Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 53

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ItreadsMωeβh̄ω (Ae + Be )2 .2 sinh ωβAjr,e = −(3.216)Combining this with Ajfl,e of (3.210) givesAjfl,e + Ajr,e = −14MωZ0h̄βdτZ0h̄β0dτ 0 e−ω|τ −τ | j(τ )j(τ 0 ) −Mωeβh̄ω/2 Ae Be .sinh(βh̄ω/2)(3.217)H. Kleinert, PATH INTEGRALS2393.8 External Source in Quantum-Statistical Path IntegralThis can be rearranged to the total source termAje = −14MωZh̄β0h̄βZdτ0dτ 0cosh ω(|τ − τ 0 | − h̄β/2)j(τ )j(τ 0 ).sinh(βh̄ω/2)(3.218)This is proved by rewriting the latter integrand asnhi01eω(τ −τ ) e−βh̄ω/2 + (ω → −ω) Θ̄(τ − τ 0 )2 sinh(βh̄ω/2)h+ eω(τ0 −τ )oie−βh̄ω/2 + (ω → −ω) Θ̄(τ 0 − τ ) j(τ )j(τ 0 ).In the second and fourth terms we replace eβh̄ω/2 by e−βh̄ω/2 + 2 sinh(βh̄ω/2) andintegrate over τ, τ 0 , with the result (3.217).The expression between the currents in (3.218) is recognized as the Euclideanversion of the periodic Green function Gpω2 (τ ) in (3.99):Gpω2 ,e (τ ) ≡ iGpω2 (−iτ )|tb −ta =−ih̄β1 cosh ω(τ − h̄β/2),2ω sinh(βh̄ω/2)=τ ∈ [0, h̄β].(3.219)In terms of (3.218), the partition function of an oscillator in the presence of thesource term is1Zω [j] = Zω exp − Aje .h̄(3.220)For completeness, let us also calculate the partition function of all paths withopen ends in the presence of the source j(t), thus generalizing the result (2.405).Integrating (3.201) over initial and final positions xa and xb we obtainZωopen [j] =swhereÃj2,e = −jj2πh̄1qe−(A2,e +Ã2,e )/h̄ ,Mω sinh[ω(τ − τ )]ba1MZ0h̄βdτZ0τ(3.221)dτ 0 j(τ )G̃ω2 (τ, τ 0 )j(τ 0 ),(3.222)withG̃ω2(τ, τ 0 )=nhi1cosh ωh̄β sinh ω(h̄β−τ ) sinh ω(h̄β−τ 0 )+sinh ωτ sinh ωτ 032ω sinh ωh̄βo+ sinh ω(h̄β −τ ) sinh ωτ 0 + sinh ω(h̄β −τ 0 ) sinh ωτ .(3.223)By some trigonometric identities, this can be simplified to1 cosh ω(h̄β − τ − τ 0 ).G̃ω2 (τ, τ ) =ωsinh ωh̄β0(3.224)2403 External Sources, Correlations, and Perturbation TheoryThe first step is to rewrite the curly brackets in (3.223) ashsinh ωτ cosh ωh̄β sinh ωτ 0 + sinh ω(h̄β −τ 0 )hii+ sinh ω(h̄β −τ 0 ) cosh ωh̄β sinh ω(h̄β −τ ) + sinh ω(h̄β − ((h̄β −τ )) .

