Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 54
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Insertingη(0) = −ζ(0) = 1/2,η(2) = ζ(2)/2 = π 2 /12,(3.256)we obtain2−h̄βh̄β2π!2 π2 1−12 42πh̄β!22(τ − h̄β/2)=τ2τh̄β− +,2h̄β 212(3.257)in agreement with (3.251).It is worth remarking that the Green function (3.251) is directly proportional tothe Bernoulli polynomial B2 (z):Gp0,e0 (τ ) =h̄βB (τ /h̄β).2 2(3.258)These polynomials are defined in terms of the Bernoulli numbers Bk as2Bn (x) =n Xnk=0kBk z n−k .(3.259)They appear in the expansion of the generating function3∞Xtn−1ezt=B (z),n!et − 1 n=0 n(3.260)and have the expansionB2n (z) = (−1)n−1∞cos(2πkz)2(2n)! X,2n(2π) k=0k 2n(3.261)B2 (z) = z 2 − z + 1/6, .
. . .(3.262)with the special casesB1 (z) = z − 1/2,By analogy with (3.248), the antiperiodic Green function can be obtained froman antiperiodic repetition23Gaω2 ,e (τ ) =(−1)n −ω|τ +nh̄β|en=−∞ 2ω=1 sinh ω(τ − h̄β/2),2ω cosh(βh̄ω/2)∞Xτ ∈ [0, h̄β],(3.263)I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.620.ibid. Formula 9.621.H. Kleinert, PATH INTEGRALS2453.8 External Source in Quantum-Statistical Path Integralwhich is an analytic continuation of (3.113) to imaginary times. In contrast to(3.249), this has a finite ω → 0 limitGaω2 ,e (τ ) =h̄βτ−,24τ ∈ [0, h̄β].(3.264)For a plot of the antiperiodic Green function for different frequencies ω see againFig.
3.4.The limiting expression (3.264) can again be derived using an expansion of thetype (3.253). The spectral representation in terms of odd Matsubara frequencies(3.104)∞1 X1 −iωmf τf2eh̄β m=−∞ ωmGaω2 ,e (τ ) ≡(3.265)is rewritten as∞∞1(−1)m1 X1 Xffcos(ωτ)=mf2f 2 sin[ωm (τ − h̄β/2)].h̄β m=−∞ ωmh̄β m=−∞ ωm(3.266)Expanding the sin function yields2h̄βh̄β2π!2#n(−1)(n−1)/2 2π(τ − h̄β/2)n!h̄βn=1,3,5,..."X∞Xm=0(−1)mm+122−n .(3.267)The sum over m at the end is 22−n times Riemann’s beta function4 β(2 − n), whichis defined asβ(z) ≡∞1 X(−1)mz ,2z m=0 m + 12(3.268)and is related to Riemann’s zeta function∞X(3.269)(−1)m2−zζ(z, (q + 1)/2),z = ζ(z, q) − 2(m+q)m=0(3.270)ζ(z, q) ≡1z.m=0 (m + q)Indeed, we see immediately that∞Xso thati1 h2−zζ(z,1/2)−2ζ(z,3/4).2zNear z = 1, the function ζ(z, q) behaves like5β(z) ≡ζ(z, q) =451− ψ(q) + O(z − 1),z−1M. Abramowitz and I.
Stegun, op. cit., Formula 23.2.21.I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.533.2.(3.271)(3.272)2463 External Sources, Correlations, and Perturbation Theorywhere ψ(z) is the Digamma function (2.555). Thus we obtain in the limit z → 1:i11hπζ(z, 1/2) − 21−z ζ(z, 3/4) = [−ψ(1/2) + ψ(3/4) + log 2] = .z→1 224(3.273)The last result follows from the specific values [compare (2.557)]:π(3.274)ψ(1/2) = −γ − 2 log 2, ψ(3/4) = −γ − 3 log 2 + .2For negative odd arguments, the beta function (3.268) vanishes, so that there are nofurther contributions.
