Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 36

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As in thequantum-mechanical partition function in (2.61), the path integral Dx now standsforINY+1 Z ∞dxnqDx ≈.(2.371)2πh̄/Mn=1 −∞qIt contains no extra 1/ 2πh̄/M factor, as in (2.368), due to the trace integrationover the exterior x.The condition x(h̄β) = x(0) is most easily enforced by expanding x(τ ) into aFourier seriesx(τ ) =∞Xm=−∞√1e−iωm τ xm ,N +1(2.372)1442 Path Integrals — Elementary Properties and Simple Solutionswith the Matsubara frequenciesωm ≡ 2πmkB T /h̄ =2πm,h̄βm = 0, ±1, ±2, . . .

.(2.373)When considered as functions on the entire τ -axis, the paths are periodic in h̄β atany τ , i.e.,x(τ ) = x(τ + h̄β).(2.374)Thus the path integral for the quantum-statistical partition function comprises allperiodic paths with a period h̄β. In the time-sliced path integral (2.368), the coordinates x(τ ) are needed only at the discrete times τn = n. Correspondingly, thesum over m in (2.372) can be restricted to run from m = −N/2 to N/2 for even Nand from −(N − 1)/2 to (N + 1)/2 for odd N (see Fig. 2.3). In order to have a realx(τn ), we must require thatxm = x∗−m(modulo N + 1).(2.375)Note that the Matsubara frequencies in the expansion of the paths x(τ ) are nowtwice as big as the frequencies νm in the quantum fluctuations (2.105) (after analyticcontinuation of tb − ta to −ih̄/kB T ). Still, they have about the same total number,since they run over positive and negative integers.

An exception is the zero frequencyωm = 0, which is included here, in contrast to the frequencies νm in (2.105) whichrun only over positive m = 1, 2, 3, . . . . This is necessary to describe paths witharbitrary nonzero endpoints xb = xa = x (included in the trace).2.10Quantum Statistics of Harmonic OscillatorThe harmonic oscillator is a good example for solving the quantum-statistical path integral. Theτ -axis is sliced at τn = n, with ≡ h̄β/(N + 1) (n = 0, . .

. , N + 1), and the partition function isgiven by the N → ∞ -limit of the product of integrals"Z#N∞YdxnpZωN =exp −AN(2.376)e /h̄ ,2πh̄/M−∞n=0where ANe is the time-sliced Euclidean oscillator actionANeN +1M X=xn (−2 ∇∇ + 2 ω 2 )xn .2 n=1(2.377)Integrating out the xn ’s, we find immediately1ZωN = q.detN +1 (−2 ∇∇ + 2 ω 2 )(2.378)Let us evaluate the fluctuation determinant via the product of eigenvalues which diagonalizethe matrix −2 ∇∇ + 2 ω 2 in the sliced action (2.377). They are2 Ωm Ωm + 2 ω 2 = 2 − 2 cos ωm + 2 ω 2 ,(2.379)H. Kleinert, PATH INTEGRALS1452.10 Quantum Statistics of Harmonic OscillatorFigure 2.3 Illustration of the eigenvalues (2.379) of the fluctuation matrix in the action(2.377) for even and odd N .with the Matsubara frequencies ωm .

For ω = 0, the eigenvalues are pictured in Fig. 2.3. Theaction (2.377) becomes diagonal after going to the Fourier components xm . To do this we arrangethe real and imaginary parts Re xm and Im xm in a row vector(Re x1 , Im x1 ; Re x2 , Im x2 ; . . . ; Re xn , Im xn ; . . .),and see that it is related to the time-sliced positions xn = x(τn ) by a transformation matrix withthe rowsTmn xn=(Tm )n xnr2 1mm√ , cos=2π · 1, sin2π · 1,N +1N +1N +12mmcos2π · 2, sin2π · 2, .

. .N +1N +1mm. . . , cos2π · n, sin2π · n, . . . xn .(2.380)N +1N +1nFor each row index m = 0, . . . , N, the column index n runs from zero to N/2 for even N , and to(N + 1)/2 for odd N . In the odd case, the last column sin Nm+1 2π · n with n = (N + 1)/2 vanishesidentically and must be dropped, so that the number of columns in Tmn is in both cases N + 1,as it should be. For odd N , the second-last column of Tmn is an alternating sequence √±1. Thus,for a proper normalization, it has to be multiplied by an extra normalization factor 1/ 2, just asthe elements in the first column.

