Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 51

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In this way we obtainf∞∞XX1 1e−iωm (t2 +t1 )e−iωm (t2 +t1 )1Gω2 (t, t0 ) =−2f 22  tb − ta m=−∞ ωmtb − ta m=−∞ ωm− ω2− ω2f∞∞XXe−iωm (t2 −t1 )e−iωm (t2 −t1 ) 11−+. (3.141)2f 2tb − ta m=−∞ ωmtb − ta m=−∞ ωm− ω2− ω2 Inserting on the right-hand side the periodic and antiperiodic Green functions (3.99)and (3.108), we obtain the decompositionGω2 (t, t0 ) =1 p[G (t + t1 ) − Gaω (t2 + t1 ) − Gpω (t2 − t1 ) + Gaω (t2 − t1 )] . (3.142)2 ω 2Using (3.99) and (3.113) we find thatGpω (t2 + t1 ) − Gpω (t2 − t1 ) =sin ω[t2 − (tb − ta )/2] sin ωt1,ω sin[ω(tb − ta )/2](3.143)H. Kleinert, PATH INTEGRALS2293.4 Summing Spectral Representation of Green FunctionGaω (t2 + t1 ) − Gaω (t2 − t1 ) = −such that (3.142) becomesGω2 (t, t0 ) =cos ω[t2 − (tb − ta )/2] sin ωt1,ω cos[ω(tb − ta )/2](3.144)1sin ωt2 sin ωt1 ,ω sin ω(tb − ta )(3.145)in agreement with the earlier result (3.36).An important limiting case ista → −∞,tb → ∞.(3.146)Then the boundary conditions become irrelevant and the Green function reduces toGω2 (t, t0 ) = −i −iω|t−t0 |e,2ω(3.147)which obviously satisfies the second-order differential equation(−∂t2 − ω 2)Gω2 (t, t0 ) = δ(t − t0 ).(3.148)The periodic and antiperiodic Green functions Gpω2 (t, t0 ) and Gaω2 (t, t0 ) at finitetb − ta in Eqs.

(3.99) and (3.113) are obtained from Gω2 (t, t0 ) by summing over allperiodic repetitions [compare (3.106)]Gpω2 (t, t0 ) =Gaω2 (t, t0 ) =∞Xn=−∞∞XG(t + n(tb − ta ), t0 ),(−1)n Gω2 (t + n(tb − ta ), t0 ).(3.149)n=−∞For completeness let us also sum the spectral representation with the normalizedwave functions [compare (3.98)–(3.69)]x0 (t) =s1,tb − taxn (t) =s2cos νn (t − ta ),tb − ta(3.150)which reads:0GNω 2 (t, t )∞X12cos νn (t − ta ) cos νn (t0 − ta )− 2+.=tb − ta2ωνn2 − ω 2n=1"#(3.151)It satisfies the Neumann boundary conditions0∂t GNω 2 (t, t )t=tb= 0,0∂t0 GNω 2 (t, t ) 0t =ta= 0.(3.152)The spectral representation (3.151) can be summed by a decomposition (3.140), ifthat the lowest line has a plus sign between the exponentials, and (3.142) becomes0GNω 2 (t, t ) =1 p[G (t + t1 ) − Gaω (t2 + t1 ) + Gpω (t2 − t1 ) − Gaω (t2 − t1 )] .

(3.153)2 ω 22303 External Sources, Correlations, and Perturbation TheoryUsing now (3.99) and (3.113) we find thatGpω (t2 + t1 ) + Gpω (t2 − t1 ) = −cos ω[t2 − (tb − ta )/2] cos ωt1,ω sin[ω(tb − ta )/2](3.154)Gaω (t2 + t1 ) + Gaω (t2 − t1 ) = −sin ω[t2 − (tb − ta )/2] cos ωt1,ω cos[ω(tb − ta )/2](3.155)and we obtain instead of (3.145):0GNω 2 (t, t ) = −1cos ω(tb − t> ) cos ω(t< − ta ),ω sin ω(tb − ta )(3.156)which has the small-ω expansion0GNω 2 (t, t ) 2≈ −ω ≈03.51tb −ta 11100202−|t−t|−(t+t)++t+t.

(3.157)3222(tb −ta )(tb −ta )ω 2Wronski Construction for Periodic and AntiperiodicGreen FunctionsThe Wronski construction in Subsection 3.2.1 of Green functions with timedependent frequency Ω(t) satisfying the differential equation (3.27)[−∂t2 − Ω2 (t)]GΩ2 (t, t0 ) = δ(t − t0 )(3.158)0can easily be carried over to the Green functions Gp,aΩ2 (t, t ) with periodic and antiperiodic boundary conditions.

