Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 38
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(2.345). Apart from this prefactor, the Neumann partition functioncoincides precisely with the open-end partition function Zωopen in Eq. (2.405).What is the reason for this coincidence up to a trivial factor, even though thepaths satisfying Neumann boundary conditions do not comprise all paths with openH. Kleinert, PATH INTEGRALS1552.13 Classical Limitends. Moreover, the integrals over the endpoints in the defining equation (2.405)does not force the endpoint velocities, but rather endpoint momenta to vanish.Indeed, recalling Eq. (2.182) for the time evolution amplitude in momentum space wecan see immediately that the partition function with open ends Zωopen in Eq.
(2.405)is identical to the imaginary-time amplitude with vanishing endpoint momenta:Zωopen = (pb h̄β|pa 0)|pb =pa =0 .(2.450)Thus, the sum over all paths with arbitrary open ends is equal to the sum of allpaths satisfying Dirichlet boundary conditions in momentum space. Only classically,the vanishing of the endpoint momenta implies the vanishing of the endpoint velocities. From the general discussion of the time-sliced path integral in phase space inSection 2.1 we know that fluctuating paths have M ẋ 6= p.
The fluctuations of thedifference are controlled by a Gaussian exponential of the type (2.51). This leads tothe explanation of the trivial factor between Zωopen and ZωN . The difference betweenM ẋ and p appears only in the last short-time intervals at the ends. But at shorttime, the potential does not influence the fluctuations in (2.51).
This is the reasonwhy the fluctuations at the endpoints contribute only a trivial overall factor le (h̄β)to the partition function ZωN .2.13Classical LimitThe alternative measure of the last section serves to show, somewhat more convincingly than before, that in the high-temperature limit the path integral representationof any quantum-statistical partition function reduces to the classical partition function as stated in Eq. (2.338). We start out with the Lagrangian formulation (2.368).Inserting the Fourier decomposition (2.435), the kinetic term becomesZh̄β0dτ∞M 2Mh̄ Xω 2 |x |2 ,ẋ =2kB T m=1 m m(2.451)and the partition function readsZ=IZ∞∞XM X1 h̄/kB T220Dx exp −ωm |xm | −dτ V (x0 +xm e−iωm τ ) .
(2.452)kB T m=1h̄ 0m=−∞"#The summation symbol with a prime implies the absence of the m = 0 -term. Themeasure is the product (2.439) of integrals of all Fourier components.We now observe that for large temperatures, the Matsubara frequencies for m 6= 0diverge like 2πmkB T /h̄ . This has the consequence that the Boltzmann factor for thexm6=0 fluctuations becomes sharply peaked around xm = 0. The average size of xm isqq0−iωm τkB T /M /ωm = h̄/2πm MkB T . If the potential V x0 + 0 ∞is am=−∞ xm esmooth function of its arguments, we can approximate it by V (x0 ), terms containinghigher powers of xm . For large temperatures, these are small on the average andP1562 Path Integrals — Elementary Properties and Simple Solutionscan be ignored. The leading term V (x0 ) is time-independent. Hence we obtain inthe high-temperature limitT →∞Z−−−→I∞1M X2V (x0 ) .ωm|xm |2 −Dx exp −kB T m=1kB T#"(2.453)The right-hand side is quadratic in the Fourier components xm .
With the measureof integration (2.439), we perform the integrals over xm and obtainT →∞Z−−−→ Zcl =Z∞−∞dx0 −V (x0 )/kB Te.le (h̄β)(2.454)This agrees with the classical statistical partition function (2.344).The derivation reveals an important prerequisite for the validity of the classicallimit: It holds only for sufficiently smooth potentials. We shall see in Chapter 8that for singular potentials such as −1/|x| (Coulomb), 1/|x|2 (centrifugal barrier),1/ sin2 θ (angular barrier), this condition is not fulfilled and the classical limit is nolonger given by (2.454). The particle distribution ρ(x) at a fixed x does not havethis problem. It always tends towards the naively expected classical limit (2.346):T →∞ρ(x) −−−→ Zcl−1 e−V (x)/kB T .(2.455)The convergence is nonuniform in x, which is the reason why the limit does notalways carry over to the integral (2.454).
