Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 66
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. . , τn ).Actually, we shall never make use of the full correlation function (3.652), but onlyof the integral over τ in (3.652). This can be obtained directly from the generatingfunctional Z[j] of all correlation functions by differentiation with respect to thesquare mass in addition to the source termsZdτ G(1,n)(τ, τ1 , . .
. , τn ) = − Z−1∂δδ···Z[j].2δj(τ)δj(τ)M∂ω1nj=0(3.653)By going over to the generating functional W [j], we obtain in a similar way theconnected parts:ZdτGc(1,n) (τ, τ1 , . . . , τn )δ∂δ=−···W[j].2δj(τn )M∂ω δj(τ1 )j=0(3.654)The right-hand side can be rewritten asZ18dτ Gc(1,n) (τ, τ1 , . . . , τn ) = −L.D.
Landau, J.E.T.P. 7 , 627 (1937).∂G(n) (τ1 , . . . , τn ).M∂ω 2 c(3.655)3063 External Sources, Correlations, and Perturbation TheoryThe connected correlation functions Gc(1,n) (τ, τ1 , . . . , τn ) can be decomposed intotree diagrams consisting of lines and one-particle irreducible vertex functionsΓ(1,n) (τ, τ1 , . . . , τn ). If integrated over τ , these are defined from Legendre transform(3.619) by a further differentiation with respect to Mω 2 :Zdτ Γ(1,n)δδ∂ ,···Γ[X](τ, τ1 , . . . , τn ) = −δX(τn )M∂ω 2 δX(τ1 )X0(3.656)implying the relationZ3.23dτ Γ(1,n) (τ, τ1 , . . . , τn ) = −∂(n)(τ1 , . . .
, τn ).2ΓM∂ω(3.657)Path Integral Calculation of Effective Actionby Loop ExpansionPath integrals give the most direct access to the effective action of a theory avoidingthe cumbersome Legendre transforms. The derivation will proceed diagrammaticallyloop by loop, which will turn out to be organized by the powers of the Planckconstant h̄. This will now be kept explicit in all formulas. For later applications toquantum mechanics we shall work with real time.3.23.1General FormalismConsider the generating functional of all Green functionsZ[j] = eiW [j]/h̄ ,(3.658)where W [j] is the generating functional of all connected Green functions. The vacuum expectation of the field, the averageX(t) ≡ hx(t)i,(3.659)is given by the first functional derivativeX(t) = δW [j]/δj(t).(3.660)This can be inverted to yield j(t) as a functional of X(t):j(t) = j[X](t),(3.661)which leads to the Legendre transform of W [j]:Γ[X] ≡ W [j] −Zdt j(t)X(t),(3.662)H.
Kleinert, PATH INTEGRALS3073.23 Path Integral Calculation of Effective Action by Loop Expansionwhere the right-hand side is replaced by (3.661). This is the effective action of thetheory. The effective action for time independent X(t) ≡ X defines the effectivepotential1V eff (X) ≡ −Γ[X].(3.663)tb − taThe first functional derivative of the effective action gives back the currentδΓ[X]= −j(t).δX(t)(3.664)The generating functional of all connected Green functions can be recovered fromthe effective action by the inverse Legendre transformW [j] = Γ[X] +Zdt j(t)X(t).(3.665)We now calculate these quantities from the path integral formula (3.558) for thegenerating functional Z[j]:Z[j] =ZDx(t)e(i/h̄){A[x]+Rdt j(t)x(t)}.(3.666)With (3.658), this amounts to the path integral formula for Γ[X]:e h̄ {Γ[X]+iRdt j(t)X(t)}=ZDx(t)e(i/h̄){A[x]+dt j(t)x(t)}R.(3.667)The action quantum h̄ is a measure for the size of quantum fluctuations.
Under manyphysical circumstances, quantum fluctuations are small, which makes it desirable todevelop a method of evaluating (3.667) as an expansion in powers of h̄.3.23.2Mean-Field ApproximationFor h̄ → 0, the path integral over the path x(t) in (3.666) is dominated by theclassical solution xcl (t) which extremizes the exponentδA[x] = −j(t),δx(t) x=xcl (t)(3.668)and is a functional of j(t) which may be written, more explicitly, as xcl (t)[j]. Atthis level we can identifyW [j] = Γ[X] +Zdt j(t)X(t) ≈ A[xcl [j]] +Zdt j(t)xcl (t)[j].(3.669)By differentiating W [j] with respect to j, we have from the general first part ofEq. (3.659):X=δΓ δXδXδW=+X +j.δjδX δjδj(3.670)3083 External Sources, Correlations, and Perturbation TheoryInserting the classical equation of motion (3.668), this becomesX=δxδA δxcl+ xcl + j cl = xcl .δxcl δjδj(3.671)Thus, to this approximation, X(t) coincides with the classical path xcl (t).
Replacingxcl (t) → X(t) on the right-hand side of Eq. (3.669), we obtain the lowest-order result,which is of zeroth order in h̄, the classical approximation to the effective action:Γ0 [X] = A[X].(3.672)For an anharmonic oscillator in N dimensions with unit mass and an interactionx , where x = (x1 , .