As in Eq. (3.53) we decompose00000Gp,aΩ2 (t, t ) = Θ̄(t − t )∆(t, t ) + a(t )ξ(t) + b(t )η(t),(3.159)with independent solutions of the homogenous equations ξ(t) and η(t), and insertthis into (3.27), where δ p,a (t − t0 ) is the periodic version of the δ-functionδp,a0(t − t ) ≡∞Xn=−∞0δ(t − t − nh̄β)(1(−1)n),(3.160)and Ω(t) is assumed to be periodic or antiperiodic in tb − ta . This yields again for∆(t, t0 ) the homogeneous initial-value problem (3.46), (3.45),[−∂t2 − Ω2 (t)]∆(t, t0 ) = 0;∆(t, t) = 0, ∂t ∆(t, t0 )|t0 =t = −1.(3.161)The periodic boundary conditions lead to the system of equationsa(t)[ξ(tb ) ∓ ξ(ta )] + b(t)[η(tb ) ∓ η(ta )] = −∆(tb , t),˙ ) ∓ ξ(t˙ )] + b(t)[η̇(t ) ∓ η̇(t )] = −∂ ∆(t , t).a(t)[ξ(tbabatb(3.162)H. Kleinert, PATH INTEGRALS2313.6 Time Evolution Amplitude in Presence of Source TermDefining now the constant 2 × 2 -matricesξ(tb ) ∓ ξ(ta ) η(tb ) ∓ η(ta )˙ ) ∓ ξ(t˙ ) η̇(t ) ∓ η̇(t )ξ(tbabap,aΛ̄ (ta , tb ) =!,(3.163)the condition analogous to (3.58),with¯ p,a (t , t ) 6= 0,det Λ̄p,a (ta , tb ) = W ∆a b(3.164)¯ p,a (t , t ) = 2 ± ∂ ∆(t , t ) ± ∂ ∆(t , t ),∆a bta btb a(3.165)enables us to obtain the unique solution to Eqs.

(3.162). After some algebra usingthe identities (3.51) and (3.52), the expression (3.159) for Green functions withperiodic and antiperiodic boundary conditions can be cast into the form00Gp,aΩ2 (t, t ) = GΩ2 (t, t ) ∓[∆(t, ta ) ± ∆(tb , t)][∆(t0 , ta ) ± ∆(tb , t0 )],¯ p,a (t , t )∆(t , t )∆a ba b(3.166)where GΩ2 (t, t0 ) is the Green function (3.59) with Dirichlet boundary conditions. Asin (3.59) we may replace the functions on the right-hand side by the solutions Da (t)and Db (t) defined in Eqs.

(2.221) and (2.222) with the help of (3.60).The right-hand side of (3.166) is well-defined unless the operator K(t) = −∂t2 −Ω2 (t) has a zero-mode, say η(t), with periodic or antiperiodic boundary conditionsη(tb ) = ±η(ta ), η̇(tb ) = ±η̇(ta ), which would make the determinant of the 2 × 2-matrix Λ̄p,a vanish.3.6Time Evolution Amplitude in Presence of Source TermGiven the Green function Gω2 (t, t0 ), we can write down an explicit expression forthe time evolution amplitude.

The quadratic source contribution to the fluctuationfactor (3.21) is given explicitly bytbtb1dtdt0 Gω2 (t, t0 ) j(t)j(t0 )(3.167)2M tataZ tbZ t11dt dt0 sin ω(tb − t) sin ω(t0 − ta )j(t)j(t0 ).=−M ω sin ω(tb − ta ) tataZAj,fl = −ZAltogether, the path integral in the presence of an external source j(t) reads(xb tb |xa ta )jω=ZM 2i Z tb(ẋ − ω 2x2 ) + jxdtDx exph̄ ta2= e(i/h̄)Aj,cl Fω,j (tb , ta ),with a total classical actionhi1Mω(x2b + x2a ) cos ω(tb − ta )−2xb xaAj,cl =2 sin ω(tb − ta )Z tb1+dt[xa sin ω(tb − t) + xb sin ω(t − ta )]j(t),sin ω(tb − ta ) ta(3.168)(3.169)2323 External Sources, Correlations, and Perturbation Theoryand the fluctuation factor composed of (2.164) and a contribution from the currentterm eiAj,fl /h̄ :iAj,fl /h̄Fω,j (tb , ta ) = Fω (tb , ta )e(i× exp −h̄Mω sin ω(tb − ta )1=q2πih̄/MtbZtadtZttasωsin ω(tb − ta )000)dt sin ω(tb − t) sin ω(t − ta )j(t)j(t ) .

