Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 65
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. , τn ) ≡...Γ[X] .(3.625)δX(τ1 )δX(τn )We shall see that the proper vertex functions are obtained from these functionsPby a Fourier transform and a simple removal of an overall factor (2π)D δ ( ni=1 ωi )to ensure momentum conservation. The functions Γ(n) (τ1 , . . . , τn ) will therefore becalled vertex functions, without the adjective proper which indicates the absenceof the δ-function.
In particular, the Fourier transforms of the vertex functionsΓ(2) (τ1 , τ2 ) and Γ(4) (τ1 , τ2 , τ3 , τ4 ) are related to their proper versions byΓ(2) (ω1 , ω2 ) = 2πδ (ω1 + ω2 ) Γ̄(2) (ω1 ),Γ(4) (ω1 , ω2 , ω3 , ω4 ) = 2πδ4Xi=1(3.626)!ωi Γ̄(4) (ω1 , ω2 , ω3 , ω4 ).(3.627)For the functional derivatives (3.625) we shall use the same short-hand notation asfor the functional derivatives (3.569) of W [j], settingΓX(τ1 )...X(τn ) ≡δδ...Γ[X] .δX(τ1 )δX(τn )(3.628)The arguments τ1 , .
. . , τn will usually be suppressed.In order to derive relations between the derivatives of the effective action andthe connected correlation functions, we first observe that the connected one-pointfunction G(1)c at a nonzero source j is simply the path expectation X [recall (3.578)]:G(1)c = X.(3.629)Second, we see that the connected two-point function at a nonzero source j is givenbyG(2)c=(1)GjδX= Wjj ==δjδjδX!−1= Γ−1XX .(3.630)17In higher dimensions there can be crystal- or quasicrystal-like modulations.
See, for example, H. Kleinert and K. Maki, Fortschr. Phys. 29, 1 (1981) (http://www.physik.fu-berlin.de/~kleinert/75). This paper was the first to investigate in detail icosahedral quasicrystalline structures discovered later in aluminum.H. Kleinert, PATH INTEGRALS3.22 Generating Functional of Connected Correlation Functions301The inverse symbols on the right-hand side are to be understood in the functionalsense, i.e., Γ−1XX denotes the functional matrix:−1ΓX(τ)X(y)δ2Γ≡δX(τ )δX(τ 0 )"#−1,(3.631)which satisfiesZ−100dτ 0 ΓX(τ)X(τ 0 ) ΓX(τ 0 )X(τ 00 = δ(τ − τ ).(3.632)Relation (3.630) states that the second derivative of the effective action determines directly the connected correlation function G(2)c (ω) of the interacting theoryin the presence of the external source j.
Since j is an auxiliary quantity, whicheventually be set equal to zero thus making X equal to X0 , the actual physicalpropagator is given byG(2)c j=0−1= ΓXXX=X0.(3.633)By Fourier-transforming this relation and removing a δ-function for the overall momentum conservation, the full propagator Gω2 (ω) is related to the vertex functionΓ(2) (ω), defined in (3.626) byGω2 (ω) ≡ Ḡ(2) (k) =1.Γ̄ (ω)(2)(3.634)The third derivative of the generating functional W [j] is obtained by functionallydifferentiating Wjj in Eq. (3.630) once more with respect to j, and applying the chainrule:Wjjj = −Γ−2XX ΓXXXδX(2) 3= −Γ−3ΓXXX .XX ΓXXX = −Gcδj(3.635)This equation has a simple physical meaning.
The third derivative of W [j] on theleft-hand side is the full three-point function at a nonzero source j, so that3(2)G(3)ΓXXX .c = Wjjj = −Gc(3.636)This equation states that the full three-point function arises from a third derivativeof Γ[X] by attaching to each derivation a full propagator, apart from a minus sign.We shall express Eq. (3.636) diagrammatically as follows:where3023 External Sources, Correlations, and Perturbation Theorydenotes the connected n-point function, andthe negative n-point vertex function.For the general analysis of the diagrammatic content of the effective action, weobserve that according to Eq.
(3.635), the functional derivative of the correlationfunction G with respect to the current j satisfies3(3)(2)ΓXXX .G(2)c j = Wjjj = Gc = −Gc(3.637)This is pictured diagrammatically as follows:(3.638)This equation may be differentiated further with respect to j in a diagrammaticway.
From the definition (3.557) we deduce the trivial recursion relationG(n)c (τ1 , . . . , τn ) =δGc(n−1) τ1 , . . . , τn−1 ,δj(τn )(3.639)which is represented diagrammatically asBy applying δ/δj repeatedly to the left-hand side of Eq. (3.637), we generate allhigher connected correlation functions. On the right-hand side of (3.637), the chainrule leads to a derivative of all correlation functions G = G(2)with respect to j,cthereby changing a line into a line with an extra three-point vertex as indicated inthe diagrammatic equation (3.638). On the other hand, the vertex function ΓXXXmust be differentiated with respect to j. Using the chain rule, we obtain for anyn-point vertex function:ΓX...Xj = ΓX...XXδX= ΓX...XX G(2)c ,δj(3.640)which may be represented diagrammatically asH.
