Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 72
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(3.856)2Mβ m=1 ωm + Ω2 (x0 )Performing the sum gives once more the subtracted correlation function Eq. (3.839),whose generating functional was calculated in (3.844).The calculation of the harmonic averages in (3.851) leads to a similar loop expansion as for the effective potential in Subsection 3.23.6 using the background fieldmethod. The path average x0 takes over the role of the background X and the nonzero Matsubara frequency part of the paths η(τ ) corresponds to the fluctuations.The only difference with respect to the earlier calculations is that the correlationfunctions of η(τ ) contain no zero-frequency contribution. Thus they are obtainedfrom the subtracted Green functions GpΩ20 (x0 ),e(τ ) defined in Eq.
(3.802).All Feynman diagrams in the loop expansion are one-particle irreducible, just asin the loop expansion of the effective potential. The reducible diagrams are absentsince there is no linear term in the interaction (3.850). This trivial absence is an advantage with respect to the somewhat involved proof required for the effective actionin Subsection 3.23.6. The diagrams in the two expansions are therefore precisely thesame and can be read off from Eqs.
(3.741) and (3.781). The only difference liesin the replacement X → x0 in the analytic expressions for the lines and vertices.In addition, there is the final integral over x0 to obtain the partition function Z inEq. (3.808). This is in contrast to the partition function expressed in terms of theeffective potential V eff (X), where only the extremum has to be taken.3.25.7Effective Potential and Magnetization CurvesThe effective classical potential V eff cl (x0 ) in the Boltzmann factor (3.807) allows usto estimate the effective potential defined in Eq.
(3.663). It can be derived from thegenerating functional Z[j] restricted to time-independent external source j(τ ) ≡ j,in which case Z[j] reduces to a mere function of j:Z(j) =Z(Dx(τ ) exp −Z0β)1dτ ẋ2 + V (x(τ )) + βj x̄ ,2(3.857)3363 External Sources, Correlations, and Perturbation Theorywhere x̄ is the path average of x(τ ). The function Z(j) is obtained from the effectiveclassical potential by a simple integral over x0 :Z(j) =Z∞−∞eff cldx√ 0 e−β[V (x0 )−jx0 ] .2πβ(3.858)The effective potential V eff (X) is equal to the Legendre transform of W (j) =log Z(j):1(3.859)V eff (X) = − W (j) + Xj,βwhere the right-hand side is to be expressed in terms of X usingX = X(j) =1 dW (j).β dj(3.860)To picture the effective potential, we calculate the average value of x(τ ) from theintegralZ ∞nodx√ 0 x0 exp −β[V eff cl (x0 ) − jx0 ]X = Z(j)−1(3.861)2πβ−∞and plot X = X(j).
By exchanging the axes we display the inverse j = j(X) whichis the slope of the effective potential:j(X) =dV eff (X).dX(3.862)The curves j(X) are shown in Fig. 3.15 for the double-well potential with a couplingstrength g = 0.4 at various temperatures.Note that the x0 -integration makes j(X) necessarily a monotonous function ofX. The effective potential is therefore always a convex function of X, no matterwhat the classical potential looks like. This is in contrast to j(X) before fluctuationsare taken into account, the mean-field approximation to (3.862) [recall the discussionin Subsection 3.23.1], which is given byj = dV (X)/dX.(3.863)For the double-well potential, this becomesj = −X + gX 3.(3.864)Thus, the mean-field effective potential coincides with the classical potential V (X),which is obviously not convex.In magnetic systems, j is a constant magnetic field and X its associated magnetization. For this reason, plots of j(X) are referred to as magnetization curves.H.
Kleinert, PATH INTEGRALS3373.25 Effective Classical PotentialFigure 3.15 Magnetization curves in double-well potential V (x) = −x2 /2 + gx4 /4 withg = 0.4, at various inverse temperatures β. The integral over these curves returns theeffective potential V eff (X). The curves arising from the approximate effective potentialW1 (x0 ) are labeled by β1 (- - -) and the exact curves (found by solving the Schrödingerequation numerically) by βex (—–). For comparison we have also drawn the classical curves(· · ·) obtained by using the potential V (x0 ) in Eqs. (3.861) and (3.858) rather than W1 (x0 ).They are labeled by βV .
Our approximation W1 (x0 ) is seen to render good magnetizationcurves for all temperatures above T = 1/β ∼ 1/10. The label β carries several subscriptsif the corresponding curves are indistinguishable on the plot. Note that all approximationsare monotonous, as they should be (except for the mean field, of course).3.25.8First-Order Perturbative ResultTo first order in the interaction V int (x0 ; η), the perturbation expansion (3.851) becomesEx01DB(x0 ) = 1 −Aint,e+ . . . BΩ (x0 ),(3.865)Ωh̄and we have to calculate the harmonic expectation value of Aint,e. Let us assumethat the interaction potential possesses a Fourier transformZ ∞dk ik(x0 +η(τ )) intinteṼ (k).(3.866)V (x0 ; η(τ )) =−∞ 2πThen we can write the expectation of (3.848) asDEx0Aint,e [x0 ; η]Ω=Zh̄β0dτZ∞−∞DEx0dk intṼ (k)eikx0 eikη(τ ).Ω2π(3.867)3383 External Sources, Correlations, and Perturbation TheoryWe now use Wick’s rule in the form (3.304) to calculateDeikη(τ )Ex0Ωx022= e−k hη (τ )iΩ /2 .(3.868)We now use Eq.
