Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 69
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Thiswill be shown in Appendix 3C.5.3.23.6Background Field Method for Effective ActionIn order to find the rules for the loop expansion to any order, let us separate thetotal effective action into a sum of the classical action A[X] and a term Γfl [X] whichcollects the contribution of all quantum fluctuations:Γ[X] = A[X] + Γfl [X].(3.768)To calculate the fluctuation part Γfl [X], we expand the paths x(t) around somearbitrarily chosen background path X(t):20x(t) = X(t) + δx(t),(3.769)and calculate the generating functional W [j] by performing the path integral overthe fluctuations: ZiiW [j] = Dδx expA [X + δx] + j[X](X + δx) .(3.770)exph̄h̄20In the theory of fluctuating fields, this is replaced by a more general background field whichexplains the name of the method.H. Kleinert, PATH INTEGRALS3213.23 Path Integral Calculation of Effective Action by Loop ExpansionFrom W [j] we find a j-dependent expectation value Xj = hxij as Xj = δW [j]/δj,and the Legendre transform Γ[X] = W [j] − jXj .
In terms of Xj , Eq. (3.770) can berewritten asiexpΓ[Xj ] + j[Xj ] Xj =h̄ZDδx exp ih̄A [X + δx] + j[X](X + δx). (3.771)The expectation value Xj has the property of extremizing Γ[X], i.e., it satisfies theequationδΓ[X] j=−= −ΓX [Xj ].(3.772)∂X X=XjWe now choose j in such a way that Xj equals the initially chosen X, and findoiinA [X+δx] − ΓX [X]δx .expΓ[X] = Dδx exph̄h̄Z(3.773)This is a functional integro-differential equation for the effective action Γ[X] whichwe can solve perturbatively order by order in h̄.
This is done diagrammatically. Thediagrammatic elements are lines representing the propagator (3.686)δ 2 A[X]= Gab [X] ≡ ih̄δXa δXb"#−1,(3.774)aband verticesn6152=δ n A[X].δXa1 δXa2 . . . δXan(3.775)43From the explicit calculations in the last two subsections we expect the effectiveaction to be the sum of all one-particle irreducible vacuum diagrams formed withthese propagators and vertices. This will now be proved to all orders in perturbationtheory.hiWe introduce an auxiliary generating functional W̃ X, j̃ which governs the correlation functions of the fluctuations δx around the above fixed backgound X:nhioexp iW̃ X, j̃ /h̄ ≡ZiDδx expà [X, δx] +h̄Zdt j̃(t) δx(t),(3.776)with the action of fluctuationsÃ[X, δx] = A[X + δx] − A[X] − AX [X]δx,(3.777)3223 External Sources, Correlations, and Perturbation Theorywhose expansion in powers of δx(t) starts out with a quadratic term.
A source j̃(t)is coupled to the fluctuations δx(t). By comparing (3.776) with (3.773) we see thatfor the special choice of the currentj̃ = −ΓX [X] + AX [X] = −Γ̃X [X],(3.778)the right-hand sides coincide, such that the auxiliary functional W̃ [X, j̃] contains precisely the diagrams in Γfl [X] which we want to calculate.
We now form the Legendretransform of W̃ [X, j̃], which is an auxiliary effective action with two arguments:hiΓ̃ X, X̃ ≡ W̃ [X, j̃] −Zdt j̃ X̃,(3.779)with the auxiliary conjugate variableX̃ =δ W̃ [X, j̃]= X̃[X, j̃].δ j̃(3.780)This is the expectation value of the fluctuations hδxi in the path integral (3.776). Ifj̃ has the value (3.778), this expectation vanishes, i.e. X̃ = 0. The auxiliary actionΓ̃ [X, 0] coincides with the fluctuating part Γfl [X] of the effective action which wewant to calculate.The functional derivatives of W̃ [X, j̃] with respect to j̃ yield all connected correlationhifunctions of the fluctuating variables δx(t). The functional derivatives ofΓ̃ X, X̃ with respect to X̃ select from these the one-particle irreducible correlationfunctions.
For X̃ = 0, only vacuum diagrams survive.Thus we have proved that the full effective action is obtained from the sum of theclassical action Γ0 [X] = A[X], the one-loop contribution Γ1 [X] given by the traceof the logarithm in Eq. (3.702), the two-loop contribution Γ2 [X] in (3.741), and thesum of all connected one-particle irreducible vacuum diagrams with more than twoloopsi X nih̄ Γn [X] =h̄ n≥3(3.781)Observe that in the expansion of Γ[X]/h̄, each line carries a factor h̄, whereas eachn-point vertex contributes a factor h̄−1 .
The contribution of an n-loop diagram toΓ[X] is therefore of order h̄n . The higher-loop diagrams are most easily generatedby a recursive treatment of the type developed in Subsection 3.22.3.For a harmonic oscillator, the expansion stops after the trace of the logarithm(3.702), and reads simply, in one dimension:iΓ[X] = A[X] + h̄ Tr log Γ(2) (tb , ta )"2#Z tbM 2 Mω 2 2i=dtẊ −X + h̄ Tr log −∂t2 − ω 2 .222ta(3.782)H. Kleinert, PATH INTEGRALS3233.24 Nambu-Goldstone TheoremEvaluating the trace of the logarithm we find for a constant X the effective potential(3.663):V eff (X) = V (X) −ilog{2πi sin[ω(tb − ta )] /Mω}.2(tb − ta )(3.783)If the boundary conditions are periodic, so that the analytic continuation of theresult can be used for quantum statistical calculations, the result isV eff (X) = V (X) −ilog{2i sin[ω(tb − ta )/2]}.(tb − ta )(3.784)It is important to keep in mind that a line in the above diagrams containsan infinite series of fundamental Feynman diagrams of the original perturbationexpansion, as can be seen by expanding the denominators in the propagator Gab inEqs.
