Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 73
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It is important to realize that thanks to thesubtraction in the Green function (3.882) caused by the z(0)-fluctuations, the limits ta → −∞ andtb → ∞ can be taken in (3.885) without any problems.3.26.2Application to Scattering AmplitudeWe can now apply this result to the path integral (2.723). With the abbreviation (3.884) we writeit asZpfpb pa =d2 b e−iqb/h̄2πih̄ Z ∞hZiiM(3.889)×D3 yz expyz [Ḡ00 (t, t0 )]−1 yz eiχb,p [yz ] − 1 ,dth̄ −∞2where [Ḡ00 (t, t0 )]−1 is the functional inverse of the subtracted Green function (3.888), and χb,p [yz ]the integral over the interaction potential V (x):Z1 ∞pχb,p [yz ] ≡ −dt V b +t + yz (t) .(3.890)h̄ −∞M3.26.3First Correction to Eikonal ApproximationThe first correction to the eikonal approximation (2.726) is obtained by expanding (3.890) to firstorder in yz (t).
This yieldsZp 1 ∞dt ∇V b + t yz (t).(3.891)χb,p [y] = χei−b ,ph̄ −∞MThe additional terms can be considered as an interactionZ1 ∞−dt yz (t) j(t),h̄ −∞(3.892)with the currentj(t) =p t .∇V b +M(3.893)H. Kleinert, PATH INTEGRALS3.26 Perturbative Approach to Scattering Amplitude341Using the generating functional (3.885), this is seen to yield an additional scattering phaseZ ∞Z ∞p 1p tt2 t> .dt∇Vb+(3.894)∆1 χei=dt∇Vb+112b,p2M h̄ −∞MM−∞To evaluate this we shall always change, as in (2.728), the time variables t1,2 to length variablesz1,2 ≡ p1,2 t/M along the direction of p.√For spherically symmetric potentials V (r) with r ≡ |x| = b2 + z 2 , we may express thederivatives parallel and orthogonal to the incoming particle momentum p as follows:∇k V = z V 0 /r,∇⊥ V = b V 0 /r.(3.895)Then (3.894) reduces to∆1 χeib ,pM2=2h̄p3Z∞−∞dz1Z∞V 0 (r1 ) V 0 (r2 ) 2b + z1 z2 z1 .r1r2dz2−∞(3.896)The part of the integrand before the bracket is obviously symmetric under z → −z and under theexchange z1 ↔ z2 .
For this reason we can rewriteZZM2 ∞V 0 (r2 ) 2V 0 (r1 ) ∞(3.897)b − z22 .∆1 χei=dzdzz211b,p3h̄p −∞r1r2−∞Now we use the relations (3.895) in the opposite direction aszV 0 /r = ∂z V,bV 0 /r = ∂b V,(3.898)and performing a partial integration in z1 to obtain21∆1 χeib ,pM2= − 3 (1 + b∂b )h̄pZ∞−∞dz V 2pb2 + z 2 .(3.899)Compared to the leading eikonal phase (2.729), this √is suppressed by a factor V (0)M/p2 .2Note that for the Coulomb potential where V ( b2 + z 2 ) ∝ 1/(b2 + z 2 ), the integral is proportional to 1/b which is annihilated by the factor 1 + b∂b .
Thus there is no first correction to theeikonal approximation (1.504).3.26.4Rayleigh-Schrödinger Expansion of Scattering AmplitudeIn Section 1.16 we have introduced the scattering amplitude as the limiting matrixelement [see (1.514)]hpb |Ŝ|pa i ≡limtb −ta →∞ei(Eb −Ea )tb /h̄ (pb 0|pa ta )e−iEa ta /h̄ .(3.900)A perturbation expansion for these quantities can be found via a Fourier transformation of the expansion (3.474).
We only have to set the oscillator frequency of theharmonic part of the action equal to zero, since the particles in a scattering processare free far away from the scattering center. Since scattering takes usually place inthree dimensions, all formulas will be written down in such a space.21This agrees with results from Schrödinger theory by S.J. Wallace, Ann. Phys.
78 , 190 (1973);S. Sarkar, Phys. Rev. D 21 , 3437 (1980). It differs from R. Rosenfelder’s result (see Footnote 37on p. 191) who derives a prefactor p cos(θ/2) instead of the incoming momentum p.3423 External Sources, Correlations, and Perturbation TheoryWe shall thus consider the perturbation expansion of the amplitudeZ(pb 0|pa ta ) =d3 xb d3 xa e−ipb xb (xb 0|xa ta )eipa xa ,(3.901)where (xb 0|xa ta ) is expanded as in (3.474). The immediate result looks as in theexpansion (3.497), if we replace the external oscillator wave functions ψn (xb ) andψa (xb ) by free-particle plane waves e−ipb xb and eipa xa :(pb 0|pa ta ) = (pb 0|pa ta )0ii12+ hpb |Aint |pa i0 −hp |A3 |p i + .
