τ2 , τ3 ) = Z −1 [j]n(3.580)= Z −1 [j] Wj(τ1 )j(τ2 )j(τ3 ) + Wj(τ1 ) Wj(τ2 )j(τ3 ) + Wj(τ2 ) Wj(τ1 )j(τ3 )o+ Wj(τ3 ) Wj(τ1 )j(τ2 ) + Wj(τ1 ) Wj(τ2 ) Wj(τ3 ) Z[j]hi(1)(2)(1)(1)(1)= G(3)c (τ1 , τ2 , τ3 ) + Gc (τ1 )Gc (τ2 , τ3 ) + 2 perm + Gc (τ1 )Gc (τ2 )Gc (τ3 ),and for the four-point functionhi(3)(1)G(4) (τ1 , .
. . , τ4 ) = G(4)c (τ1 , . . . , τ4 ) + Gc (τ1 , τ2 , τ3 ) Gc (τ4 ) + 3 permhi(2)+ G(2)c (τ1 , τ2 ) Gc (τ3 , τ4 ) + 2 permh(1)(1)+ G(2)c (τ1 , τ2 ) Gc (τ3 )Gc (τ4 ) + 5 perm(1)+ G(1)c (τ1 ) · · · Gc (τ4 ).i(3.581)In the pure x4 -theory there are no odd correlation functions, because of the symmetry of the potential.For the general correlation function G(n) , the total number of terms is most easilyretrieved by dropping all indices and differentiating with respect to j (the argumentsτ1 , . .
. , τn of the currents are again suppressed):G(1) = e−W eWG(2) = e−W eWG(3) = e−W eWG(4) = e−W eWj= Wj = G(1)cjj(1) 2= Wjj + Wj 2 = G(2)c + Gcjjj(2) (1)(1)3= Wjjj + 3Wjj Wj + Wj 3 = G(3)c + 3Gc Gc + Gcjjjj= Wjjjj + 4Wjjj Wj + 3Wjj 2 + 6Wjj Wj 2 + Wj 4(1)2(2)2(3) (1)+ G(1)4+ 6G(2)= G(4)c .c Gcc + 4Gc Gc + 3Gc(3.582)All equations follow from the recursion relation(n−1)G(n) = G j+ G(n−1) G(1)c ,n ≥ 2,(3.583)(n−1)= G(n)and the initial relation G(1) = G(1)if one uses Gc jcc . By comparing thefirst four relations with the explicit expressions (3.579)–(3.581) we see that thenumerical factors on the right-hand side of (3.582) refer to the permutations of thearguments τ1 , τ2 , τ3 , . .
. of otherwise equal expressions. Since there is no problem inH. Kleinert, PATH INTEGRALS3.22 Generating Functional of Connected Correlation Functions293reconstructing the explicit permutations we shall henceforth write all compositionlaws in the short-hand notation (3.582).The formula (3.582) and its generalization is often referred to as cluster decomposition, or also as the cumulant expansion, of the correlation functions.We can now prove that the connected correlation functions collect precisely allconnected diagrams in the n-point functions.
For this we observe that the decomposition rules can be inverted by repeatedly differentiating both sides of the equationW [j] = log Z[j] functionally with respect to the current j:G(1)cG(2)cG(3)cG(4)c====G(1)G(2) − G(1) G(1)G(3) − 3G(2) G(1) + 2G(1)3G(4) − 4G(3) G(1) + 12G(2) G(1)2 − 3G(2)2 − 6G(1)4 .(3.584)Each equation follows from the previous one by one more derivative with respect toj, and by replacing the derivatives on the right-hand side according to the rule(n)G j = G(n+1) − G(n) G(1) .(3.585)Again the numerical factors imply different permutations of the arguments and thesubscript j denotes functional differentiations with respect to j.Note that Eqs.
(3.584) for the connected correlation functions are valid for symmetric as well as asymmetric potentials V (x). For symmetric potentials, the equations simplify, since all terms involving G(1) = X = hxi vanish.It is obvious that any connected diagram contained in G(n) must also be contained(n)in G(n)c , since all the terms added or subtracted in (3.584) are products of G j s, andthus necessarily disconnected. Together with the proof in Section 3.22.1 that thecorrelation functions G(n)contain only the connected parts of G(n) , we can now becsure that G(n)contains precisely the connected parts of G(n) .c3.22.3Functional Generation of Vacuum DiagramsThe functional differential equation (3.570) for W [j] contains all information on theconnected correlation functions of the system.
