Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 50
Текст из файла (страница 50)
Kleinert, PATH INTEGRALS2233.3 Green Functions of First-Order Differential EquationAs a further step we consider another Green function Gaω (t, t0 ). It fulfills thesame first-order differential equation i∂t − ω as Gpω (t, t0 ):(i∂t − ω)Gaω (t, t0 ) = iδ(t − t0 ),t − t0 ∈ [0, tb − ta ),(3.101)but in contrast to Gpω (t, t0 ) it satisfies the antiperiodic boundary conditionGaω (t, t0 ) ≡ Gaω (t − t0 ) = −Gaω (t − t0 + tb − ta ).(3.102)As for periodic boundary conditions, the Green function Gaω (t, t0 ) depends only onthe time difference t − t0 . In contrast to Gpω (t, t0 ), however, Gaω (t, t0 ) changes signunder a shift t → t + (tb − ta ). The Fourier expansion of Gaω (t − t0 ) isGaω (t) =∞X1fi,e−iωm t ftb − ta m=−∞ωm − ω(3.103)where the frequency sum covers the odd Matsubara-like frequenciesfωm=π(2m + 1).tb − ta(3.104)The superscript f stands for fermionic since these frequencies play an importantrole in the statistical mechanics of particles with Fermi statistics to be explained inSection 7.10 [see Eq.
(7.407)].The antiperiodic Green functions are obtained from a sum similar to (3.82), butmodified by an additional phase factor eiπn = (−)n . When inserted into the Poissonsummation formula (3.81), such a phase is seen to select the half-integer numbersin the integral instead of the integer ones:∞Xf (m + 1/2) =m=−∞Z∞−∞dµ∞X(−)n e2πiµn f (µ).(3.105)n=−∞Using this formula, we can expand∞ ZXGaω (t) =∞n=−∞ −∞∞Xn=dω 00i(−)n e−iω [t−(tb −ta )n] 02πω − ω + iη(−) Gω (t − (tb − ta )n),(3.106)n=−∞or, more explicitly,Gaω (t)=∞Xn=−∞e−iω[t−(tb −ta )n] (−)n Θ̄(t − (tb − ta )n).(3.107)For t ∈ [0, tb − ta ), this givesGaω (t) =0Xe−iω[t−(tb −ta )n] (−)n =n=−∞−iω[t−(tb −ta )/2]=e,2 cos[ω(tb − ta )/2]eiωt1 + e−iω(tb −ta )t ∈ [0, tb − ta ).(3.108)2243 External Sources, Correlations, and Perturbation TheoryOutside the interval t ∈ [0, tb − ta ), the function is defined by its antiperiodicity.The τ -behavior of the antiperiodic Green function Gaω,e (τ ) is also shown in Fig.
3.2.In the limit ω → 0, the right-hand side of (3.108) is equal to 1/2, and theantiperiodicity implies thatGa0 (t) =1(t),2t ∈ [−(tb − ta ), (tb − ta )].(3.109)Antiperiodic Green functions play an important role in the quantum statisticsof Fermi particles. After analytically continuing t to the imaginary time −iτ withtb − ta → −ih̄/kB T , the expression (3.108) takes the formGaω,e (τ ) =1−h̄ω/kB T1+ee−ωτ ,τ ∈ [0, h̄β).(3.110)The prefactor is related to the average Fermi occupation number of a state of energyh̄ω, given by the Fermi-Dirac distribution functionnfω =1h̄ω/kB Te+1.(3.111)In terms of it,Gaω,e (τ ) = (1 − nfω )e−ωτ ,τ ∈ [0, h̄β).(3.112)With the help of Gaω (t), we form the antiperiodic analog of (3.97), (3.99), i.e., theantiperiodic Green function associated with the second-order differential operator−∂t2 − ω 2 :∞Xf11e−iωm t f 2tb − ta m=0ωm − ω 21=[Gaω (t) − Ga−ω (t)]2ωi1 sin ω[t − (tb − ta )/2]= −,2ω cos[ω(tb − ta )/2]Gaω2 (t) =t ∈ [0, tb − ta ].(3.113)Outside the basic interval t ∈ [0, tb − ta ], the Green function is determined by itsantiperiodicity.