(3.840) to write this asDeikη(τ )Ex0Ω−k 2 a2Ω(x ) /2=e0.(3.869)Thus we find for the expectation value (3.867):Z h̄βZ ∞DEx0dk intikx −k 2 a2Ω(x ) /20Ṽ (k)e 0.(3.870)Aint,e [x0 ; η]=dτΩ0−∞ 2πDue to the periodic boundary conditions satisfied by the correlation function andthe associated invariance under time translations, this result is independent of τ , sothat the τ -integral can be performed trivially, yielding simply a factor h̄β. We nowreinsert the Fourier coefficients of the potentialṼ int (k) =Z∞−∞dx V int (x0 ; η) e−ik(x0+η) ,(3.871)perform the integral over k via a quadratic completion, and obtainDVint(x(τ ))Ex0DΩ≡intVa2 (x0 )Ω=Ex0Zdx00∞−∞intq−η2 /2a2Ω(x2πa2Ω(x0 )e0)V int (x0 ; η).(3.872)The expectation V int (x(τ ))≡ Va2 (x0 ) of the potential arises therefore from aΩΩconvolution integral of the original potential with a Gaussian distribution of squarewidth a2Ω(x0 ) .
The convolution integral smears the original interaction potentialintVa2Ω (x0 ) out over a length scale aΩ(x0 ) . In this way, the approximation accounts forthe quantum-statistical path fluctuations of the particle.As a result, we can write the first-order Boltzmann factor (3.865) as follows:B(x0 ) ≈noΩ(x0 )h̄βintexp −βMΩ(x0 )2 x20 /2 − βVa2 (x0 ) .Ω2 sin[Ω(x0 )h̄β/2](3.873)Recalling the harmonic effective classical potential (3.831), this may be written asa Boltzmann factor associated with the first-order effective classical potentialinteff clV eff cl (x0 ) ≈ VΩ(x(x0 ) + Va2 (x0 ).0)(3.874)ΩGiven the power series expansion (3.850) of the interaction potentialV int (x0 ; η) =we may use the integral formulaZ∞√dη−η2 /2a2 k(∞X1 (k)V (x0 )h̄k ,k!k=3(k − 1)!! ak0)(3.875)(eη =2πa2we find the explicit smeared potential∞X(k − 1)!! (k)V (x0 )ak (x0 ).Vaint(x)=20k!k=4,6,...−∞)evenfor k =,odd(3.876)(3.877)H.
Kleinert, PATH INTEGRALS3.26 Perturbative Approach to Scattering Amplitude3.26339Perturbative Approach to Scattering AmplitudeIn Eq. (2.723) we have derived a path integral representation for the scattering amplitude. Itinvolves calculating a path integral of the general form Z tbZZZZM 2id3 ya d3 za D3 y D3 z exp(3.878)dtẏ − ż2 F [y(t) − z(0)],h̄ ta2where the paths y(t) and z(t) vanish at the final time t = tb whereas the initial positions areintegrated out. In lowest approximation, we may neglect the fluctuations in y(t) and z(0) andobtain the eikonal approximation (2.726). In order to calculate higher-order corrections to pathintegrals of the form (3.878) we find the generating functional of all correlation functions of y(t) −z(0).3.26.1Generating FunctionalFor the sake of generality we calculate the harmonic path integral over y: Z tb ZZM 2i332 2Z[jy ] ≡ d ya D y expdtẏ − ω y − jy y .h̄ ta2(3.879)This differs from the amplitude calculated in (3.168) only by an extra Fresnel integral over theinitial point and a trivial extension to three dimensions.
This yieldsZjZ[jy ] =d3 ya (yb tb |ya ta )ωy Z tb1iyb (sin [ω(t − ta )] + sin [ω(tb − t)]) jydt= exph̄ tasin ω(tb − ta )Z tb Z ti h̄000dt× exp − 2dt jy (t)Ḡω2 (t, t )jy (t ) ,(3.880)h̄ M tatawhere Ḡω2 (t, t0 ) is obtained from the Green function (3.36) with Dirichlet boundary conditions byadding the result of the quadratic completion in the variable yb − ya preceding the evaluation ofthe integral over d3 ya :Ḡω2 (t, t0 ) =1sin ω(tb − t> ) [sin ω(t< − ta ) + sin ω(tb − t< )] .ω sin ω(tb − ta )(3.881)We need the special case ω = 0 whereḠω2 (t, t0 ) = tb − t> .(3.882)In contrast to Gω2 (t, t0 ) of (3.36), this Green function vanishes only at the final time.
This reflectsthe fact that the path integral (3.878) is evaluated for paths y(t) which vanish at the final timet = tb .A similar generating functional for z(t) leads to the same result with opposite sign in theexponent. Since the variable z(t) appears only with time argument zero in (3.878), the relevantgenerating functional isZZZZ333Z[j] ≡d ya d za D y D 3 z Z tb M 2i2222,(3.883)dtẏ − ż − ω y − z − j yz× exph̄ ta2with yb = zb = 0, where we have introduced the subtracted variableyz (t) ≡ y(t) − z(0),(3.884)3403 External Sources, Correlations, and Perturbation Theoryfor brevity. From the above calculations we can immediately write down the resultZ tb Z tb1i h̄0000Z[j] =exp − 2dtdt j(t)Ḡω2 (t, t )j(t ) ,Dωh̄ 2M tata(3.885)where Dω is the functional determinant associated with the Green function (3.881) which is obtained by integrating (3.881) over t ∈ (tb , ta ) and over ω 2 :Dω =1expcos2 [ω(tb − ta )]Ztb −ta0dt(cos ωt − 1) ,t(3.886)and Ḡ0ω2 (t, t0 ) is the subtracted Green function (3.881):Ḡ0ω2 (t, t0 ) ≡ Ḡω2 (t, t0 ) − Ḡω2 (0, 0).(3.887)For ω = 0 where Dω = 1, this is simplyḠ00 (t, t0 ) ≡ −t> ,(3.888)where t> denotes the larger of the times t and t0 .
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