However, it does not tell us anythingabout the vacuum diagrams of the theory. These are contained in W [0], whichremains an undetermined constant of functional integration of these equations.In order to gain information on the vacuum diagrams, we consider a modificationof the generating functional (3.558), in which we set the external source j equal tozero, but generalize the source j(τ ) in (3.558) coupled linearly to x(τ ) to a bilocalform K(τ, τ 0 ) coupled linearly to x(τ )x(τ 0 ):Z[K] =Zwhere Ae [x, K] is the Euclidean actionAe [x, K] ≡ A0 [x] + Aint [x] +12Dx(τ ) e−Ae [x,K],(3.586)Z(3.587)dτZdτ 0 x(τ )K(τ, τ 0 )x(τ 0 ).2943 External Sources, Correlations, and Perturbation TheoryWhen forming the functional derivative with respect to K(τ, τ 0 ) we obtain the correlation function in the presence of K(τ, τ 0 ):G(2) (τ, τ 0 ) = −2Z −1 [K]δZ.δK(τ, τ 0 )(3.588)At the end we shall set K(τ, τ 0 ) = 0, just as previously the source j.
When differentiating Z[K] twice, we obtain the four-point functionG(4) (τ1 , τ2 , τ3 , τ4 ) = 4Z −1 [K]δ2Z.δK(τ1 , τ2 )δK(τ3 , τ4 )(3.589)As before, we introduce the functional W [K] ≡ log Z[K]. Inserting this into (3.588)and (3.589), we findδW,(3.590)δK(τ, τ 0 )#"δWδWδ2W(4)+. (3.591)G (τ1 , τ2 , τ3 , τ4 ) = 4δK(τ1 , τ2 )δK(τ3 , τ4 ) δK(τ1 , τ2 ) δK(τ3 , τ4 )G(2) (τ, τ 0 ) = 2With the same short notation as before, we shall use again a subscript K to denotefunctional differentiation with respect to K, and writeG(2) = 2WK ,G(4) = 4 [WKK + WK WK ] = 4WKK + G(2) G(2) .(3.592)From Eq. (3.582) we know that in the absence of a source j and for a symmetricpotential, G(4) has the connectedness structure(2) (2)G(4) = G(4)c + 3Gc Gc .(3.593)This shows that in contrast to Wjjjj , the derivative WKK does not directly yield aconnected four-point function, but two disconnected parts:(2) (2)4WKK = G(4)c + 2Gc Gc ,(3.594)the two-point functions being automatically connected for a symmetric potential.More explicitly, (3.594) reads4δ 2 WδK(τ1 , τ2 )δK(τ3 , τ4 )(2)(2)(2)(2)= G(4)c (τ1 , τ2 , τ3 , τ4 ) + Gc (τ1 , τ3 )Gc (τ2 , τ4 ) + Gc (τ1 , τ4 )Gc (τ2 , τ3 ).
(3.595)Let us derive functional differential equations for Z[K] and W [K]. By analogy with(3.560) we start out with the trivial functional differential equationZDx x(τ )δe−Ae [x,K] = −δ(τ − τ 0 )Z[K],δx(τ 0 )(3.596)H. Kleinert, PATH INTEGRALS3.22 Generating Functional of Connected Correlation Functions295which is immediately verified by a functional integration by parts. Performing thefunctional derivative yieldsZDx x(τ )δAe [x, K] −Ae [x,K]e= δ(τ − τ 0 )Z[K],δx(τ 0 )(3.597)orZDxZdτZdτ 0()λ00x(τ )x3 (τ 0 ) e−Ae [x,K] = δ(τ − τ 0 )Z[K].x(τ )G−10 (τ, τ )x(τ ) +3!(3.598)For brevity, we have absorbed the source in the free-field correlation function G0 :−1G0 → [G−10 − K] .(3.599)The left-hand side of (3.598) can obviously be expressed in terms of functionalderivatives of Z[K], and we obtain the functional differential equation whose shortform reads1λ(3.600)G−10 ZK + ZKK = Z.32Inserting Z[K] = eW [K], this becomes1λG−10 WK + (WKK + WK WK ) = .32It is useful to reconsider the functional W [K] as a functional W [G0 ].δG0 /δK = G20 , and the derivatives of W [K] becomeWK = G20 WG0 ,WKK = 2G30 WG0 + G40 WG0 G0 ,(3.601)Then(3.602)and (3.601) takes the formλ1G0 WG0 + (G40 WG0 G0 + 2G30 WG0 + G40 WG0 WG0 ) = .32(3.603)This equation is represented diagrammatically in Fig.
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