Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 56
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(3.327)−4Mωsinh(βh̄ω)ωh̄β0"#For the subtracted off-diagonal correlation functions (3.325) we findh̄ωB(2) 0(2) 0(2)Gω2 ,B,xy(τ, τ 0 ) = −Gω2 ,B,yx(τ, τ 0 ) = Gω2 ,B,xy +(τ − τ 0 ).2Miω+ ω−(3.328)For more details see the literature.73.12Correlation Functions in Canonical Path IntegralSometimes it is desirable to know the correlation functions of position and momentum variables(m,n)Gω2(τ1 , . .
. , τm ; τ1 , . . . , τn ) ≡ hx(τ1 )x(τ2 ) · · · x(τm )p(τ1 )p(τ2 ) · · · p(τn )i(3.329)ZZ1Dp(τ )x(τ1 )x(τ2 ) · · · x(τm )p(τ ) p(τ1 )p(τ2 ) · · · p(τn ) exp − Ae .≡ Z −1 Dx(τ )2πh̄These can be obtained from a direct extension of the generating functional (3.233) by anothersource k(τ ) coupled linearly to the momentum variable p(τ ):ZZ[j, k] = Dx(τ ) e−Ae [j,k]/h̄ .(3.330)7M. Bachmann, H.
Kleinert, and A. Pelster, Phys. Rev. A 62 , 52509 (2000) (quant-ph/0005074);Phys. Lett. A 279 , 23 (2001) (quant-ph/0005100).H. Kleinert, PATH INTEGRALS2553.12 Correlation Functions in Canonical Path Integral3.12.1Harmonic Correlation FunctionsFor the harmonic oscillator, the generating functional (3.330) is denoted by Zω [j, k] and its Euclidean action readsZ h̄β 1 2 M 2 2Ae [j, k] =dτ −ip(τ )ẋ(τ ) +p +ω x − j(τ )x(τ ) − k(τ )p(τ ) ,(3.331)2M20the partition function is denoted by Zω [j, k]. Introducing the vectors in phase space V(τ ) =(p(τ ), x(τ )) and J(τ ) = (j(τ ), k(τ )), this can be written in matrix form asZ h̄β 1 TAe [J] =dτV Dω2 ,e V − VT J ,(3.332)20where Dω2 ,e (τ, τ 0 ) is the functional matrix0Dω2 ,e (τ, τ ) ≡M ω2i∂τ−i∂τM −1!δ(τ − τ 0 ),τ − τ 0 ∈ [0, h̄β].(3.333)Its functional inverse is the Euclidean Green function,0Gpω2 ,e (τ, τ 0 ) = Gpω2 ,e (τ −τ 0 ) = D−1ω 2 ,e (τ, τ )!M −1 −i∂τ=(−∂τ2 + ω 2 )−1 δ(τ − τ 0 ),i∂τM ω2(3.334)with the periodic boundary condition.
After performing a quadratic completion as in (3.238) byshifting the path:V → V0 = V + Gpω2 ,e J,(3.335)the Euclidean action takes the formZZ h̄βZ h̄β1 h̄β1 0T0dτdτ 0 JT (τ 0 )Gpω2 ,e (τ − τ 0 )J(τ 0 ).Ae [J] =dτ V Dω2 ,e V −22 000(3.336)The fluctuations over the periodic paths V0 (τ ) can now be integrated out and yield for J(τ ) ≡ 0the oscillator partition function−1/2Zω = Det Dω2 ,e .(3.337)A Fourier decomposition into Matsubara frequenciesDω2 ,e (τ, τ 0 )=∞01 XDp 2 (ωm )e−iωm (τ −τ ) ,h̄β m=−∞ ω ,e(3.338)has the componentsDpω2 ,e (ωm )with the determinants=M −1ωm−ωmM ω2!,(3.339)2det Dpω2 ,e (ωm ) = ωm+ ω2,(3.340)and the inversesGpe (ωm )=[Dpω2 ,e (ωm )]−1=M ω2ωm−ωmM−1!2ωm1.+ ω2(3.341)2563 External Sources, Correlations, and Perturbation TheoryThe product of determinants (3.340) for all ωm required in the functional determinant of Eq. (3.337)is calculated with the rules of analytic regularization in Section 2.15, and yields the same partitionfunction as in (3.241), and thus the same partition function (3.242):11p=.222 sinh(βh̄ω/2)ωm + ωm=1Zω = Q ∞(3.342)We therefore obtain for arbitrary sources J(τ ) = (j(τ ), k(τ )) 6= 0 the generating functional1[J],(3.343)Z[J] = Zω exp − AJh̄ ewith the source termAJe [J] = −12Z0h̄βdτZ0h̄βdτ 0 JT (τ )Gpω2 ,e (τ, τ 0 )J(τ 0 ).(3.344)The Green function Gpω2 ,e (τ, τ 0 ) follows immediately from Eq.
