Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 33
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Kleinert, PATH INTEGRALS1272.4 Gelfand-Yaglom FormulaThe homogeneous differential equation and the initial conditions are obviously satisfied by thepartial derivative matrix Dij (t) = ∂xi (t)/∂ ẋja , so that the explicit representations of Dij (t) interms of the general solution of the classical equations of motion −∂t2 δij − Ω2ij (t) xj (t) = 0become∂xi∂xiDren = det jb = −det aj .(2.240)∂ ẋa∂ ẋbA further couple of formulas for functional determinants can be found by constructing a solutionof the homogeneous differential equation (2.211) which passes through specific initial and finalpoints xa and xb at ta and tb , respectively:x(xb , xa ; t) =Db (t)Da (t)xa +xb .Db (ta )Da (tb )(2.241)The Gelfand-Yaglom functions Da (t) and Db (t) can therefore be obtained from the partial derivatives∂x(xb , xa ; t)Db (t)∂x(xb , xa ; t)Da (t)=,=.(2.242)Da (tb )∂xbDb (ta )∂xaAt the endpoints, Eqs.
(2.241) yieldḊb (ta )1xa +xb ,Db (ta )Da (tb )ẋa=ẋb= −(2.243)Ḋa (tb )1xa +xb ,Db (ta )Da (tb )(2.244)so that the fluctuation determinant Dren = Da (tb ) = Db (ta ) is given by the formulasDren =∂ ẋa∂xb−1=−∂ ẋb∂xa−1,(2.245)where ẋa and ẋb are functions of the independent variables xa and xb . The equality of these expressions with the previous ones in (2.236) and (2.238) is a direct consequence of the mathematicalidentity for partial derivatives !−1∂ ẋa ∂xb =.(2.246)∂ ẋa xa∂xb xaLet us emphasize that all functional determinants calculated in this Chapter apply to thefluctuation factor of paths with fixed endpoints. In mathematics, this property is referred toas Dirichlet boundary conditions.
In the context of quantum statistics, we shall also need suchdeterminants for fluctuations with periodic boundary conditions, for which the Gelfand-Yaglommethod must be modified. We shall see in Section 2.11 that this causes considerable complicationsin the lattice derivation, which will make it desirable to find a simpler derivation of both functionaldeterminants. This will be found in Section 3.27 in a continuum formulation.In general, the homogenous differential equation (2.211) with time-dependent frequency Ω(t)cannot be solved analytically. The equation has the same form as a Schrödinger equation fora point particle in one dimension moving in a one dimensional potential Ω2 (t), and there areonly a few classes of potentials for which the solutions are known in closed form.
Fortunately,however, the functional determinant will usually arise in the context of quadratic fluctuationsaround classical solutions in time-independent potentials (see in Section 4.3). If such a classicalsolution is known analytically, it will provide us automatically with a solution of the homogeneousdifferential equation (2.211).
Some important examples will be discussed in Sections 17.4 and17.11.1282.4.62 Path Integrals — Elementary Properties and Simple SolutionsGeneralization to D DimensionsThe above formulas have an obvious generalization to a D-dimensional version ofthe fluctuation action (2.194)A=ZtbtadtM[(δ ẋ)2 − δxT22(t)δx],(2.247)where 2 (t) is a D × D matrix with elements Ω2ij (t). The fluctuation factor (2.195)generalizes to1NF (tb , ta ) = qD2πh̄i(tb − ta )/M"detN (−2 ∇∇ − 2detN (−2 ∇∇)2)#−1/2.(2.248)The fluctuation determinant is found by Gelfand-Yaglom’s construction from a formulaDren = det Da (tb ) = det Db (ta ),(2.249)with the matrices Da (t) and Db (t) satisfying the classical equations of motion andinitial conditions corresponding to (2.221) and (2.222):[∂t2 +[∂t2 +22(t)]Da (t) = 0 ;(t)]Db (t) = 0 ;Da (ta ) = 0,Db (tb ) = 0,Ḋa (ta ) = 1,Ḋb (tb ) = −1,(2.250)(2.251)where 1 is the unit matrix in D dimensions.
We can then repeat all steps in the lastsection and find the D-dimensional generalization of formulas (2.245):Dren2.5∂ ẋi= det aj∂xb!−1∂ ẋi= − det jb∂xa!−1.(2.252)Harmonic Oscillator with Time-Dependent FrequencyThe results of the last section put us in a position to solve exactly the path integralof a harmonic oscillator with arbitrary time-dependent frequency Ω(t).