(3.686)–(3.688) in powers of X2 . This expansion produces a sum of diagramswhich can be obtained from the loop diagrams in the expansion of the trace of thelogarithm in (3.707) by cutting the loop.If the potential is a polynomial in X, the effective potential at zero temperaturecan be solved most efficiently to high loop orders with the help of recursion relations.This is shown in detail in Appendix 3C.5.3.24Nambu-Goldstone TheoremThe appearance of a zero-frequency mode as a consequence of a nonzero expectation value X can easily be proved for any continuous symmetry and to all ordersin perturbation theory by using the full effective action.
To be more specific weconsider as before the case of O(N)-symmetry, and perform infinitesimal symmetrytransformations on the currents j in the generating functional W [j]:ja → ja − icd (Lcd )ab jb ,(3.785)where Lcd are the N(N −1)/2 generators of O(N)-rotations with the matrix elements(Lcd )ab = i (δca δdb − δda δcb ) ,(3.786)and ab are the infinitesimal angles of the rotations. Under these, the generatingfunctional is assumed to be invariant:δW [j] = 0 =ZdtδW [j]i (Lcd )ab jb cd = 0.δja (x)(3.787)Expressing the integrand in terms of Legendre-transformed quantities viaEqs. (3.620) and (3.622), we obtainZdtXa (t)i (Lcd )abδΓ[X] = 0.δXb (t) cd(3.788)3243 External Sources, Correlations, and Perturbation TheoryThis expresses the infinitesimal invariance of the effective action Γ[X] under infinitesimal rotationsXa → Xa − icd (Lcd )ab Xb .The invariance property (3.788) is called the Ward-Takakashi identity for the functional Γ[X]. It can be used to find an infinite set of equally named identities for allvertex functions by forming all Γ[X] functional derivatives of Γ[X] and setting Xequal to the expectation value at the minimum of Γ[X].
The first derivative of Γ[X]gives directly from (3.788) (dropping the infinitesimal parameter cd )δΓ[X]δX(t)b(Lcd )ab jb (t) = (Lcd )ab= −Z00dt Xa0 (t ) (Lcd )a0 bδ 2 Γ[X].δXb (t0 )δXn (t)(3.789)Denoting the expectation value at the minimum of the effective potential by X̄, thisyieldsZ00dt X̄a0 (t ) (Lcd )a0 bδ 2 Γ[X]= 0.δXb (t0 )δXa (t) X(t)=X̄(3.790)Now the second derivative is simply the vertex function Γ(2) (t0 , t) which is the functional inverse of the correlation function G(2) (t0 , t).
The integral over t selects thezero-frequency component of the Fourier transformΓ̃(2) (ω 0 ) ≡Z0dt0 eiω t Γ(2) (t0 , t).(3.791)If we define the Fourier components of Γ(2) (t0 , t) accordingly, we can write (3.790)in Fourier space as0Xa00 (Lcd )a0 b G̃−1ba (ω = 0) = 0.(3.792)Inserting the matrix elements (3.786) of the generators of the rotations, this equationshows that for X̄ 6= 0, the fully interacting transverse propagator has to possess asingularity at ω 0 = 0. In quantum field theory, this implies the existence of N − 1massless particles, the Nambu-Goldstone boson. The conclusion may be drawnonly if there are no massless particles in the theory from the outset, which may be“eaten up” by the Nambu-Goldstone boson, as explained earlier in the context ofEq.
(3.688).As mentioned before at the end of Subsection 3.23.1, the Nambu-Goldstone theorem does not have any consequences for quantum mechanics since fluctuations aretoo violent to allow for the existence of a nonzero expectation value X. The effectiveaction calculated to any finite order in perturbation theory, however, is incapable ofreproducing this physical property and does have a nonzero extremum and ensuingtransverse zero-frequency modes.H. Kleinert, PATH INTEGRALS3253.25 Effective Classical Potential3.25Effective Classical PotentialThe loop expansion of the effective action Γ[X] in (3.768), consisting of the traceof the logarithm (3.702) and the one-particle irreducible diagrams (3.741), (3.781)and the associated effective potential V (X) in Eq.
(3.663), can be continued in astraightforward way to imaginary times setting tb −ta → −ih̄β to form the Euclideaneffective potential Γe [X]. For the harmonic oscillator, where the expansion stopsafter the trace of the logarithm and the effective potential reduces to the simpleexpression (3.782), we find the imaginary-time versionVeff!βh̄ω1(X) = V (X) + log 2 sinh.β2(3.793)Since the effective action contains the effect of all fluctuations, the minimum of theeffective potential V (X) should yield directly the full quantum statistical partitionfunction of a system:Z = exp[−βV (X) ].(3.794)minInserting the harmonic oscillator expression (3.793) we find indeed the correct result(2.399).For anharmonic systems, we expect the loop expansion to be able to approximate V (X) rather well to yield a good approximation for the partition functionvia Eq.
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