(3.334) and (3.237):0Gpω2 ,e (τ, τ 0 ) = Gpω2 ,e (τ −τ 0 ) = D−1ω 2 ,e (τ, τ ) =M −1i∂τ−i∂τM ω2!Gpω2 ,e (τ − τ 0 ),(3.345)where Gpω2 ,e (τ − τ 0 ) is the simple periodic Green function (3.248). From the functional derivativesof (3.343) with respect to j(τ )/h̄ and k(τ )/h̄ as in (3.299), we now find the correlation functionsh̄ pG 2 (τ − τ 0 ),M ω ,e(2)Gω2 ,e,xp (τ, τ 0 ) ≡ hx(τ )p(τ 0 )i = −ih̄Ġpω2 ,e (τ − τ 0 ),(2)Gω2 ,e,xx(τ, τ 0 ) ≡ hx(τ )x(τ 0 )i =(2)Gω2 ,e,px (τ, τ 0 )(2)Gω2 ,e,pp (τ, τ 0 )(3.346)(3.347)≡ hp(τ )x(τ 0 )i = ih̄Ġpω2 ,e (τ − τ 0 ),(3.348)≡ hp(τ )p(τ 0 )i =(3.349)h̄M ω 2 Gpω2 ,e (τ− τ 0 ).The correlation function hx(τ )x(τ 0 )i is the same as in the pure configuration space formulation(3.301). The mixed correlation function hp(τ )x(τ 0 )i is understood immediately by rewriting thecurrent-free part of the action (3.331) asZ h̄β M 212(3.350)Ae [0, 0] =dτ(p − iM ẋ) +ẋ + ω 2 x2 ,2M20which shows that p(τ ) fluctuates harmonically around the classical momentum for imaginary timeiM ẋ(τ ).
It is therefore not surprising that the correlation function hp(τ )x(τ 0 )i comes out to bethe same as that of iM hẋ(τ )x(τ 0 )i. Such an analogy is no longer true for the correlation functionhp(τ )p(τ 0 )i. In fact, the correlation function hẋ(τ )ẋ(τ 0 )i is equal tohẋ(τ )ẋ(τ 0 )i = −h̄M ∂τ2 Gpω2 ,e (τ − τ 0 ).(3.351)Comparison with (3.349) reveals the relationhp(τ )p(τ 0 )ih̄−∂τ2 + ω 2 Gpω2 ,e (τ − τ 0 )Mh̄0= hẋ(τ )ẋ(τ )i +δ(τ − τ 0 ).M= hẋ(τ )ẋ(τ 0 )i +(3.352)The additional δ-function on the right-hand side is the consequence of the fact that p(τ ) is notequal to iM ẋ, but fluctuates around it harmonically.H.
Kleinert, PATH INTEGRALS2573.12 Correlation Functions in Canonical Path IntegralFor the canonical path integral of a particle in a uniform magnetic field solved in Section 2.18,there are analogous relations. Here we write the canonical action (2.619) with a vector potential(2.616) in the Euclidean form asAe [p, x] =Zh̄βdτ0i2 M1 he2 2p − B × x − iM ẋ +ω x ,2Mc2(3.353)showing that p(τ ) fluctuates harmonically around the classical momentum pcl (τ ) = (e/c)B × x −iM ẋ. For a magnetic field pointing in the z-direction we obtain, with the frequency ωB = ωL /2of Eq.