(3.170)This expression is easily generalized to arbitrary time-dependent frequencies.Using the two independent solutions Da (t) and Db (t) of the homogenous differentialequations (3.48), which were introduced in Eqs. (2.221) and (2.222), we find for theaction (3.169) theR general expression, composed of the harmonic action (2.261) andthe current term ttab dtxcl (t)j(t) with the classical solution (2.241):Aj,cliM h 21=xb Ḋa (tb )−x2a Ḋb (ta )−2xb xa +2Da (tb )Da (tb )Ztbtadt [xb Da (t)+xa Db (t)]j(t).(3.171)The fluctuation factor is composed of the expression (2.256) for the current-freeaction, and the generalization of (3.167) with the Green function (3.61):iAj,fl /h̄Fω,j (tb , ta ) = Fω (tb , ta )e×ZtbtadtZttah0(1iqexp −=q2h̄MDa (tb )2πih̄/M Da (tb )10000i0dt j(t) Θ̄(t − t )Db (t)Da (t ) + Θ̄(t − t)Da (t)Db (t ) j(t ) . (3.172)For applications to statistical mechanics which becomes possible after an analyticcontinuation to imaginary times, it is useful to write (3.169) and (3.170) in anotherform.

We introduce the Fourier transforms of the currenttb1dte−iω(t−ta ) j (t) ,A(ω) ≡Mω ta1 Z tbB(ω) ≡dte−iω(tb −t) j(t) = −e−iω(tb −ta ) A(−ω),Mω taZ(3.173)(3.174)and see that the classical source term in the exponent of (3.168) can be written asAj,cl = −inhioMωxb (eiω(tb −ta ) A − B) + xa (eiω(tb −ta ) B − A) .sin ω(tb − ta )(3.175)The source contribution to the quadratic fluctuations in Eq. (3.167), on the otherhand, can be rearranged to yieldAj,fl =i4MωZtbtaZdttbtb0dt0 e−iω|t−t | j(t)j(t0 )−hiMωeiω(tb −ta ) (A2 +B 2 )−2AB .2 sin ω(tb −ta )(3.176)H. Kleinert, PATH INTEGRALS2333.6 Time Evolution Amplitude in Presence of Source TermThis is seen as follows: We write the Green function between j(t), j(t0 ) in (3.168) ashi− sin ω(tb − t) sin ω(t0 − ta )Θ̄(t − t0 ) + sin ω(tb − t0 ) sin ω(t − ta )Θ̄(t0 − t) i1 h iω(tb −ta ) −iω(t−t0 )0=ee+ cc − eiω(tb +ta ) e−iω(t+t ) + cc Θ̄(t − t0 )4no+ t ↔ t0 .(3.177)Using Θ̄(t − t0 ) + Θ̄(t0 − t) = 1, this becomes1 n iω(tb +ta ) −iω(t0 +t)− ee+ cc400+eiω(tb −ta ) e−iω(t−t ) Θ̄(t − t0 ) + e−iω(t −t) Θ̄(t0 − t)h00+e−iω(tb −ta ) eiω(t−t ) (1 − Θ̄(t0 − t)) + eiω(t −t) (1 − Θ̄(t − t0 ))(3.178)io.A multiplication by j(t), j(t0 ) and an integration over the times t, t0 yield1h− eiω(tb −ta ) 4M 2 ω 2 (B 2 + A2 )4Z+ eiω(tb −ta ) − e−iω(tb −ta )tbta(3.179)dtZtbtbi0dt0 e−iω|t−t | j(t)j(t0 ) + 4M 2 ω 2 2AB ,thus leading to (3.176).If the source j(t) is time-independent, the integrals in the current terms of theexponential of (3.169) and (3.170) can be done, yielding the j-dependent exponentiiiAj,cl + Aj,fl =Aj =h̄h̄h̄(1[1 − cos ω(tb − ta )](xb + xa )jω sin ω(tb − ta )"# )cos ω(tb − ta ) − 1 21ω(tb − ta ) + 2j .+sin ω(tb − ta )2Mω 3(3.180)Substituting (1−cos α) by sin α tan(α/2), this yields the total source action becomes"#1ω(tb − ta ) 2ω(tb − ta )1Aj = tanj .

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