Inserting this into (3.267) the only surviving e n = 1 -termyields once more (3.264).Note that the relation (3.273) could also have been found directly from theexpansion (2.556) of the Digamma function, which yieldsβ(1) = lim1[ψ(3/2) − ψ(1/4)] ,(3.275)4and is equal to (3.273) due to ψ(1/4) = −γ − 3 log 2 − π/2.For currents j(τ ), which are periodic in h̄β, the source term (3.244) can also bewritten more simply:β(1) =Aje [j] = −14MωZh̄β0dτZ∞−∞0dτ 0 e−ω|τ −τ | j(τ )j(τ 0 ).(3.276)This follows directly by rewriting (3.276), by analogy with (3.149), as a sum overall periodic repetitions of the zero-temperature Green function (3.246):Gpω2 ,e (τ ) =∞1 Xe−ω|τ +nh̄β| .2ω n=−∞(3.277)When inserted into (3.244), the factors e−nβh̄ω can be removed by an irrelevantperiodic temporal shift in the current j(τ 0 ) → j(τ 0 − nh̄β) leading to (3.276).For a time-dependent periodic or antiperiodic potential Ω(τ ), the Green functionp,aGω2 ,e (τ ) solving the differential equation0p,a[∂τ2 − Ω2 (τ )]Gp,a(τ − τ 0 ),ω 2 ,e (τ, τ ) = δ(3.278)with the periodic or antiperiodic δ-functionδp,a0(τ − τ ) =∞Xn=−∞0δ(τ − τ − nh̄β)(1(−1)n),(3.279)can be expressed6 in terms of two arbitrary solutions ξ(τ ) and η(τ ) of the homogenous differential equation in the same way as the real-time Green functions inSection 3.5:00Gp,aω 2 ,e (τ, τ ) = Gω 2 ,e (τ, τ ) ∓[∆(τ, τa ) ± ∆(τb , τ )][∆(τ 0 , τa ) ± ∆(τb , τ 0 )], (3.280)¯ p,a (τ , τ )∆(τ , τ )∆a ba b6See H.
Kleinert and A. Chervyakov, Phys. Lett. A 245 , 345 (1998) (quant-ph/9803016);J. Math. Phys. B 40 , 6044 (1999) (physics/9712048).H. Kleinert, PATH INTEGRALS2473.8 External Source in Quantum-Statistical Path Integralwhere Gω2 ,e (τ, τ 0 ) is the imaginary-time Green function with Dirichlet boundaryconditions corresponding to (3.206):Θ̄(τ − τ 0 )∆(τb , τ )∆(τ 0 , τa ) + Θ̄(τ − τ 0 )∆(τb , τ 0 )∆(τ, τa )Gω2 ,e (τ, τ ) =,∆(τa , τb )0(3.281)with∆(τ, τ 0 ) =andi1 hξ(τ )η(τ 0 ) − ξ(τ 0 )η(τ ) ,W˙ )η(τ ),W = ξ(τ )η̇(τ ) − ξ(τ¯ p,a (τ , τ ) = 2 ± ∂ ∆(τ , τ ) ± ∂ ∆(τ , τ ).∆a bτa bτb a(3.282)(3.283)Let also write down the imaginary-time versions of the periodic or antiperiodicGreen functions for time-dependent frequencies.
Recall the expressions for constantfrequency Gpω (t) and Gaω (t) of Eqs. (3.94) and (3.112) for τ ∈ (0, h̄β):Gpω,e (τ ) =1 X −iωm τ −11e= e−ω(τ −h̄β/2)h̄β miωm − ω2 sinh(βh̄ω/2)= (1 + nbω )e−ωτ ,(3.284)andGaω,e (τ ) =11 X −iωmf τ−1= e−ω(τ −h̄β/2)efh̄β m2 cosh(βh̄ω/2)iωm − ω= (1 − nfω )e−ωτ ,(3.285)the first sum extending over the even Matsubara frequencies, the second over theodd ones. The Bose and Fermi distribution functions nb,fω were defined in Eqs. (3.93)and (3.111).For τ < 0, periodicity or antiperiodicity determinep,aGp,aω,e (τ ) = ±Gω,e (τ + h̄β).(3.286)The generalization of these expressions to time-dependent periodic and antiperiodic frequencies Ω(τ ) satisfying the differential equations0p,a[−∂τ − Ω(τ )]Gp,a(τ − τ 0 )Ω,e (τ, τ ) = δhas for β → ∞ the form0Gp,aΩ,e (τ, τ )0−= Θ̄(τ − τ )eRτ0dτ 0 Ω(τ 0 ).(3.287)(3.288)Its periodic superposition yields for finite β a sum analogous to (3.277):0Gp,aΩ,e (τ, τ )=∞Xn=0−eR τ +nh̄β0dτ 0 Ω(τ 0 )(1(−1)n),h̄β > τ > τ 0 > 0,which reduces to (3.284), (3.285) for a constant frequency Ω(τ ) ≡ ω.(3.289)2483.93 External Sources, Correlations, and Perturbation TheoryLattice Green FunctionAs in Chapter 2, it is easy to calculate the above results also on a sliced time axis.
