Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 40
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Recall the discussion of the iηprescription in Section 3.3. The integral (2.524) can be split into the integralsZ∞−∞dω 0ωlog[ω 0 ± (ω − iη)] = i ,2π2ω ≥ 0.(2.525)Hence formula (2.521) can be generalized to arbitrary complex frequencies ω =ωR + iωI as follows:Z ∞dω 0ωlog(ω 0 ± iω) = ∓(ωR ) ,(2.526)2−∞ 2πandZ∞−∞ωdω 0log(ω 0 ± ω) = −i(ωI ) ,2π2(2.527)where (x) = Θ(x)−Θ(−x) = x/|x| is the antisymmetric Heaviside function (1.312),which yields the sign of its argument. The formulas (2.526) and (2.527) are thelarge-time limit of the more complicated sumskB Th̄∞X"#k Th̄ωlog(ωm ± iω) = B log 2(ωR ) sinh,h̄2kB Tm=−∞(2.528)andkB Th̄∞X"#k Th̄ωlog(ωm ± ω) = B log −2i(ωI ) sin.h̄2kB Tm=−∞(2.529)The first expression is periodic in the imaginary part of ω, with period 2πkB T , thesecond in the real part. The determinants possess a meaningful large-time limit onlyif the periodic parts of ω vanish.
In many applications, however, the fluctuations willinvolve sums of logarithms (2.529) and (2.528) with different complex frequenciesω, and only the sum of the imaginary or real parts will have to vanish to obtain ameaningful large-time limit. On these occasions we may use the simplified formulas(2.526) and (2.527). Important examples will be encountered in Section 18.9.2.H. Kleinert, PATH INTEGRALS2.15 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 1672.15.4Gradient Expansion of One-Dimensional TracelogFormula (3.133) may be used to calculate the trace of the logarithm of a secondorder differential equation with arbitrary frequency as a semiclassical expansion.
Weintroduce the Planck constant h̄ and the potential w(τ ) ≡ h̄Ω(τ ), and factorize asin (2.520):hiDet −h̄∂τ2 + w 2 (τ ) = Det [−h̄∂τ − w̄(τ )] × Det [h̄∂τ − w̄(τ )] ,(2.530)where the function w̄(τ ) satisfies Riccati differential equation:23h̄∂τ w̄(τ ) + w̄ 2 (τ ) = w 2(τ ).(2.531)By solving this we obtain the trace of the logarithm from (3.133):Tr logh−h̄2 ∂τ2("1+ w (τ ) = 2 log 2 sinh2h̄i2Zh̄β00000dτ w̄(τ )#).(2.532)The solution to the Riccati equation (2.531) can be found as a power series in h̄:w̄(τ ) =∞Xw̄n (τ )h̄n ,(2.533)n=0which provides us with a so-called gradient expansion of the trace of the logarithm.The lowest-order coefficient function w̄0 (τ ) is obviously equal to w(τ ).
The higherones obey the recursion relationn−1X1˙w̄n (τ ) = −w̄n−1 (τ ) +w̄n−k (τ )w̄k (τ ) ,2w(τ )k=1!n ≥ 1.(2.534)These are solved for n = 0, 1, 2, 3 by(pv 0 (τ )v(τ ), −,4 v(τ )−1105 v 0(τ )5 v 0 (τ )25/232 v(τ )411/22048 v(τ )where v(τ ) ≡ w2 (τ ).2.15.5−+v 00 (τ )8 v(τ )−,3/22+221 v 0(τ ) v 00 (τ )9/2256 v(τ )−15 v 0 (τ )3464 v(τ )19 v 00 (τ )27/2128 v(τ )−+9 v 0 (τ ) v 00 (τ )32 v(τ )37 v 0 (τ ) v (3) (τ )7/232 v(τ )−+v (3) (τ )16 v(τ )2,v (4) (τ )5/232 v(τ )),(2.535)The series can, of course, be trivially extended any desired orders.Duality Transformation and Low-Temperature ExpansionThere exists another method of calculating the finite-temperature part of the freeenergy (2.483) which is worth presenting at this place, due to its broad applicabilityin statistical mechanics.
For this we rewrite (2.508) in the formk T∆Fω = B223∞Xh̄−kB Tm=−∞Z∞−∞!dωm2log(ωm+ ω 2 ).2π(2.536)Recall the general form of the Riccati differential equation y 0 = f (τ )y + g(τ )y 2 + h(y), whichis an inhomogeneous version of the Bernoulli differential equation y 0 = f (τ )y + g(τ )y n for n = 2.1682 Path Integrals — Elementary Properties and Simple SolutionsChanging the integration variable to m, this becomes∞Xk T∆Fω = B2!2πkB Tdm log −h̄−∞m=−∞Z∞!22m +ω2.(2.537)Within analytic regularization, this expression is rewritten with the help of formula(2.498) ask T∆Fω = − B2∞Z0dττ∞Xm=−∞−∞Z−∞!dm e−τ [(2πkB T /h̄)2 m2 +ω 2].(2.538)The duality transformation proceeds by performing the sum over the MatsubaraRfrequencies with the help of Poisson’s formula (1.205) as an integral dµ using anextra sum over integer numbers n. This brings (2.538) to the form (expressing thetemperature in terms of β),1∆Fω = −2βZdττ∞0Z∞−∞dµ2πµni − τ!22πµnien=−∞The parentheses contain the sum 2exponent2πh̄β∞Xµ2 = −τP∞n=12πh̄β!− 1 e−τ [(2π/h̄β)2 µ2 +ω 2].(2.539)e2πµni .