(3.225)The first bracket is equal to sinh βh̄ω cosh ωτ , the second to sinh βh̄ω cosh ω(h̄β −τ 0 ),so that we arrive athisinh ωh̄β sinh ωτ cosh ωτ 0 + sinh ω(h̄β −τ ) cosh ω(h̄β −τ 0 ) .The bracket is now rewritten as(3.226)i1hsinh ω(τ + τ 0 ) + sinh ω(τ − τ 0 ) + sinh ω(2h̄β − τ − τ 0 ) + sinh ω(τ 0 − τ ) , (3.227)2which is equal toi1hsinh ω(h̄β + τ + τ 0 − h̄β) + sinh ω(h̄β + h̄β − τ − τ 0 ) ,2(3.228)i1h2 sinh ωh̄β cosh ω(h̄β − τ − τ 0 ) ,2(3.229)and thus tosuch that we arrive indeed at (3.224). The source action in the exponent in (3.221)is therefore:(Aj2,e + Ãj2,e ) = −1MZ0h̄βdτZ0τ00dτ 0 j(τ )Gopenω 2 ,e (τ, τ )j(τ ),(3.230)with (3.205)cosh ω(h̄β − |τ − τ 0 |) + cosh ω(h̄β − τ − τ 0 )2ω sinh ωh̄βcosh ω(h̄β − τ> ) cosh ωτ<=.ω sinh ωh̄β0Gopenω 2 ,e (τ, τ ) =(3.231)This Green function coincides precisely with the Euclidean version of Green function0GNω 2 (t, t ) in Eq.

(3.151) using the relation (3.208). This coincidence should have beenexpected after having seen in Section 2.12 that the partition function of all pathswith open ends can be calculated, up to a trivial factor le (h̄β) of Eq. (2.345), as a sumover all paths satisfying Neumann boundary conditions (2.443), which is calculatedusing the measure (2.446) for the Fourier components.In the limit of small-ω, the Green function (3.231) reduces to0Gopenω 2 ,e (τ, τ ) ≈2ω ≈0β 111 210200,τ+τ+−|τ−τ|−(τ+τ)+22ββω 2 3 2(3.232)which is the imaginary-time version of (3.157).H. Kleinert, PATH INTEGRALS2413.8 External Source in Quantum-Statistical Path Integral3.8.2Calculation at Imaginary TimeLet us now see how the partition function with a source term is calculated directlyin the imaginary-time formulation, where the periodic boundary condition is usedfrom the outset.

Thus we considerZω [j] =ZDx(τ ) e−Ae [j]/h̄,(3.233)with the Euclidean actionAe [j] =Zh̄β0dτM 2(ẋ + ω 2x2 ) − j(τ )x(τ ) .2(3.234)Since x(τ ) satisfies the periodic boundary condition, we can perform a partial integration of the kinetic term without picking up a boundary term xẋ|ttba . The actionbecomesAe [j] =Zh̄β0Mdτx(τ )(−∂τ2 + ω 2)x(τ ) − j(τ )x(τ ) .2(3.235)Let De (τ, τ 0 ) be the functional matrixDω2 ,e (τ, τ 0 ) ≡ (−∂τ2 + ω 2 )δ(τ − τ 0 ),τ − τ 0 ∈ [0, h̄β].(3.236)Its functional inverse is the Euclidean Green function,Gpω2 ,e (τ, τ 0 ) = Gpω2 ,e (τ − τ 0 ) = Dω−12 ,e (τ, τ 0 ) = (−∂τ2 + ω 2 )−1 δ(τ − τ 0 ),(3.237)with the periodic boundary condition.Next we perform a quadratic completion by shifting the path:x → x0 = x −1 PG 2 j.M ω ,e(3.238)This brings the Euclidean action to the formAe [j] =Z0h̄βdτM 01x (−∂τ2 + ω 2)x0 −22MZh̄β0dτZ0h̄βdτ 0 j(τ )Gpω2 ,e (τ − τ 0 )j(τ 0 ).(3.239)The fluctuations over the periodic paths x0 (τ ) can now be integrated out and yieldfor j(τ ) ≡ 0−1/2Zω = Det Dω2 ,e .(3.240)As in Subsection 2.15.2, we find the functional determinant by rewriting the productof eigenvalues asDet Dω2 ,e =∞Ym=−∞2(ωm+ ω 2 ) = exp"∞Xm=−∞#2log(ωm+ ω2) ,(3.241)2423 External Sources, Correlations, and Perturbation Theoryand evaluating the sum in the exponent according to the rules of analytic regularization.