If, for example, t ∈ [−(tb − ta ), 0], one merely has to replace t by |t|.Note that the Matsubara sumsGpω2 ,e (0) =∞11 X,2h̄β m=−∞ ωm + ω 2Gpω,e (0) =∞11 X,f2h̄β m=−∞ ωm + ω 2(3.114)can also be calculated from the combinations of the simple Green functions (3.79)and (3.103):ii1 h p1 h p1 Gω,e (η) + Gpω,e (−η) =Gω,e (η) + Gpω,e (h̄β −η) =1 + nbω 1 + e−βω2ω2ω2ωH. Kleinert, PATH INTEGRALS2253.3 Green Functions of First-Order Differential Equationh̄ωβ1coth,(3.115)2ω2ii1 h a1 1 h aGω,e (η) + Gaω,e (−η) =Gω,e (η) − Gaω,e (h̄β −η) =1 − nfω 1 − e−βω2ω2ω2ω1h̄ωβ=tanh,(3.116)2ω2=where η is an infinitesimal positive number needed to specify on which side of thejump the Green functions Gp,aω,e (τ ) at τ = 0 have to be evaluated (see Fig.
3.2).3.3.2Time-Dependent FrequencyThe above results (3.89) and (3.108) for the periodic and antiperiodic Green functions of the first-order differential operator (i∂t − ω) can easily be found also forarbitrary time-dependent frequencies Ω(t), thus solving (3.75).
We shall look forthe retarded version which vanishes for t < t0 . This property is guaranteed by theansatz containing the Heaviside function (1.310):GΩ (t, t0 ) = Θ̄(t − t0 )g(t, t0 ).(3.117)Using the property (1.304) of the Heaviside function, that its time derivative yieldsthe δ-function, and normalizing g(t, t) to be equal to 1, we see that g(t, t0 ) mustsolve the homogenous differential equation[i∂t − Ω(t)] g(t, t0) = 0.The solution is00−ig(t, t ) = K(t )eThe condition g(t, t) = 1 fixes K(t) = eiRtcRtc(3.118)dt00 Ω(t00 )dt00 Ω(t00 )GΩ (t, t0 ) = Θ̄(t − t0 )e−i.(3.119), so that we obtainRtdt00 Ω(t00 )iRtt0.(3.120)The most general Green function is a sum of this and an arbitrary solution of thehomogeneous equation (3.118):hGΩ (t, t0 ) = Θ̄(t − t0 ) + C(t0 ) e−it0dt00 Ω(t00 ).(3.121)For a periodic frequency Ω(t) we impose periodic boundary conditions upon theGreen function, setting GΩ (ta , t0 ) = GΩ (tb , t0 ). This is ensured if for tb > t > t0 > ta :C(t0 )e−iR tat0dt00 Ω(t00 )hi= 1 + C(t0 ) e−iThis equation is solved by a t0 -independent C(t0 ):C = npΩ ≡R tb1ieR tbtadt00 Ω(t00 )t0.−1dt00 Ω(t00 ).(3.122)(3.123)2263 External Sources, Correlations, and Perturbation TheoryHence we obtain the periodic Green functionGpΩ (t, t0 )h0= Θ̄(t − t ) +npΩi−ieRtt0dt00 Ω(t00 ).(3.124)For antiperiodic boundary conditions we obtain the same equation with npΩ replacedby −naΩ where1naΩ ≡ R tb.(3.125)idt Ω(t)tae+1Note that a sign change in the time derivative of the first-order differential equation(3.75) to[−i∂t − Ω(t)] GΩ (t, t0 ) = iδ(t − t0 )(3.126)has the effect of interchanging in the time variable t and t0 of the Green functionEq.
(3.120).If the frequency Ω(t) is a matrix, all exponentials have to be replaced by timeordered exponentials [recall (1.253)]ieR tbtadt Ω(t)i→ T̂ eR tbtadt Ω(t).(3.127)As remarked in Subsection 2.15.3, this integral cannot, in general, be calculatedexplicitly. A simple formula is obtained only if the matrix Ω(t) varies only littlearound a fixed matrix Ω0 .For imaginary times τ = it we generalize the results (3.92) and (3.110) for the periodic and antiperiodic imaginary-time Green functions of the first-order differentialequation (3.76) to time-dependent periodic frequencies Ω(τ ). Here the particularGreen function (3.120) becomesGΩ (τ, τ 0 ) = Θ̄(τ − τ 0 )e−Rττ0dτ 00 Ω(τ 00 ),(3.128)and the periodic Green function (3.124):hiGΩ (τ, τ 0 ) = Θ̄(τ − τ 0 ) + nb e−whereRττ0dτ 00 Ω(τ 00 ),(3.129)1(3.130)nb ≡ R h̄β 00 00dτ Ω(τ )0e−1is the generalization of the Bose distribution function in Eq.
(3.93). For antiperiodicboundary conditions we obtain the same equation, except that the generalized Bosedistribution function is replaced by the negative of the generalized Fermi distributionfunction in Eq. (3.111):1.(3.131)nf ≡ R h̄β 00 00e 0 dτ Ω(τ ) + 1For the opposite sign of the time derivative in (3.128), the arguments τ and τ 0 areinterchanged.H.