An argument similar to (2.110), (2.111) shows that the resultingmatrix is orthogonal. Thus, we can diagonalize the sliced action in (2.377) as followsi hPN/22 222ωx+2(ΩΩ+ω)|x|forN = even,m mm0m=1M  2 2N(2.381)Ae =ω x0 + (Ω(N +1)/2 Ω(N +1)/2 + ω 2 )xN2+1i2 P(N −1)/2+ 2 m=1 (Ωm Ωm + ω 2 )|xm |2forN = odd.Q R∞Thanks to the orthogonality of Tmn , the measure n −∞ dx(τn ) transforms simply intoZN/2∞dx0−∞Z∞−∞dx0Z∞−∞dx(N +1)/2YZm=1∞d Re xm∞Z∞d Im xmforN = even,−∞−∞(2.382)(N −1)/2 Z ∞Ym=1Z−∞d Re xm−∞d Im xmforN = odd.1462 Path Integrals — Elementary Properties and Simple SolutionsBy performing the Gaussian integrals we obtain the partition function" N#−1/2YN22 2 −1/22 22Zω = detN +1 (− ∇∇ + ω )=( Ωm Ωm + ω )m=0=(NY2(1 − cos ωm ) + 2 ω 2m=0)−1/2 "=N Ym=0ωm + 2 ω 24 sin22#−1/2.(2.383)Thanks to the periodicity of the eigenvalues under the replacement n → n + N + 1, the result hasbecome a unique product expression for both even and odd N .It is important to realize that contrary to the fluctuation factor (2.155) in the real-time amplitude, the partition function (2.383) contains the square root of only positive eigenmodes as aunique result of Gaussian integrations.

There are no phase subtleties as in the Fresnel integral(1.334).To calculate the product, we observe that upon decomposingωm ωm ωm 1 − cos,(2.384)sin2= 1 + cos222the sequence of first factors1 + cosπmωm ≡ 1 + cos2N +1(2.385)runs for m = 1, . . . N through the same values as the sequence of second factors1 − cosωm πmN +1−m= 1 − cos≡ 1 + cos π,2N +1N +1(2.386)except in an opposite order. Thus, separating out the m = 0 -term, we rewrite (2.383) in the formZωN1=ω#−1 " N!#−1/2Yωm 2 ω 22 1 − cos1+.24 sin2 ωm2m=1m=1"NY(2.387)The first factor on the right-hand side is the quantum-mechanical fluctuation determinant of thefree-particle determinant detN (−2 ∇∇) = N + 1 [see (2.118)], so that we obtain for both even andodd N" N!#−1/22 ω 2kB T YN1+Zω =.(2.388)h̄ω m=14 sin2 ωm2To evaluate the remaining product, we must distinguish again between even and odd cases of N .For even N , where every eigenvalue occurs twice (see Fig.

2.3), we obtain!−1N/22 2YωkTB .1+ZωN =(2.389)h̄ω m=14 sin2 Nmπ+1For odd N , the term with m = (N + 1)/2 occurs only once and must be treated separately so that!−1−1)/22 22 2 1/2 (NY ωkB T  ω .1+ZωN =1+(2.390)2 πmh̄ω44sinN +1m=1We now introduce the parameter ω̃e , the Euclidean analog of (2.156), via the equationssin iωω̃e ≡i ,22sinhω̃e ω≡.22(2.391)H. Kleinert, PATH INTEGRALS1472.10 Quantum Statistics of Harmonic OscillatorIn the odd case, the product formula16"#(N −1)/2Ysin2 x2 sin[(N + 1)x]1−=2 mπsin 2x (N + 1)sin (N +1)m=1[similar to (2.158)] yields, with x = ω̃e /2,−1kB Tsinh[(N + 1)ω̃e /2]1ZωN =.h̄ω sinh(ω̃e /2)N +1In the even case, the formula17"N/2Y1−m=1sin2 xsin2 (Nmπ+1)#=1 sin[(N + 1)x],sin x (N + 1)(2.392)(2.393)(2.394)produces once more the same result as in Eq. (2.393).