This will be an important point in derivingin Chapter 12 a new path integral formula valid for singular potentials. At first, weshall ignore such subtleties and continue with the conventional discussion valid forsmooth potentials.2.14Calculation Techniques on Sliced Time Axis.Poisson FormulaIn the previous sections we have used tabulated product formulas such as (2.117), (2.158), (2.166),(2.392), (2.394) to find fluctuation determinants on a finite sliced time axis. With the recentinterest in lattice models of quantum field theories, it is useful to possess an efficient calculationaltechnique to derive such product formulas (and related sums). Consider, as a typical example, thequantum-statistical partition function for a harmonic oscillator of frequency ω on a time axis withN + 1 slices of width ,Z=NY[2(1 − cos ωm ) + 2 ω 2 ]−1/2 ,(2.456)m=0with the product running over all Matsubara frequencies ωm = 2πmkB T /h̄.
Instead of dealingwith this product it is advantageous to consider the free energyF = −kB T log Z =NX1kB Tlog[2(1 − cos ωm ) + 2 ω 2 ].2m=0(2.457)H. Kleinert, PATH INTEGRALS1572.14 Calculation Techniques on Sliced Time Axis. Poisson FormulaWe now observe that by virtue of Poisson’s summation formula (1.205), the sum can be rewrittenas the following combination of a sum and an integral:∞ Z 2πXdλ iλn(N +1)1F = kB T (N + 1)elog[2(1 − cos λ) + 2 ω 2 ].(2.458)22πn=−∞ 0The sum over n squeezes λ to integer multiples of 2π/(N + 1) = ωm which is precisely what wewant.We now calculate integrals in (2.458):Z 2πdλ iλn(N +1)elog[2(1 − cos λ) + 2 ω 2 ].(2.459)2π0For this we rewrite the logarithm of an arbitrary positive argument as the limit Z ∞dτ −τ a/2+ log(2δ) + γ,elog a = lim −δ→0τδwhere0γ ≡ −Γ (1)/Γ(1) = limN →∞NX1− log Nnn=1!≈ 0.5773156649 .
. .is the Euler-Mascheroni constant. Indeed, the functionZ ∞dt −tE1 (x) =etx(2.460)(2.461)(2.462)is known as the exponential integral with the small-x expansion18E1 (x) = −γ − log x −∞X(−x)kk=1kk!.(2.463)With the representation (2.460) for the logarithm, the free energy can be rewritten as Z ∞Z∞dτ 2π dλ iλn(N +1)−τ [2(1−cos λ)+2 ω2 ]/21 Xlim −F =e− δn0 [log(2δ) + γ] .2 n=−∞ δ→0τ 0 2πδ(2.464)The integral over λ is now performed19 giving rise to a modified Bessel function In(N +1) (τ ):F = Z ∞∞2 2dτ1 Xlim −In(N +1) (τ )e−τ (2+ ω )/2 − δn0 [log(2δ) + γ] .2 n=−∞ δ→0τδIf we differentiate this with respect to 2 ω 2 ≡ m2 , we obtain∞ Z ∞21 X∂Fdτ In(N +1) (τ )e−τ (2+m )/2=∂m24 n=−∞ 0and perform the τ -integral, using the formula valid for Re ν > −1, Re α > Re µppZ ∞α2 − µ2 )−ν ,α2 − µ2 )ν−τ αν (α −−ν (α −ppdτ Iν (µτ )e=µ=µ,α2 − µ2α2 − µ201819I.S.
Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.214.2.I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 8.411.1 and 8.406.1.(2.465)(2.466)(2.467)1582 Path Integrals — Elementary Properties and Simple Solutionsto find"#|n|(N +1)p∞m2 + 2 − (m2 + 2)2 − 41 X∂F1p=.∂m22 n=−∞ (m2 + 2)2 − 42(2.468)From this we obtain F by integration over m2 + 1.
The n = 0 -term under the sum givesp(2.469)log[(m2 + 2 + (m2 + 2)2 − 4 )/2] + constand the n 6= 0 -terms:p1[(m2 + 2 + (m2 + 2)2 − 4 )/2]−|n|(N +1) + const ,−|n|(N + 1)(2.470)where the constants of integration can depend on n(N + 1). They are adjusted by going to thelimit m2 → ∞ in (2.465). There the integral is dominated by the small-τ regime of the Besselfunctions1 z α[1 + O(z 2 )],(2.471)Iα (z) ∼|α|! 2and the first term in (2.465) becomesZ ∞1dτ τ |n|(N +1) −τ m2 /2−e(|n|(N + 1))! δ τ 2log m2 + γ + log(2δ)n=0.≈−(m2 )−|n|(N +1) /|n|(N + 1) n =6 0(2.472)The limit m2 → ∞ in (2.469), (2.470) gives, on the other hand, log m2 + const and−(m2 )−|n|(N +1) /|n|(N + 1) + const , respectively. Hence the constants of integration must bezero.