This is usefulwhen it comes to comparing analytic results with Monte Carlo lattice simulations. We consider hereonly the Euclidean versions; the quantum-mechanical ones can be obtained by analytic continuationto real times.The Green function Gω2 (τ, τ 0 ) on an imaginary-time lattice with infinitely many lattice pointsof spacing reads [instead of the Euclidean version of (3.147)]:Gω2 (τ, τ 0 ) =0011e−ω̃e |τ −τ | =e−ω̃e |τ −τ | ,2 sinh ω̃e2ω cosh(ω̃e /2)(3.290)where ω̃e is given, as in (2.398), by2ωarsinh .2This is derived from the spectral representationZ2dω 0 −iω0 (τ −τ 0 )00eGω2 (τ, τ ) = Gω2 (τ − τ ) =2π2(1 − cos ω 0 ) + 2 ω 2ω̃e =(3.291)(3.292)by rewriting it asGω2 (τ, τ 0 ) =Z∞0dsZdω 0 −iω0 n −s[2(1−cos ω0 )+2 ω2 ]/2ee,2π(3.293)with n ≡ (τ 0 − τ )/, performing the ω 0 -integral which produces a Bessel function I(τ −τ 0 )/ (2s/2 ),and subsequently the integral over s with the help of formula (2.467).
The Green function (3.290)is defined only at discrete τn = nh̄β/(N + 1). If it is summed over all periodic repetitions n →n + k(N + 1) with k = 0, ±1, ±2, . . . , one obtains the lattice analog of the periodic Green function(3.248):Gpe (τ )==3.10∞1 X2e−iωm τh̄β m=−∞ 2(1 − cos ωm ) + 2 ω 211cosh ω̃(τ − h̄β/2),2ω cosh(ω̃/2) sinh(h̄ω̃β/2)τ ∈ [0, h̄β].(3.294)Correlation Functions, Generating Functional,and Wick ExpansionEquipped with the path integral of the harmonic oscillator in the presence of anexternal source it is easy to calculate the correlation functions of any number ofposition variables x(τ ). We consider here only a system in thermal equilibriumand study the behavior at imaginary times.
The real-time correlation functions canbe discussed similarly. The precise relation between them will be worked out inChapter 18.In general, i.e., also for nonharmonic actions, the thermal correlation functionsof n-variables x(τ ) are defined as the functional averages(n)Gω2 (τ1 , . . . , τn ) ≡ hx(τ1 )x(τ2 ) · · · x(τn )i≡ Z −1Z(3.295)1Dx x(τ1 )x(τ2 ) · · · x(τn ) exp − Ae .h̄H. Kleinert, PATH INTEGRALS2493.10 Correlation Functions, Generating Functional, and Wick ExpansionThey are also referred to as n-point functions. In operator quantum mechanics, thesame quantities are obtained from the thermal expectation values of time-orderedproducts of Heisenberg position operators x̂H (τ ):(n)nhGω2 (τ1 , .
. . , τn ) = Z −1 Tr T̂τ x̂H (τ1 )x̂H (τ2 ) · · · x̂H (τn )e−Ĥ/kB Twhere Z is the partition functionio,Z = e−F/kB T = Tr(e−Ĥ/kB T )(3.296)(3.297)and T̂τ is the time-ordering operator. Indeed, by slicing the imaginary-time evolutionoperator e−Ĥτ /h̄ at discrete times in such a way that the times τi of the n position(n)operators x(τi ) are among them, we find that Gω2 (τ1 , .
. . , τn ) has precisely the pathintegral representation (3.295).By definition, the path integral with the product of x(τi ) in the integrand iscalculated as follows. First we sort the times τi according to their time order,denoting the reordered times by τt(i) . We also set τb ≡ τt(n+1) and τa ≡ τt(0) .Assuming that the times τt(i) are different from one another, we slice the time axisτ ∈ [τa , τb ] into the intervals [τb , τt(n) ], [τt(n) , τt(n−1) ], [τt(n−2) , τt(n−3) ],. . ., [τt(4) , τt(3) ],[τt(2) , τt(1) ], [τt(1) , τa ].
For each of these intervals we calculate the time evolutionamplitude (xt(i+1) τt(i+1) |xt(i) τt(i) ) as usual. Finally, we recombine the amplitudes byperforming the intermediate x(τt(i) )-integrations, with an extra factor x(τi ) at eachτi , i.e.,(n)Gω2 (τ1 , . . . , τn )n+1Y Z ∞=i=1−∞dxτt(i) (xt(n+1) τb) |xt(n) τt(n) ) · x(τt(n) ) · . . .·(xt(i+1) τt(i+1) |xt(i) τt(i) ) · x(τt(i) ) · (xt(i) τt(i) |xt(i−1) τt(i−1) ) · x(τt(i−1) )· . . . · (xt(2) τt(2) |xt(1) τt(1) ) · x(τt(1) ) · (xt(1) τt(1) |xt(0) τa ).(3.298)We have set xt(n+1) ≡ xb = xa ≡ xt(0) , in accordance with the periodic boundarycondition.
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