Inserting Eq. (2.391) leads to the partitionfunction on the sliced imaginary time axis:ZωN =1.2 sinh(h̄ω̃e β/2)(2.395)The partition function can be expanded into the following seriesZωN = e−h̄ω̃e /2kB T + e−3h̄ω̃e /2kB T + e−5h̄ω̃e /2kB T + . . . .(2.396)By comparison with the general spectral expansion (2.321), we display the energy eigenvalues ofthe system:1h̄ω̃e .(2.397)En = n +2They show the typical linearly rising oscillator sequence withω̃e =2ωarsinh2(2.398)playing the role of the frequency on the sliced time axis, and h̄ω̃e /2 being the zero-point energy.In the continuum limit → 0, the time-sliced partition function ZωN goes over into the usualoscillator partition function1Zω =.(2.399)2 sinh(βh̄ω/2)In D dimensions this becomes, of course, [2 sinh(βh̄ω/2)]−D , due to the additivity of the action ineach component of x.Note that the continuum limit of the product in (2.388) can also be taken factor by factor.Then Zω becomes" ∞ #−1ω2kB T Y1+ 2Zω =.(2.400)h̄ω m=1ωmAccording to formula (2.166), the productand we find with x = h̄ωβ/2Zω =1617Q∞m=11+x2m2 π 2converges rapidly against sinh x/x1kB Th̄ω/2kB T=.h̄ω sinh(h̄ω/2kB T )2 sinh(βh̄ω/2)I.S.

Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.391.1.ibid., formula 1.391.3.(2.401)1482 Path Integrals — Elementary Properties and Simple SolutionsAs discussed after Eq. (2.176), the continuum limit can be taken in each factor since the productin (2.388) contains only ratios of frequencies.Just as in the quantum-mechanical case, this procedure of obtaining the continuum limit canbe summarized in the sequence of equations arriving at a ratio of differential operators−1/2ZωN=detN +1 (−2 ∇∇ + 2 ω 2 )#−1/2"−1/2 detN +1 (−2 ∇∇ + 2 ω 2 ) 02=detN +1 (− ∇∇)det0N +1 (−2 ∇∇)−1−1/2∞ 2→0+ ω2kB T det(−∂τ2 + ω 2 )kB T Y ωm.(2.402)−−−→=2h̄h̄ω m=1ωmdet0 (−∂τ2 )In the ω = 0 -determinants, the zero Matsubara frequency is excluded to obtain a finite expression.This is indicated by a prime. The differential operator −∂τ2 acts on real functions which are periodic2under the replacement τ → τ +h̄β.

Remember that each eigenvalue ωmof −∂τ2 occurs twice, exceptfor the zero frequency ω0 = 0, which appears only once.Let us finally mention that the results of this section could also have been obtained directlyfrom the quantum-mechanical amplitude (2.168) [or with the discrete times from (2.192)] by ananalytic continuation of the time difference tb − ta to imaginary values −i(τb − τa ):r1ω(xb τb |xa τa ) = p2πh̄/M sinh ω(τb − τa )Mω122× exp −[(x + xa ) cosh ω(τb − τa ) − 2xb xa ] .(2.403)2h̄ sinh ω(τb − τa ) bBy setting x = xb = xa and integrating over x, we obtain [compare (2.326)]sZ ∞ω(τb − τa )1Zω =dx (x τb |x τa ) = psinh[ω(τ2πh̄(τb − τa )/Mb − τa )]−∞p2πh̄ sinh[ω(τb − τa )]/ωM1×=.2 sinh[ω(τb − τa )/2]2 sinh[ω(τb − τa )/2](2.404)Upon equating τb − τa = h̄β, we retrieve the partition function (2.399).

A similar treatment ofthe discrete-time version (2.192) would have led to (2.395). The main reason for presenting anindependent direct evaluation in the space of real periodic functions was to display the frequencystructure of periodic paths and to see the difference with respect to the quantum-mechanical pathswith fixed ends. We also wanted to show how to handle the ensuing product expressions.For applications in polymer physics (see Chapter 15) one also needs the partition function ofall path fluctuations with open endssZ ∞Z ∞1ω(τb − τa ) 2πh̄opendxa (xb τb |xa τa ) = pZω=dxb2πh̄(τb − τa )/M sinh[ω(τb − τa )] M ω−∞−∞r2πh̄1p=.(2.405)M ω sinh[ω(τb − τa )]√The prefactor is 2π times the length scale λω of Eq. (2.296).2.11Time-Dependent Harmonic PotentialIt is often necessary to calculate thermal fluctuation determinants for the case of a time-dependentfrequency Ω(τ ) which is periodic under τ → τ + h̄β.

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