This leads directly to the partition function of the harmonic oscillator as inEq. (2.401):1Zω =.(3.242)2 sinh(βh̄ω/2)The generating functional for j(τ ) 6= 0 is therefore1Z[j] = Zω exp − Aje [j] ,h̄(3.243)1 Z h̄β Z h̄β 0dτ j(τ )Gpω2 ,e (τ − τ 0 )j(τ 0 ).dτ2M 00(3.244)with the source term:Aje [j] = −The Green function of imaginary time is calculated as follows. The eigenfunctions2of the differential operator −∂τ2 are e−iωm τ with eigenvalues ωm, and the periodicboundary condition forces ωm to be equal to the thermal Matsubara frequenciesωm = 2πm/h̄β with m = 0, ±1, ±2, . .

. . Hence we have the Fourier expansionGpω2 ,e (τ ) =∞1 X1e−iωm τ .2h̄β m=−∞ ωm + ω 2(3.245)In the zero-temperature limit, the Matsubara sum becomes an integral, yieldingGpω2 ,e (τ ) =T =0Z11 −ω|τ |dωm−iωm τ=e.22e2π ωm + ω2ω(3.246)The frequency sum in (3.245) may be written as such an integral over ωm , providedthe integrand contains an additional Poisson sum (3.81):∞Xm̄=−∞δ(m − m̄) =∞Xi2πnmen=−∞=∞Xeinωm h̄β .(3.247)n=−∞This implies that the finite-temperature Green function (3.245) is obtained from(3.246) by a periodic repetition:Gpω2 ,e (τ )∞X=1 −ω|τ +nh̄β|en=−∞ 2ω=1 cosh ω(τ − h̄β/2),2ω sinh(βh̄ω/2)τ ∈ [0, h̄β].(3.248)A comparison with (3.97), (3.99) shows that Gpω2 ,e (τ ) coincides with Gpω2 (t) at imaginary times, as it should.Note that for small ω, the Green function has the expansionGpω2 ,e (τ ) =τh̄β1τ2− ++ ...

.+22h̄β 212h̄βω(3.249)H. Kleinert, PATH INTEGRALS2433.8 External Source in Quantum-Statistical Path IntegralThe first term diverges in the limit ω → 0. Comparison with the spectral representation (3.245) shows that it stems from the zero Matsubara frequency contributionto the sum. If this term is omitted, the subtracted Green functionGpω20 ,e (τ ) ≡ Gpω2 ,e (τ ) −has a well-defined ω → 0 limit1h̄βGp0,e0 (τ ) =1h̄βω 2(3.250)1 −iωm ττ2τh̄βe=− +,22h̄β212m=±1,±2,...

ωmX(3.251)the right-hand side being correct only for τ ∈ [0, h̄β]. Outside this interval it mustbe continued periodically. The subtracted Green function Gpω20 ,e (τ ) is plotted fordifferent frequencies ω in Fig. 3.4.0.08Gpω02 ,e(τ )Gaω2 ,e (τ )0.20.060.10.040.02-1-0.5-0.020.5121.5τ /h̄β-1-0.50.511.52τ /h̄β-0.1-0.2-0.04Figure 3.4 Subtracted periodic Green function Gpω02 ,e(τ ) ≡ Gpω2 ,e (τ ) − 1/h̄βω 2 and antiperiodic Green function Gaω2 ,e (τ ) for frequencies ω = (0, 5, 10)/h̄β (with increasing dashlength). Compare Fig. 3.2.The limiting expression (3.251) can, incidentally, be derived using the methodsdeveloped in Subsection 2.15.5.

We rewrite the sum as1h̄β(−1)m −iωm (τ −h̄β/2)e2m=±1,±2,... ωmX(3.252)and expand2−h̄βh̄β2π!2#n"2π1−i(τ − h̄β/2)h̄βn=0,2,4,... n!X(−1)m−12−n .m=1 m∞X(3.253)The sum over m on the right-hand side is Riemann’s eta function1η(z) ≡1(−1)m−1,mzm=1∞XM. Abramowitz and I. Stegun, op. cit., Formula 23.2.19.(3.254)2443 External Sources, Correlations, and Perturbation Theorywhich is related to the zeta function (2.514) byη(z) = (1 − 21−z )ζ(z).(3.255)Since the zeta functions of negative integers are all zero [recall (2.567)], only theterms with n = 0 and 2 contribute in (3.253).

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