(2.624), the following relations between the correlation functions involving momenta andthose involving only coordinates given in (3.322), (3.324), (3.324):(2)(2)(2)(3.354)Gω2 ,B,xpy (τ, τ 0 ) ≡ hx(τ )py (τ 0 )i = iM ∂τ 0 Gω2 ,B,xy (τ, τ 0 ) + M ωB Gω2 ,B,xx (τ, τ 0 ),(2)(2)(2)(3.355)Gω2 ,B,zpz (τ, τ 0 ) ≡ hz(τ )pz (τ 0 )i = iM ∂τ 0 Gω2 ,B,zz (τ, τ 0 ),(2)(2)(3.356)Gω2 ,B,xpx (τ, τ 0 ) ≡ hx(τ )px (τ 0 )i = iM ∂τ 0 Gω2 ,B,xx (τ, τ 0 ) − M ωB Gω2 ,B,xy (τ, τ 0 ),(2)Gω2 ,B,px px (τ, τ 0 )(2)Gω2 ,B,px py (τ, τ 0 )≡(2)(2)hpx (τ )px (τ 0 )i = −M 2 ∂τ ∂τ 0 Gω2 ,B,xx (τ, τ 0 )−2iM 2 ωB ∂τ Gω2 ,B,xy (τ, τ 0 )(2)2+ M 2 ωBGω2 ,B,xx (τ, τ 0 ) + h̄M δ(τ − τ 0 ),0≡ hpx (τ )py (τ )i = −M2(2)∂τ ∂τ 0 Gω2 ,B,xy (τ, τ 0 )+ iM2(2)2+ M 2 ωBGω2 ,B,xy (τ, τ 0 ),(2)(2)(3.357)(2)∂τ Gω2 ,B,xx (τ, τ 0 )Gω2 ,B,pz pz (τ, τ 0 ) ≡ hpz (τ )pz (τ 0 )i = −M 2 ∂τ ∂τ 0 Gω2 ,B,zz (τ, τ 0 ) + h̄M δ(τ − τ 0 ).(3.358)(3.359)Only diagonal correlations between momenta contain the extra δ-function on the right-hand side(2)(2)according to the rule (3.352).
Note that ∂τ ∂τ 0 Gω2 ,B,ab (τ, τ 0 ) = −∂τ2 Gω2 ,B,ab (τ, τ 0 ). Each correlation function is, of course, invariant under time translations, depending only on the time differenceτ − τ 0.The correlation functions hx(τ )x(τ 0 )i and hx(τ )y(τ 0 )i are the same as before in Eqs. (3.324)and (3.325).3.12.2Relations between Various AmplitudesA slight generalization of the generating functional (3.330) contains paths with fixed endpointsrather than all periodic paths. If the endpoints are held fixed in configuration space, one defines(xb h̄β|xa 0)[j, k] =Zx(h̄β)=xbDxx(0)=xaDp1exp − Ae [j, k] .2πh̄h̄(3.360)If the endpoints are held fixed in momentum space, one defines(pb h̄β|pa 0)[j, k] =Zp(h̄β)=pbDp1exp − Ae [j, k] .2πh̄h̄(3.361)dxb e−i(pb xb −pa xa )/h̄ (xb h̄β|xa 0)[j, k] .(3.362)Dxp(0)=paThe two are related by a Fourier transformation(pb h̄β|pa 0)[j, k] =Z+∞−∞dxaZ+∞−∞We now observe that in the canonical path integral, the amplitudes (3.360) and (3.361) withfixed endpoints can be reduced to those with vanishing endpoints with modified sources.