We can therefore write down the free energy for N + 1 time steps asF==N1 Xlog[2(1 − cos(ωm )) + 2 ω 2 ]2β m=0(h ip1log 2 ω 2 + 2 + (2 ω 2 + 2)2 − 4 22(2.473)∞ i−|n|(N +1)p2 X 1 h 2 2 ω + 2 + (2 ω 2 + 2)2 − 4 2−N + 1 n=1 n)Here it is convenient to introduce the parameternhi opω̃e ≡ log 2 ω 2 + 2 + (2 ω 2 + 2)2 − 4 2 ,.(2.474)which satisfiescosh(ω̃e ) = (2 ω 2 + 2)/2,orsinh(ω̃e ) =p(2 ω 2 + 2)2 − 4/2,(2.475)sinh(ω̃e /2) = ω/2.Thus it coincides with the parameter introduced in (2.391), which brings the free energy (2.473)to the simple form"#∞X21 −ω̃e n(N +1)h̄ω̃e −eF =2(N + 1) n=1 n1h̄ω̃e + 2kB T log(1 − e−βh̄ω̃e )=21log [2 sinh(βh̄ω̃e /2)] ,(2.476)=βH. Kleinert, PATH INTEGRALS2.15 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 159whose continuum limit is→0F =2.15h̄ω11log [2 sinh (βh̄ω/2)] =+ log(1 − e−βh̄ω ).β2β(2.477)Field-Theoretic Definition of Harmonic Path Integralby Analytic RegularizationA slight modification of the calculational techniques developed in the last sectionfor the quantum partition function of a harmonic oscillator can be used to definethe harmonic path integral in a way which neither requires time slicing, as in theoriginal Feynman expression (2.64), nor a precise specification of the integrationmeasure in terms of Fourier components, as in Section 2.12.
The path integral forthe partition functionZω =I−DxeR h̄β0M [ẋ2 (τ )+ω 2 x2 (τ )]/2=IDxe−R h̄β0M x(τ )[−∂τ2 +ω 2 ]x(τ )/2(2.478)is formally evaluated as1122Zω = q= e− 2 Tr log(−∂τ +ω ) .Det(−∂τ2 + ω 2 )(2.479)Since the determinant of an operator is the product of all its eigenvalues, we maywrite, again formally,Y1qZω =.(2.480)2ω0ω0 + ω2The product runs over an infinite set of quantities which grow with ω 02 , thus beingcertainly divergent. It may be turned into a divergent sum by rewriting Zω as1Zω ≡ e−Fω /kB T = e− 2Pω0log(ω 0 2 +ω 2 ).(2.481)This expression has two unsatisfactory features.
First, it requires a proper definitionof the formal sum over a continuous set of frequencies. Second, the logarithm of2the dimensionful arguments ωm+ ω 2 must be turned into a meaningful expression.The latter problem would be removed if we were able to exchange the logarithmP2by log[(ω 0 + ω 2 )/ω 2 ]. This would require the formal sum ω0 log ω 2 to vanish. Weshall see below in Eq. (2.507) that this is indeed one of the pleasant properties ofanalytic regularization.At finite temperatures, the periodic boundary conditions along the imaginarytime axis make the frequencies ω 0 in the spectrum of the differential operator −∂τ2 +ω 2discrete, and the sum in the exponent of (2.481) becomes a sum over all Matsubarafrequencies ωm = 2πkB T /h̄ (m = 0, ±1, ±2, .
. .):∞1 X2Zω = exp −log(ωm+ ω2) .2 m=−∞"#(2.482)1602 Path Integrals — Elementary Properties and Simple SolutionsFor the free energy Fω ≡ (1/β) log Zω , this impliesFω =∞1 X12Tr log(−∂τ2 + ω 2 ) =log(ωm+ ω 2).per2β2β m=−∞(2.483)where the subscript per emphasizes the periodic boundary conditions in the τ interval (0, h̄β).2.15.1Zero-Temperature Evaluation of Frequency SumIn the limit T → 0, the sum in (2.483) goes over into an integral, and the free energybecomes1h̄Fω ≡=Tr log(−∂τ2 + ω 2)±∞2β2∞Z−∞dω 0log(ω 02 + ω 2),2π(2.484)where the subscript ±∞ indicates the vanishing boundary conditions of the eigenfunctions at τ = ±∞.
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