Kleinert, PATH INTEGRALS2273.3 Green Functions of First-Order Differential EquationFrom the Green functions (3.124) or (3.128) we may find directly the trace ofthe logarithm of the operators [−i∂t + Ω(t)] or [∂τ + Ω(τ )]. At imaginary time, wemultiply Ω(τ ) with a strength parameter g, and using the formulaTr log [∂τ + gΩ(τ )] =Zg0dg 0 Gg0 Ω (τ, τ ).(3.132)Inserting on the right-hand side (3.129), we find for g = 1:(#)"1 Z h̄β 00dτ Ω(τ 00 )Tr log [∂τ + Ω(τ )] = log 2 sinh2 0Z h̄βR h̄β 001−dτ Ω(τ 00 )00000, (3.133)=dτ Ω(τ ) + log 1 − e2 0which reduces at low temperature to1 Z h̄β 00Tr log [∂τ + Ω(τ )] =dτ Ω(τ 00 ).2 0(3.134)The result is the same for the opposite sign of the time derivative and the traceof the logarithm is sensitive only to Θ̄(τ − τ 0 ) at τ = τ 0 , where it is equal to 1/2.As an exercise for dealing with distributions it is instructive to rederive this resultin the following perturbative way.
For a positive Ω(τ ), we introduce an infinitesimalpositive quantity η and decomposehTr log [±∂τ + Ω(τ )] = Tr log [±∂τ + η] + Tr log 1 + (±∂τ + η)−1 Ω(τ )hi(3.135)i= Tr log [±∂τ + η] + Tr log 1 + (±∂τ + η)−1 Ω(τ ) .∞Thefirst term Tr log [±∂τ + η] = Tr log [±∂τ + η] = −∞dω log ω vanishes sinceR∞dωlogω=0indimensionalregularizationbyVeltman’srule[see (2.500)]. Using−∞the Green functionsR−1[±∂τ + η]0(τ, τ ) =(Θ̄(τ − τ 0 ),Θ̄(τ 0 − τ ))(3.136)the second term can be expanded in a Taylor series(−1)n+1nn=1∞XZdτ1 · · · dτn Ω(τ1 )Θ̄(τ1 −τ2 )Ω(τ2 )Θ̄(τ2 −τ3 ) · · · Ω(τn )Θ̄(τn −τ1 ).
(3.137)For the lower sign of ±∂τ , the Heaviside functions have reversed arguments τ2 −τ1 , τ3 − τ2 , . . . , τ1 − τn . The integrals over a cyclic product of Heaviside functionsin (3.137) are zero since the arguments τ1 , . . . , τn are time-ordered which makes theargument of the last factor Θ̄(τn −τ1 ) [or Θ̄(τ1 −τn )] negative and thus Θ̄(τn −τ1 ) = 0[or Θ̄(τ1 − τn )]. Only the first term survives yieldingZdτ1 Ω(τ1 )Θ̄(τ1 − τ1 ) =12Zdτ Ω(τ ),(3.138)2283 External Sources, Correlations, and Perturbation Theorysuch that we re-obtain the result (3.134).This expansion (3.133) can easily be generalized to an arbitrary matrix Ω(τ ) or atime-dependent operator, Ĥ(τ ). Since Ĥ(τ ) and Ĥ(τ 0 ) do not necessarily commute,the generalization is1TrTr log[h̄∂τ + Ĥ(τ )] =2h̄"Z0h̄β#∞XR h̄β 00001Tr T̂ e−n 0 dτ Ĥ(τ )/h̄ , (3.139)dτ Ĥ(τ ) −n=1 nwhere T̂ is the time ordering operator (1.242). Each term in the sum contains apower of the time evolution operator (1.256).3.4Summing Spectral Representation of Green FunctionAfter these preparations we are ready to perform the spectral sum (3.70) for theGreen function of the differential equation of second order with Dirichlet boundaryconditions.
Setting t2 ≡ tb − t, t1 ≡ t0 − ta , we rewrite (3.70) asGω2 (t, t0 ) ===∞2 X(−1)n+1 (eiνn t2 − e−iνn t2 )(eiνn t1 − e−iνn t1 )tb − ta n=1 (2i)2νn2 − ω 2∞[(e−iνn (t2 +t1 ) − e−iνn (t2 −t1 ) ) + cc ]1 1 X(−1)n2 tb − ta n=1νn2 − ω 2∞Xe−iνn (t2 +t1 ) − e−iνn (t2 −t1 )1 1(−1)n.2 tb − ta n=−∞νn2 − ω 2(3.140)We now separate even and odd frequencies νn and write these as bosonic andffermionic Matsubara frequencies ωm = ν2m and ωm= ν2m+1 , respectively, recallingthe definitions (3.80) and (3.104).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