Kleinert, PATH INTEGRALS3.22 Generating Functional of Connected Correlation Functions303With these diagrammatic rules, we can differentiate (3.635) any number of times,and derive the diagrammatic structure of the connected correlation functions withan arbitrary number of external legs. The result up to n = 5 is shown in Fig.
3.12.Figure 3.12 Diagrammatic differentiations for deriving tree decomposition of connectedcorrelation functions. The last term in each decomposition yields, after amputation andremoval of an overall δ-function of momentum conservation, precisely all one-particle irreducible diagrams.The diagrams generated in this way have a tree-like structure, and for this reasonthey are called tree diagrams. The tree decomposition reduces all diagrams to theirone-particle irreducible contents.The effective action Γ[X] can be used to prove an important composition theorem: The full propagator G can be expressed as a geometric series involving theso-called self-energy.
Let us decompose the vertex function asintΓ̄(2) = G−10 + Γ̄XX ,(3.641)3043 External Sources, Correlations, and Perturbation Theorysuch that the full propagator (3.633) can be rewritten asG = 1 + G0 Γ̄intXX−1G0 .(3.642)Expanding the denominator, this can also be expressed in the form of an integralequation:intintG = G0 − G0 Γ̄intXX G0 + G0 Γ̄XX G0 Γ̄XX G0 − . . . .(3.643)The quantity −Γ̄intXX is called the self-energy, commonly denoted by Σ:Σ ≡ −Γ̄intXX ,(3.644)i.e., the self-energy is given by the interacting part of the second functional derivativeof the effective action, except for the opposite sign.According to Eq. (3.643), all diagrams in G can be obtained from a repetition ofself-energy diagrams connected by a single line.
In terms of Σ, the full propagatorreads, according to Eq. (3.642):−1G ≡ [G−10 − Σ] .(3.645)This equation can, incidentally, be rewritten in the form of an integral equation forthe correlation function G:G = G0 + G0 ΣG.3.22.6(3.646)Ginzburg-Landau Approximation to GeneratingFunctionalSince the vertex functions are the functional derivatives of the effective action [see(3.625)], we can expand the effective action into a functional Taylor seriesΓ[X] =∞X1n=0 n!Zdτ1 . . . dτn Γ(n) (τ1 , . .
. , τn )X(τ1 ) . . . X(τn ).(3.647)The expansion in the number of loops of the generating functional Γ[X] collectssystematically the contributions of fluctuations. To zeroth order, all fluctuationsare neglected, and the effective action reduces to the initial action, which is themean-field approximation to the effective action. In fact, in the absence of loopdiagrams, the vertex functions contain only the lowest-order terms in Γ(2) and Γ(4) :(2)Γ0 (τ1 , τ2 ) = M −∂τ21 + ω 2 δ(τ1 −τ2 ),(3.648)Γ0 (τ1 , τ2 , τ3 , τ4 ) = λ δ(τ1 −τ2 )δ(τ1 − τ3 )δ(τ1 − τ4 ).(3.649)(4)Inserted into (3.647), this yields the zero-loop approximation to Γ[X]:Γ0 [X] =M2!Zdτ [(∂τ X)2 + ω 2 X 2 ] +λ4!Zdτ X 4 .(3.650)H.
Kleinert, PATH INTEGRALS3053.22 Generating Functional of Connected Correlation FunctionsThis is precisely the original action functional (3.559). By generalizing X(τ ) to be amagnetization vector field, X(τ ) → M(x), which depends on the three-dimensionalspace variables x rather than the Euclidean time, the functional (3.650) coincideswith the phenomenological energy functional set up by Ginzburg and Landau todescribe the behavior of magnetic materials near the Curie point, which they wroteas18Γ[M] =Z31Xm2 2 λ 42dxM + M .(∂ M) +2 i=1 i2!4!3"#(3.651)The use of this functional is also referred to as mean-field theory or mean-fieldapproximation to the full theory.3.22.7Composite FieldsSometimes it is of interest to study also correlation functions in which two fieldscoincide at one point, for instance1G(1,n) (τ, τ1 , . .
. , τn ) = hx2 (τ )x(τ1 ) · · · x(τn )i.2(3.652)If multiplied by a factor Mω 2 , the composite operator Mω 2 x2 (τ )/2 is preciselythe frequency term in the action energy functional (3.559). For this reason onespeaks of a frequency insertion, or, since in the Ginzburg-Landau action (3.651) thefrequency ω is denoted by the mass symbol m, one speaks of a mass insertion intothe correlation function G(n) (τ1 , .
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