Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 49

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It will be useful to do this in severalsteps. In the present context, some of these may appear redundant. They will,however, yield intermediate results which will be needed in Chapters 7 and 18 whendiscussing path integrals occurring in quantum field theories.2183.33 External Sources, Correlations, and Perturbation TheoryGreen Functions of First-Order Differential EquationAn important quantity of statistical mechanics are the Green functions GpΩ (t, t0 )which solve the first-order differential equation[i∂t − Ω(t)] GΩ (t, t0 ) = iδ(t − t0 ),t − t0 ∈ [0, tb − ta ),(3.75)or its Euclidean version GpΩ,e (τ, τ 00 ) which solves the differential equation, obtainedfrom (3.75) for t = −iτ :[∂τ − Ω(τ )] GΩ,e (τ, τ 0 ) = δ(τ − τ 0 ),τ − τ 0 ∈ [0, h̄β).(3.76)These can be calculated for an arbitrary function Ω(t).3.3.1Time-Independent FrequencyConsider first the simplest case of a Green function Gpω (t, t0 ) with fixed frequency ωwhich solves the first-order differential equation(i∂t − ω)Gpω (t, t0 ) = iδ(t − t0 ),t − t0 ∈ [0, tb − ta ).(3.77)The equation determines Gpω (t, t0 ) only up to a solution H(t, t0 ) of the homogeneousdifferential equation (i∂t − ω)H(t, t0) = 0.

The ambiguity is removed by imposingthe periodic boundary conditionGpω (t, t0 ) ≡ Gpω (t − t0 ) = Gpω (t − t0 + tb − ta ),(3.78)indicated by the superscript p. With this boundary condition, the Green function Gpω (t, t0 ) is translationally invariant in time. It depends only on the differencebetween t and t0 and is periodic in it.The spectral representation of Gpω (t, t0 ) can immediately be written down, assuming that tb − ta does not coincide with an even multiple of π/ω:Gpω (t − t0 ) =∞X0i1e−iωm (t−t ).tb − ta m=−∞ωm − ω(3.79)The frequencies ωm are twice as large as the previous νm ’s in (3.64):ωm ≡2πm,tb − tam = 0, ±1, ±2, ±3, .

. . .(3.80)As for the periodic orbits in Section 2.9, there are “about as many” ωm as νm ,since there is an ωm for each positive and negative integer m, whereas the νm areall positive (see the last paragraph in that section). The frequencies (3.80) are thereal-time analogs of the Matsubara frequencies (2.373) of quantum statistics withthe usual correspondence tb − ta = −ih̄/kB T of Eq. (2.323).H. Kleinert, PATH INTEGRALS3.3 Green Functions of First-Order Differential Equation219To calculate the spectral sum, we use the Poisson summation formula in theform (1.197):∞Xf (m) =m=−∞∞Z−∞dµ∞Xe2πiµn f (µ).(3.81)n=−∞Accordingly, we rewrite the sum over ωm as an integral over ω 0, followed by anauxiliary sum over n which squeezes the variable ω 0 onto the proper discrete valuesωm = 2πm/(tb − ta ):Gpω (t)=∞ ZX∞n=−∞ −∞dω 0 −iω0 [t−(tb −ta )n] ie.2πω0 − ω(3.82)At this point it is useful to introduce another Green function Gω (t−t0 ) associatedwith the first-order differential equation (3.77) on an infinite time interval:Gω (t) =Zdω 0 −iω0 t ie.2πω0 − ω(3.83)Gω (t − (tb − ta )n).(3.84)∞−∞In terms of this function, the periodic Green function (3.82) can be written as a sumwhich exhibits in a most obvious way the periodicity under t → t + (tb − ta ):Gpω (t)=∞Xn=−∞The advantage of using Gω (t − t0 ) is that the integral over ω 0 in (3.83) can easilybe done.

We merely have to prescribe how to treat the singularity at ω 0 = ω. Thisalso removes the freedom of adding a homogeneous solution H(t, t0 ). To make theintegral unique, we replace ω by ω − iη where η is a very small positive number,i.e., by the iη-prescription introduced after Eq. (2.161). This moves the pole in theintegrand of (3.83) into the lower half of the complex ω 0-plane, making the integralover ω 0 in Gω (t) fundamentally different for t < 0 and for t > 0.

For t < 0, thecontour of integration can be closed in the complex ω 0 -plane by a semicircle in the0upper half-plane at no extra cost, since e−iω t is exponentially small there (see Fig.3.1). With the integrand being analytic in the upper half-plane we can contract thecontour to zero and find that the integral vanishes. For t > 0, on the other hand,the contour is closed in the lower half-plane containing a pole at ω 0 = ω − iη. Whencontracting the contour to zero, the integral picks up the residue at this pole andyields a factor −2πi. At the point t = 0, finally, we can close the contour either way.The integral over the semicircles is now nonzero, ∓1/2, which has to be subtractedfrom the residues 0 and 1, respectively, yielding 1/2.

Hence we finddω 0 −iω0 tiGω (t) =e0ω − ω + iη−∞ 2πfor t > 0, 1for t = 0,= e−iωt × 120 for t < 0.Z∞(3.85)2203 External Sources, Correlations, and Perturbation TheoryFigure 3.1 Pole in Fourier transform of Green functions Gp,aω (t), and infinite semicirclesin the upper (lower) half-plane which extend the integrals to a closed contour for t < 0(t > 0).The vanishing of the Green function for t < 0 is the causality property of Gω (t)discussed in (1.307) and (1.308).