We shallfirst do this in coordinate space, later in momentum space.2.5.1Coordinate SpaceConsider the path integral(xb tb |xa ta ) =ZiDx expA[x] ,h̄(2.253)with the Lagrangian actionA[x] =ZtbtaiMh 2ẋ (t) − Ω2 (t)x2 (t) ,2(2.254)H. Kleinert, PATH INTEGRALS1292.5 Harmonic Oscillator with Time-Dependent Frequencywhich is harmonic with a time-dependent frequency. As in Eq. (2.14), the result canbe written as a product of a fluctuation factor and an exponential containing theclassical action:(xb tb |xa ta ) =ZDx eiA[x]/h̄ = FΩ (tb , ta )eiAcl /h̄ .(2.255)From the discussion in the last section we know that the fluctuation factor is, byanalogy with (2.164), and recalling (2.236),11qFΩ (tb , ta ) = q.2πih̄/M Da (tb )(2.256)The determinant Da (tb ) = Dren may be expressed in terms of partial derivativesaccording to formulas (2.236) and (2.245):1∂xb∂ ẋaFΩ (tb − ta ) = q2πih̄/M!−1/21=q2πih̄/M∂ ẋa∂xb!1/2,(2.257)where the first partial derivative is calculated from the function x(xa , ẋa ; t), thesecond from ẋ(xb , xa ; t).
Equivalently we may use (2.238) and the right-hand partof Eq. (2.245) to write1FΩ (tb − ta ) = q2πih̄/M∂x− a∂ ẋb!−1/21=q2πih̄/M∂ ẋ− b∂xa!1/2.(2.258)It remains to calculate the classical action Acl . This can be done in the sameway as in Eqs. (2.148) to (2.152). After a partial integration, we have as beforeAcl =M(x ẋ − xa ẋa ).2 b b(2.259)Exploiting the linear dependence of ẋb and ẋa on the endpoints xb and xa , we mayrewrite this asMAcl =2!∂ ẋ∂ ẋ∂ ẋ∂ ẋxb b xb − xa a xa + xb b xa − xa a xb .∂xb∂xa∂xa∂xb(2.260)Inserting the partial derivatives from (2.243) and (2.244) and using the equality ofDa (tb ) and Db (ta ), we obtain the classical actionAcl =iM h 2xb Ḋa (tb ) − x2a Ḋb (ta ) − 2xb xa .2Da (tb )(2.261)Note that there exists another simple formula for the fluctuation determinant Dren :Dren = Da (tb ) = Db (ta ) = −M∂2A∂xb ∂xa cl!−1.(2.262)1302 Path Integrals — Elementary Properties and Simple SolutionsFor the harmonic oscillator with time-independent frequency ω, the GelfandYaglom function Da (t) of Eq.
(2.228) has the property (2.224) due to time reversalinvariance, and (2.261) reproduces the known result (2.152).The expressions containing partial derivatives are easily extended to D dimensions: We simply have to replace the partial derivatives ∂xb /∂ ẋa , ∂ ẋb /∂ ẋa , . . .
bythe corresponding D × D matrices, and write the action as the associated quadraticform.The D-dimensional versions of the fluctuation factors (2.257) are1FΩ (tb − ta ) = qD2πih̄/M∂xidet jb∂ ẋa"#−1/21=qD2πih̄/M∂ ẋidet aj∂xb"#1/2.(2.263)All formulas for fluctuation factors hold initially only for sufficiently short timestb − ta . For larger times, they carry phase factors determined as before in (2.162).The fully defined expression may be written as1FΩ (tb − ta ) = qD2πih̄/Mdet−1/2∂xib ∂ ẋja −iνπ/2e1=qD2πih̄/Mdet1/2∂ ẋia ∂xjb e−iνπ/2 ,(2.264)where ν is the Maslov-Morse index.
In the one-dimensional case it counts the turning points of the trajectory, in the multidimensional case the number of zeros indeterminant det ∂xib /∂ ẋja along the trajectory, if the zero is caused by a reductionof the rank of the matrix ∂xib /∂ ẋja by one unit. If it is reduced by more than oneunit, ν increases accordingly. In this context, the number ν is also called the Morseindex of the trajectory.The zeros of the functional determinant are also called conjugate points.
Theyare generalizations of the turning points in one-dimensional systems. The surfacesin x-space, on which the determinant vanishes, are called caustics. The conjugatepoints are the places where the orbits touch the caustics.13Note that for infinitesimally short times, all fluctuation factors and classical actions coincide with those of a free particle. This is obvious for the time-independentharmonic oscillator, where the amplitude (2.170) reduces to that of a free particlein Eq.
(2.125) in the limit tb → ta . Since a time-dependent frequency is constantover an infinitesimal time, this same result holds also here. Expanding the solutionof the equations of motion for infinitesimally short times asxb ≈ (tb − ta )ẋa + xa ,xa ≈ −(tb − ta )ẋb + xb ,(2.265)we have immediately∂xib= δij (tb − ta ),∂ ẋja13∂xa= −δij (tb − ta ).∂ ẋjb(2.266)See M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, Berlin, 1990.H. Kleinert, PATH INTEGRALS1312.5 Harmonic Oscillator with Time-Dependent FrequencySimilarly, the expansionsẋb ≈ ẋa ≈lead to∂ ẋib1,j = −δijtb − ta∂xaxb − xatb − ta(2.267)1∂ ẋia.j = δijtb − ta∂xb(2.268)Inserting the expansions (2.266) or (2.267) into (2.259) (in D dimensions), the actionreduces approximately to the free-particle actionM (xb − xa )2.Acl ≈2 tb − ta2.5.2(2.269)Momentum SpaceLet us also find the time evolution amplitude in momentum space.
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