After a quadratic completion of the!2 "nh̄2 β 2µ−i4πτ#2−1(h̄βn)2 ,4τ(2.540)the integral over µ can be performed, with the result∞2h̄ Z ∞ dτ −1/2 X2∆Fω = − √τe−(nh̄β) /4τ −τ ω .2 π 0 τn=1(2.541)Now we may use the integral formula [compare (1.344)]24Z0∞ νadτ ν −a2 /τ −b2 ττ e=2τbK−ν (2ab),(2.542)to obtain the sum over modified Bessel functions∞√h̄ω X∆Fω = − √2 (nβh̄ω)−1/2 2K1/2 (nβh̄ω).2 π n=1(2.543)The modified Bessel functions with index 1/2 are particularly simple:K1/2 (z) =24rπ −ze .2z(2.544)I.S.
Gradshteyn and I.M. Ryzhik, ibid., Formula 3.471.9.H. Kleinert, PATH INTEGRALS2.15 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 169Inserting this into (2.543), the sum is a simple geometric one, and may be performedas follows:1 X 1 −βh̄ωn1∆Fω = −(2.545)e= log 1 − e−βh̄ω ,β n=1 nβin agreement with the previous result (2.518).The effect of the duality tranformation may be rephrased in another way. Itconverts the original sum over m in the expression (2.509):S(βh̄ω) =∞X22ωmωm+1−loglogω2ω2#(2.546)∞Xβh̄ω1 −nβh̄ω− log βh̄ω −e.2n=1 n(2.547)m=1"!into a sum over n:S(βh̄ω) =The first sum (2.546) converges fast at high temperatures, where it can be expandedin powers of ω 2 :(−1)kS(βh̄ω) = −kk=1∞X∞X!1βh̄ω2k2πm=1 m!2 k.(2.548)The expansion coefficients are equal to Riemann’s zeta function ζ(z) of Eq.
(2.514)at even arguments z = 2k, so that we may write(−1)kβh̄ωζ(2k)S(βh̄ω) = −k2πm=1∞X!2k.(2.549)At even positive arguments, the zeta function is related to the Bernoulli numbersby25(2π)2n|B |,ζ(2n) =2(2n)! 2nζ(1 − 2n) = −B2n.2n(2.550)The right-hand side states a similar relation for ζ(z) at odd negative integer valueswhich arises from the even positive ones by the identity26ζ(z) = 2z π z−1 sin(πz/2)Γ(1 − z)ζ(1 − z),(2.551)which can also be written asζ(z) = 2z−1π z ζ(1 − z)/Γ(z) cos2526zπ.2ibid., Formulas 9.542 and 9.535.I.S. Gradshteyn and I.M.
Ryzhik, op. cit., Formula 9.535.2.(2.552)1702 Path Integrals — Elementary Properties and Simple SolutionsThe first few of them are27π4π6π2, . . . , ζ(∞) = 1.ζ(2) = , ζ(4) = , ζ(5) =690945(2.553)In contrast to the original sum (2.546) and its expansion (2.549), the duallytransformed sum (2.547) converges rapidly for low temperatures. It converges everywhere except at very large temperatures, where it diverges logarithmically.
Theprecise behavior can be calculated as follows: For large T there exists a large numberN which is still much smaller than 1/βh̄ω, such that e−βh̄ωN is close to unity. Thenwe split the sum as−1∞X11 −nβh̄ω1 −nβh̄ω NXe≈+e.n=1 nn=1 nn=N n∞X(2.554)Since N is large, the second sum can be approximated by an integralZ∞Ndn −nβh̄ωe=nZ∞N βh̄ωdx −xe ,xwhich is an exponential integral E1 (Nβh̄ω) of Eq. (2.462) with the large-argumentexpansion −γ − log(Nβh̄ω) of Eq. (2.463).The first sum in (2.554) is calculated with the help of the Digamma functionψ(z) ≡Γ0 (z).Γ(z)(2.555)This has an expansion28ψ(z) = −γ −∞ Xn=011−,n+z n+1(2.556)which reduces for integer arguments toψ(N) = −γ +N−1Xn=11,n(2.557)and has the large-z expansionψ(z) ≈ log z −∞X1B2n−.2z n=1 2nz 2n(2.558)Combining this with (2.463), the logarithm of N cancels, and we find for the sumin (2.554) the large-T behavior∞X1 −nβh̄ωe≈ − log βh̄ω.T →∞n=1 n(2.559)27Other often-needed values are ζ(0) = −1/2, ζ 0 (0) = − log(2π)/2, ζ(−2n) = 0, ζ(3) ≈1.202057, ζ(5) ≈ 1.036928, .
. . .28I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.362.1.H. Kleinert, PATH INTEGRALS2.15 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 171This cancels the logarithm in (2.547).The low-temperature series (2.547) can be used to illustrate the power of analyticregularization. Suppose we want to extract from it the large-T behavior, where thesum∞X1 −nβh̄ωe(2.560)g(βh̄ω) ≡n=1 nconverges slowly.
We would like to expand the exponentials in the sum into powersof β = 1/kB T , but this gives rise to sums over positive powers of n. We therefore proceed as in the evaluation of the sums of the Matsubara frequency by firstperforming an integral over n and subsequently the difference between sum andintegral:g(βh̄ω) =Z∞0Z ∞∞X1 −nβh̄ω1 −nβh̄ωdn e−+e.nn0n=1!(2.561)The integral diverges for n → 0. Thus we introduce for a moment a regularizationparameter ν 6= 1, and consider the more general functiongν (βh̄ω) =Z∞1Z ∞∞X1 −nβh̄ω1 −nβh̄ω−dn ν e+e.nnν0n=1!(2.562)We now expand the exponential in the second term in powers of β and obtain theformal expression(−1)k(βh̄ω)k .k!00n=1(2.563)The integral is now convergent, and yields Γ(1 − ν) (βh̄ω)ν via integral formula(2.490).
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