It is a general property of functions whose Fouriertransforms are analytic in the upper half-plane.The three cases in (3.85) can be collected into a single formula using the Heavisidefunction Θ(t) of Eq. (1.310):Gω (t) = e−iωt Θ̄(t).(3.86)The periodic Green function (3.84) can then be written as∞XGpω (t) =n=−∞e−iω[t−(tb −ta )n] Θ̄(t − (tb − ta )n).(3.87)Being periodic in tb − ta , its explicit evaluation can be restricted to the basic intervalt ∈ [0, tb − ta ).(3.88)Inside the interval (0, tb − ta ), the sum can be performed as follows:Gpω (t) =0Xe−iω[t−(tb −ta )n] =n=−∞e−iω[t−(tb −ta )/2],= −i2 sin[ω(tb − ta )/2]e−iωt1 − e−iω(tb −ta )t ∈ (0, tb − ta ).(3.89)At the point t = 0, the initial term with Θ̄(0) contributes only 1/2 so that1Gpω (0) = Gpω (0+) − .(3.90)2Outside the basic interval (3.88), the Green function is determined by its periodicity.For instance,Gpω (t) = −ie−iω[t+(tb −ta )/2],2 sin[ω(tb − ta )/2]t ∈ (−(tb − ta ), 0).(3.91)H.

Kleinert, PATH INTEGRALS2213.3 Green Functions of First-Order Differential EquationNote that as t crosses the upper end of the interval [0, tb −ta ), the sum in (3.87) picksup an additional term (the term with n = 1). This causes a jump in Gpω (t) whichenforces the periodicity.

At the upper point t = tb − ta , there is again a reductionby 1/2 so that Gpω (tb − ta ) lies in the middle of the jump, just as the value 1/2 liesin the middle of the jump of the Heaviside function Θ̄(t).The periodic Green function is of great importance in the quantum statistics ofBose particles (see Chapter 7). After a continuation of the time to imaginary values,t → −iτ , tb − ta → −ih̄/kB T , it takes the formGpω,e (τ ) =1−h̄ω/kB T1−ee−ωτ ,τ ∈ (0, h̄β),(3.92)where the subscript e records the Euclidean character of the time.

The prefactoris related to the average boson occupation number of a particle state of energy h̄ω,given by the Bose-Einstein distribution functionnbω =1h̄ω/kB TeIn terms of it,−1Gpω,e (τ ) = (1 + nbω )e−ωτ ,.(3.93)τ ∈ (0, h̄β).The τ -behavior of the subtracted periodic Green functionis shown in Fig. 3.2.0Gpω,e(τ )10.80.60Gpω,e(τ )(3.94)≡Gpω,e (τ )−1/h̄βωGaω,e (τ )0.50.40.2-1-0.5-0.20.511.52τ /h̄β-1-0.4-0.50.511.52τ /h̄β-0.5-10 ≡ Gp (τ )−1/h̄βω and antiperiodicFigure 3.2 Subtracted periodic Green function Gpω,eω,eGreen function Gaω,e(τ ) for frequencies ω = (0, 5, 10)/h̄β (with increasing dash length).The points show the values at the jumps of the three functions (with increasing point size)corresponding to the relation (3.90).As a next step, we consider a Green function Gpω2 (t) associated with the secondorder differential operator −∂t2 − ω 2 ,Gpω2 (t, t0 ) = (−∂t2 − ω 2 )−1 δ(t − t0 ),t − t0 ∈ [ta , tb ),(3.95)Gpω2 (t, t0 ) ≡ Gpω2 (t − t0 ) = Gpω2 (t − t0 + tb − ta ).(3.96)which satisfies the periodic boundary condition:2223 External Sources, Correlations, and Perturbation TheoryFigure 3.3 Two poles in Fourier transform of Green function Gp,aω 2 (t).Just like Gpω (t, t0 ), this periodic Green function depends only on the time differencet − t0 .

It obviously has the spectral representationGpω2 (t)∞X11=,e−iωm t 2tb − ta m=−∞ωm − ω 2(3.97)which makes sense as long as tb − ta is not equal to an even multiple of π/ω. Atinfinite tb − ta , the sum becomes an integral over ωm with singularities at ±ω whichmust be avoided by an iη-prescription, which adds a negative imaginary part tothe frequency ω [compare the discussion after Eq. (2.161)]. This fixes also thecontinuation from small tb − ta beyond the multiple values of π/ω. By decomposing11=0222iωω − ω + iη!ii,− 00ω − ω + iη ω + ω − iη(3.98)the calculation of the Green function (3.97) can be reduced to the previous case.The positions of the two poles of (3.98) in the complex ω 0-plane are illustrated inFig.

3.3. In this way we find, using (3.89),i1 h pGω (t) − Gp−ω (t)2ωi1 cos ω[t − (tb − ta )/2],= −2ω sin[ω(tb − ta )/2]Gpω2 (t) =t ∈ [0, tb − ta ).(3.99)In Gp−ω (t) one must keep the small negative imaginary part attached to the frequencyω. For an infinite time interval tb − ta , this leads to a Green function Gpω2 (t − t0 ):alsoG−ω (t) = −e−iωt Θ̄(−t).(3.100)The directional change in encircling the pole in the ω 0 -integral leads to the exchangeΘ̄(t) → −Θ̄(−t).Outside the basic interval t ∈ [0, tb − ta ), the function is determined by itsperiodicity. For t ∈ [−(tb − ta ), 0), we may simply replace t by |t|.H.

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