Themodification consists in shifting the current k(τ ) in the action by the source term ixb δ(τb − τ ) −2583 External Sources, Correlations, and Perturbation Theoryixa δ(τ − τa ) and observe that this produces in (3.361) an overall phase factor in the limit τb ↑ h̄βand τa ↓ 0:lim lim (pb h̄β|pa 0)[j(τ ), k(τ ) + ixb δ(τb − τ ) − ixa δ(τ − τa )]i= exp(pb xb − pa xa ) (pb h̄β|pa 0)[j(τ ), k(τ )] .h̄τb ↑h̄β τa ↓0(3.363)By inserting (3.363) into the inverse of the Fourier transformation (3.362),ZZ +∞dpa +∞ dpb i(pb xb −pa xa )/h̄(xb h̄β|xa 0)[j, k] =e(pb h̄β|pa 0)[j, k],−∞ 2πh̄ −∞ 2πh̄(3.364)we obtain(xb h̄β|xa 0)[j, k] = lim lim (0 h̄β|0 0)[j(τ ), k(τ ) + ixb δ(τb −τ ) − ixa δ(τ −τa )] .τb ↑h̄β τa ↓0(3.365)In this way, the fixed-endpoint path integral (3.360) can be reduced to a path integral with vanishing endpoints but additional δ-terms in the current k(τ ) coupled to the momentum p(τ ).There is also a simple relation between path integrals with fixed equal endpoints and periodicpath integrals.
The measures of integration are related byZx(h̄β)=xx(0)=xDxDp=2πh̄IDxDpδ(x(0) − x) .2πh̄(3.366)Using the Fourier decomposition of the delta function, we rewrite (3.366) asZx(h̄β)=xx(0)=xDxDp= limτa0 ↓02πh̄Z+∞dpa ipa x/h̄e2πh̄IDxDp −ie2πh̄R h̄βdτ pa δ(τ −τa0 )x(τ )/h̄.(3.367)×Z [j(τ ) − ipa δ(τ − τa0 ), k(τ ) + ixb δ(τb − τ ) − ixa δ(τ − τa )] ,(3.368)−∞0Inserting now (3.367) into (3.365) leads to the announced desired relationZ +∞dpa(xb h̄β|xa 0)[k, j] = lim lim lim0τb ↑h̄β τa ↓0 τa ↓0 −∞ 2πh̄where Z[j, k] is the thermodynamic partition function (3.330) summing all periodic paths.
Whenusing (3.368) we must be careful in evaluating the three limits. The limit τa0 ↓ 0 has to be evaluatedprior to the other limits τb ↑ h̄β and τa ↓ 0.3.12.3Harmonic Generating FunctionalsHere we write down explicitly the harmonic generating functionals with the above shifted sourceterms:k̃(τ ) = k(τ ) + ixb δ(τb − τ ) − ixa δ(τ − τa ) ,j̃(τ ) = j(τ ) − ipδ(τ − τa0 ),(3.369)leading to the factorized generating functionalZω [k̃, j̃] = Zω(0) [0, 0]Zω(1) [k, j]Zωp [k, j] .(3.370)The respective terms on the right-hand side of (3.370) read in detail1 2 p 0 0p0p0(0)Zω [0, 0] = Zω exp2 −p Gxx (τa , τa ) − 2p xa Gxp (τa , τa ) + xb Gxp (τa , τb )2h̄H.
Kleinert, PATH INTEGRALS2593.12 Correlation Functions in Canonical Path IntegralZω(1) [k, j] =Zωp [k, j] =−x2a Gppp (τa , τa ) − x2b Gppp (τb , τb ) + 2xa xb Gppp (τa , τb ) ,(3.371) Z h̄β1expdτ j(τ )[−ipGpxx (τ, τa0 ) + ixb Gpxp (τ, τb ) − ixa Gpxp (τ, τa )]2h̄ 0(3.372)+k(τ )[−ipGpxp (τ, τa0 ) + ixb Gppp (τ, τb ) − ixa Gppp (τ, τa )] ,(Z h̄βZ h̄β1dτexpdτ2 [(j(τ1 ), k(τ2 ))12h̄2 00 pGxx (τ1 , τ1 ) Gpxp (τ1 , τ2 )j(τ2 ),(3.373)×k(τ2 )Gppx (τ1 , τ2 ) Gppp (τ1 , τ2 )where Zω is given by (3.342) and Gpxp (τ1 , τ2 ) etc. are the periodic Euclidean Green functions(2)Gω2 ,e,ab (τ1 , τ2 ) defined in Eqs.
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