. . .2 hpb |Aint |pa i0 −h̄2!h̄3!h̄3 b int a 0Here2(pb 0|pa ta )0 = (2πh̄)3 δ (3) (pb − pa )eipb ta /2M h̄(3.902)(3.903)is the free-particle time evolution amplitude in momentum space [recall (2.133)] andthe matrix elements are defined byhpb | . . . |pa i0 ≡Z33−ipb xbd xb d xa eZ3iA0 /h̄D x...eeipa xa .(3.904)In contrast to (3.497) we have not divided out the free-particle amplitude (3.903) inthis definition since it is too singular. Let us calculate the successive terms in theexpansion (3.902). Firsthpb |Aint |pa i0 = −Z0tadt1Zd3 xb d3 xa d3 x1 e−ipb xb (xb 0|x1 t1 )0× V (x1 )(x1 t1 |xa ta )0 eipa xa .(3.905)SinceZZ2d3 xb e−ipb xb (xb tb |x1 t1 )0 = e−ipb x1 e−ipb (tb −t1 )/2M h̄ ,2d3 xa (x1 t1 |xa ta )0 e−ipb xb = e−ipa x1 eipa (t1 −ta )/2M h̄ ,(3.906)this becomeshpb |Aint |pa i0 = −Z0ta222dt1 ei(pb −pa )t1 /2M h̄ Vpb pa eipa ta /2M h̄ ,(3.907)Z(3.908)whereVpb pa ≡ hpb |V̂ |pa i =d3 xei(pb −pa )x/h̄ V (x) = Ṽ (pb − pa )[recall (1.492)].
Inserting a damping factor eηt1 into the time integral, and replacingp2 /2M by the corresponding energy E, we obtaini1hpb |Aint|pa i0 = −VeiEa ta .h̄Eb − Ea − iη pb pa(3.909)H. Kleinert, PATH INTEGRALS3.27 Functional Determinants from Green Functions343Inserting this together with (3.903) into the expansion (3.902), we find for the scattering amplitude (3.900) the first-order approximationhpb |Ŝ|pa i ≡limtb −ta →∞i(Eb −Ea )tb /h̄e"#1V(2πh̄) δ (pb − pa ) −Eb − Ea − iη pb pa(3.910)3 (3)corresponding precisely to the first-order approximation of the operator expression(1.517), the Born approximation.Continuing the evaluation of the expansion (3.902) we find that Vpb pa in (3.910)is replaced by the T -matrix [recall (1.475)]1d 3 pc= Vpb pa −V(3.911)3 Vpb pcEc − Ea − iη pc pa(2πh̄)Z11d 3 pc Z d 3 pdVpc pdV+ ...
.+33 Vpb pcEc − Ea − iηEd − Ea − iη pd pa(2πh̄)(2πh̄)ZTpb paThis amounts to an integral equationTpb pa = Vpb pa −Zd 3 pc1T,3 Vpb pcEc − Ea − iη pc pa(2πh̄)(3.912)which is recognized as the Lippmann-Schwinger equation (1.523) for the T -matrix.3.27Functional Determinants from Green FunctionsIn Subsection 3.2.1 we have seen that there exists a simple method, due to Wronski,for constructing Green functions of the differential equation (3.27),O(t)Gω2 (t, t0 ) ≡ [−∂t2 − Ω2 (t)]Gω2 (t, t0 ) = δ(t − t0 ),(3.913)with Dirichlet boundary conditions.
That method did not require any knowledgeof the spectrum and the eigenstates of the differential operator O(t), except for thecondition that zero-modes are absent. The question arises whether this method canbe used to find also functional determinants.22 The answer is positive, and we shallnow demonstrate that Gelfand and Yaglom’s initial-value problem (2.206), (2.207),(2.208) with the Wronski construction (2.218) for its solution represents the mostconcise formula for the functional determinant of the operator O(t).
Starting pointis the observation that a functional determinant of an operator O can be written asDet O = eTr log O ,(3.914)and that a Green function of a harmonic oscillator with an arbitrary time-dependentfrequency has the integralTr22Z01dg Ω2 (t)[−∂t2 − gΩ2 (t)]−1 δ(t − t0 )See the reference in Footnote 6 on p. 246.= −Tr {log[−∂t2 − Ω2 (t)]δ(t − t0 )}+Tr {log[−∂t2 ]δ(t − t0 )}.(3.915)3443 External Sources, Correlations, and Perturbation TheoryIf we therefore introduce a strength parameter g ∈ [0, 1] and an auxiliary Greenfunction Gg (t, t0 ) satisfying the differential equationOg (t)Gg (t, t0 ) ≡ [−∂t2 − gΩ2 (t)]Gg (t, t0 ) = δ(t − t0 ),(3.916)we can express the ratio of functional determinants Det O1 /Det O0 asDet (O0−1 O1 ) = e−R10dg Tr [Ω2 (t)Gg (t,t0 )].(3.917)Knowing of the existence of Gelfand-Yaglom’s elegant method for calculating functional determinants in Section 2.4, we now try to relate the right-hand side in (3.917)to the solution of the Gelfand-Yaglom’s equations (2.208), (2.206), and (2.207):Og (t)Dg (t) = 0; Dg (ta ) = 0, Ḋg (ta ) = 1.(3.918)By differentiating these equations with respect to the parameter g, we obtain forthe g-derivative Dg0 (t) ≡ ∂g Dg (t) the inhomogeneous initial-value problemOg (t)Dg0 (t) = Ω2 (t)Dg (t); Dg0 (ta ) = 0, Ḋg0 (ta ) = 0.(3.919)The unique solution of equations (3.918) can be expressed as in Eq.