. . , xN ), which is symmetric under N-dimensional rotationsO(N), the lowest-order effective action reads4Γ0 [X] =Z1 2g 2 2dtXa,Ẋa − ω 2 Xa2 −24!(3.673)where repeated indices a, b, . . . are summed from 1 to N following Einstein’s summation convention. The effective potential (3.663) is simply the initial potentialV0eff (X) = V (X) =ω 2 2 g 2 2Xa .X +2 a 4!(3.674)For ω 2 > 0, this has a minimum at X ≡ 0, and there are only two non-vanishingvertex functions Γ(n) (t1 , . . . , tn ):For n = 2:(2)Γ (t1 , t2 )ab ≡δ2Aδ2Γ=δXa (t1 )δXb (t2 ) Xa =0xa (t1 )xb (t2 ) xa =Xa =0= (−∂t2 − ω 2 )δab δ(t1 − t2 ).(3.675)This determines the inverse of the propagator:Γ(2) (t1 , t2 )ab = [ih̄G−1 ]ab (t1 , t2 ).(3.676)Thus we find to this zeroth-order approximation that Gab (t1 , t2 ) is equal to the freepropagator:Gab (t1 , t2 ) = G0ab (t1 , t2 ).(3.677)For n = 4:Γ(4) (t1 , t2 , t3 , t4 )abcd ≡withδ4Γ= gTabcd ,δXa (t1 )δXb (t2 )δXc (t3 )δXd (t4 )1Tabcd = (δab δcd + δac δbd + δad δbc ).3(3.678)(3.679)H.
Kleinert, PATH INTEGRALS3093.23 Path Integral Calculation of Effective Action by Loop ExpansionAccording to the definition of the effective action, all diagrams of the theorycan be composed from the propagator Gab (t1 , t2 ) and this vertex via tree diagrams.Thus we see that in this lowest approximation, we recover precisely the subset ofall original Feynman diagrams with a tree-like topology.
These are all diagramswhich do not involve any loops. Since the limit h̄ → 0 corresponds to the classicalequations of motion with no quantum fluctuations we conclude: Classical theorycorresponds to tree diagrams.For ω 2 < 0 the discussion is more involved since the minimum of the potential(3.674) lies no longer at X = 0, but at a nonzero vector X0 with an arbitrarydirection, and a lengthq|X0| = −6ω 2 /g.(3.680)The second functional derivative (3.675) at X is anisotropic and readsω2 > 0V eff (X)ω2 < 0V eff (X)X2X2X1X1Figure 3.13 Effective potential for ω 2 > 0 and ω 2 < 0 in mean-field approximation,pictured for the case of two components X1 , X2 . The right-hand figure looks like a Mexicanhat or a champaign bottle..(2)Γ (t1 , t2 )abδ2Aδ2Γ=≡δXa (t1 )δXb (t2 ) Xa 6=0xa (t1 )xb (t2 ) xa =Xa 6=0g222= −∂t − ω −δ X + 2Xa Xb δab δ(t1 − t2 ).6 ab c(3.681)This is conveniently separated into longitudinal and transversal derivatives withrespect to the direction X̂ = X/|X|.
We introduce associated projection matrices:PLab (X̂) = X̂a X̂b ,PT ab (X̂) = δab − X̂a X̂b ,(3.682)and decompose(2)(2)Γ(2) (t1 , t2 )ab = ΓL (t1 , t2 )ab PLab (X̂) + ΓT (t1 , t2 )ab PT ab (X̂),(3.683)3103 External Sources, Correlations, and Perturbation Theorywhereg(2)ΓL (t1 , t2 )ab = −∂t2 − ω 2 + X26andg=− ω + 3 X26This can easily be inverted to find the propagator(2)ΓL (t1 , t2 )abhG(t1 , t2 )ab = ih̄ Γ(2) (t1 , t2 )i−1ab−∂t22δ(t1 − t2 ),(3.684)δ(t1 − t2 ).= GL (t1 , t2 )ab PLab (X̂)+GT (t1 , t2 )ab PT ab (X̂),(3.685)(3.686)whereih̄ih̄=,2ΓL (t1 , t2 )−∂t − ωL2 (X)ih̄ih̄==2(2)−∂t − ωT2 (X)ΓT (t1 , t2 )GL (t1 , t2 )ab =(3.687)GT (t1 , t2 )ab(3.688)are the longitudinal and transversal parts of the Green function.
For convenience,we have introduced the X-dependent frequencies of the longitudinal and transversalGreen functions:ggωL2 (X) ≡ ω 2 + 3 X2 , ωT2 (X) ≡ ω 2 + X2 .66(3.689)To emphasize the fact that this propagator is a functional of X we represent it bythe calligraphic letter G. For ω 2 > 0, we perform the fluctuation expansion aroundthe minimum of the potential (3.663) at X = 0, where the two Green functionscoincide, both having the same frequency ω:GL (t1 , t2 )ab |X=0 = GT (t1 , t2 )ab |X=0 = G(t1 , t2 )ab |X=0 =ih̄,− ω2−∂t2(3.690)For ω 2 < 0, however, where the minimum lies at the vector X0 of length (3.680),they are different:GL (t1 , t2 )ab |X=X0 =−∂t2ih̄ih̄.2 , GT (t1 , t2 )ab |X=X0 =+ 2ω−∂t2(3.691)Since the curvature of the potential at the minimum in radial direction of X ispositive at the minimum, the longitudinal part has now the positive frequency −2ω 2 .The movement along the valley of the minimum, on the other hand, does not increasethe energy.
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