(2.214) in termsof an arbitrary set of solutions ηg (t) and ξg (t) as followsξg (ta )ηg (t) − ξg (t)ηg (ta )= ∆g (t, ta ),WgDg (t) =(3.920)where Wg is the constant Wronski determinantWg = ξg (t)η̇g (t) − ηg (t)ξ˙g (t).(3.921)DetΛg= ∆g (tb , ta ),Wg(3.922)We may also writeDg (tb ) =where Λg is the constant 2 × 2 -matrixΛg =ξg (ta ) ηg (ta )ξg (tb ) ηg (tb )!.(3.923)With the help of the solution ∆g (t, t0 ) of the homogenous initial-value problem(3.918) we can easily construct a solution of the inhomogeneous initial-value problem(3.919) by superposition:Dg0 (t) =Zttadt0 Ω2 (t0 )∆g (t, t0 )∆g (t0 , ta ).(3.924)Comparison with (3.59) shows that at the final point t = tbDg0 (tb )= ∆g (tb , ta )Ztbtadt0 Ω2 (t0 )Gg (t0 , t0 ).(3.925)H.
Kleinert, PATH INTEGRALS3453.27 Functional Determinants from Green FunctionsTogether with (3.922), this implies the following equation for the integral over theGreen function which solves (3.913) with Dirichlet’s boundary conditions:det ΛgTr [Ω (t)Gg (t, t )] = −∂g logWg20!= −∂g log Dg (tb ).(3.926)Inserting this into (3.915), we find for the ratio of functional determinants the simpleformulaDet (O0−1 Og ) = C(tb , ta )Dg (tb ).(3.927)The constant of g-integration, which still depends in general on initial and finaltimes, is fixed by applying (3.927) to the trivial case g = 0, where O0 = −∂t2 andthe solution to the initial-value problem (3.918) isD0 (t) = t − ta .(3.928)At g = 0, the left-hand side of (3.927) is unity, determining C(tb , ta ) = (tb − ta )−1and the final result for g = 1:Det (O0−1 O1 )det Λ1=W1,DetΛ0D (t )= 1 b ,W0tb − ta(3.929)in agreement with the result of Section 2.7.The same method permits us to find the Green function Gω2 (τ, τ 0 ) governingquantum statistical harmonic fluctuations which satisfies the differential equation022p,a0p,aOg (τ )Gp,a(τ − τ 0 ),g (τ, τ ) ≡ [∂τ − gΩ (τ )]Gg (τ, τ ) = δ(3.930)with periodic and antiperiodic boundary conditions, frequency Ω(τ ), and δ-function.The imaginary-time analog of (3.915) for the ratio of functional determinants readsDet (O0−1 O1 ) = e−R10dgTr [Ω2 (τ )Gg (τ,τ 0 )].(3.931)0The boundary conditions satisfied by the Green function Gp,ag (τ, τ ) are0p,a0Gp,ag (τb , τ ) = ±Gg (τa , τ ),0p,a0Ġp,ag (τb , τ ) = ±Ġg (τa , τ ).(3.932)According to Eq.
(3.166), the Green functions are given by00Gp,ag (τ, τ ) = Gg (τ, τ ) ∓[∆g (τ, τa ) ± ∆g (τb , τ )][∆g (τ 0 , τa ) ± ∆g (τb , τ 0 )], (3.933)¯ p,a (τ , τ ) · ∆ (τ , τ )∆ga bg a bwhere [compare (3.49)]∆(τ, τ 0 ) =i1 hξ(τ )η(τ 0 ) − ξ(τ 0 )η(τ ) ,W(3.934)3463 External Sources, Correlations, and Perturbation Theory˙ )η(τ ), and [compare (3.165)]with the Wronski determinant W = ξ(τ )η̇(τ ) − ξ(τ¯ p,a (τ , τ ) = 2 ± ∂ ∆ (τ , τ ) ± ∂ ∆ (τ , τ ).∆ga bτ g a bτ g b a(3.935)The solution is unique provided that¯ p,adet Λ̄p,ag = Wg ∆g (τa , τb ) 6= 0.(3.936)The right-hand side is well-defined unless the operator Og (t) has a zero-mode withηg (tb ) = ±ηg (ta ), η̇g (tb ) = ±η̇g (ta ), which would make the determinant of the 2 × 2-matrix Λ̄p,ag vanish.We are now in a position to rederive the functional determinant of the operatorO(τ ) = ∂τ2 −Ω2 (τ ) with periodic or antiperiodic boundary conditions more elegantlythan in Section 2.11.
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