Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 32
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To calculate the classical action on thelattice, we insert (2.190) into (2.188). After some trigonometry, and replacing 2 ω 2 by 4 sin2 (ω̃/2),the action resembles closely the continuum expression (2.152):ANcl = 2sin ω̃M(xb + x2a ) cos ω̃(tb − ta ) − 2xb xa .2 sin ω̃(tb − ta )(2.191)The total time evolution amplitude on the sliced time axis isN(xb tb |xa ta ) = eiAcl /h̄ FωN (tb − ta ),(2.192)with sliced action (2.191) and the sliced fluctuation factor (2.162).H.
Kleinert, PATH INTEGRALS1212.4 Gelfand-Yaglom Formula2.4Gelfand-Yaglom FormulaIn many applications one encounters a slight generalization of the oscillator fluctuation problem: The action is harmonic but contains a time-dependent frequencyΩ2 (t) instead of the constant oscillator frequency ω 2 . The associated fluctuationfactor isF (tb , ta ) =iDδx(t) expA ,h̄Z(2.193)with the actionA=ZtbtadtM[(δ ẋ)2 − Ω2 (t)(δx)2 ].2(2.194)Since Ω(t) may not be translationally invariant in time, the fluctuation factor depends now in general on both the initial and final times.
The ratio formula (2.179)holds also in this more general case, i.e.,detN (−2 ∇∇ − 2 Ω2 )F (tb , ta ) = qdetN (−2 ∇∇)2πh̄i(tb − ta )/M"1N#−1/2.(2.195)Here Ω2 (t) denotes the diagonal matrixΩ2 (t) = with the matrix elements Ω2n = Ω2 (tn ).2.4.1Ω2N...Ω21,(2.196)Recursive Calculation of Fluctuation DeterminantIn general, the full set of eigenvalues of the matrix −∇∇ − Ω2 (t) is quite difficultto find, even in the continuum limit.
It is, however, possible to derive a powerfuldifference equation for the fluctuation determinant which can often be used to find itsvalue without knowing all eigenvalues. The method is due to Gelfand and Yaglom.11Let us denote the determinant of the N × N fluctuation matrix by DN , i.e.,DN ≡ detN −2 ∇∇ − 2 Ω22 − 2 Ω2N−10 ... 0002 200−12 − ΩN −1 −1 . . . 0.....≡.2 2000 . . . −1 2 − Ω2−1000 ... 0−12 − 2 Ω2111I.M. Gelfand and A.M. Yaglom, J. Math. Phys. 1 , 48 (1960).(2.197).1222 Path Integrals — Elementary Properties and Simple SolutionsBy expanding this along the first column, we obtain the recursion relationDN = (2 − 2 Ω2N )DN −1 − DN −2 ,(2.198)which may be rewritten as2"1DN − DN −1 DN −1 − DN −2−!+Ω2N DN −1#= 0.(2.199)Since the equation is valid for all N, it implies the determinant DN to satisfy thedifference equation(∇∇ + Ω2N +1 )DN = 0.(2.200)In this notation, the operator −∇∇ is understood to act on the dimensional labelN of the determinant.
The determinant DN may be viewed as the discrete values ofa function of D(t) evaluated on the sliced time axis. Equation (2.200) is called theGelfand-Yaglom formula. Thus the determinant as a function of N is the solutionof the classical difference equation of motion and the desired result for a given N isobtained from the final value DN = D(tN +1 ). The initial conditions areD1 = (2 − 2 Ω21 ),D2 = (2 − 2 Ω21 )(2 − 2 Ω22 ) − 1.2.4.2(2.201)ExamplesAs an illustration of the power of the Gelfand-Yaglom formula, consider the known case of aconstant Ω2 (t) ≡ ω 2 where the Gelfand-Yaglom formula reads(∇∇ + ω 2 )DN = 0.(2.202)This is solved by a linear combination of sin(N ω̃) and cos(N ω̃), where ω̃ is given by (2.156).
Thesolution satisfying the correct boundary condition is obviouslyDN =sin(N + 1)ω̃.sin ω̃(2.203)Indeed, the two lowest elements areD1= 2 cos ω̃,D2= 4 cos2 ω̃ − 1,(2.204)which are the same as (2.201), since 2 Ω2 ≡ 2 ω 2 =2(1 − cos ω̃).The Gelfand-Yaglom formula becomes especially easy to handle in the continuum limit → 0.Then, by considering the renormalized functionDren (tN ) = DN ,(2.205)the initial conditions D1 = 2 and D2 = 3 can be re-expressed as(D)1 = Dren (ta ) = 0,(2.206)→0D2 − D1= (∇D)1 −−−→ Ḋren (ta ) = 1.(2.207)H.
Kleinert, PATH INTEGRALS1232.4 Gelfand-Yaglom FormulaFigure 2.2 Solution of equation of motion with zero initial value and unit initial slope.Its value at the final time is equal to 1/ times the fluctuation determinant.The difference equation for DN turns into the differential equation for Dren (t):[∂t2 + Ω2 (t)]Dren (t) = 0.(2.208)The situation is pictured in Fig. 2.2. The determinant DN is 1/ times the value of the functionDren (t) at tb . This value is found by solving the differential equation starting from ta with zerovalue and unit slope.As an example, consider once more the harmonic oscillator with a fixed frequency ω. Theequation of motion in the continuum limit is solved byDren (t) =1sin ω(t − ta ),ω(2.209)which satisfies the initial conditions (2.207).
Thus we find the fluctuation determinant to become,for small ,→0−−→det(−2 ∇∇ − 2 ω 2 ) −1 sin ω(tb − ta ),ω(2.210)in agreement with the earlier result (2.203). For the free particle, the solution is Dren (t) = t − taand we obtain directly the determinant detN (−2 ∇∇) = (tb − ta )/.For time-dependent frequencies Ω(t), an analytic solution of the Gelfand-Yaglom initial-valueproblem (2.206), (2.207), and (2.208) can be found only for special classes of functions Ω(t). Infact, (2.208) has the form of a Schrödinger equation of a point particle in a potential Ω2 (t), andthe classes of potentials for which the Schrödinger equation can be solved are well-known.2.4.3Calculation on Unsliced Time AxisIn general, the most explicit way of expressing the solution is by linearly combiningDren =DN from any two independent solutions ξ(t) and η(t) of the homogeneousdifferential equation[∂t2 + Ω2 (t)]x(t) = 0.(2.211)The solution of (2.208) is found from a linear combinationDren (t) = αξ(t) + βη(t).(2.212)The coefficients are determined from the initial condition (2.207), which implyαξ(ta ) + βη(ta ) = 0,˙ ) + β η̇(t ) = 1,αξ(taa(2.213)1242 Path Integrals — Elementary Properties and Simple Solutionsand thusξ(t)η(ta ) − ξ(ta )η(t).˙ )η(t ) − ξ(t )η̇(t )ξ(taaaa(2.214)˙W ≡ ξ(t) ∂ t η(t) ≡ ξ(t)η̇(t) − ξ(t)η(t)(2.215)Dren (t) =The denominator is recognized as the time-independent Wronski determinant of thetwo solutions↔at the initial point ta .
The right-hand side is independent of t.The Wronskian is an important quantity in the theory of second-order differentialequations. It is defined for all equations of the Sturm-Liouville type"#ddy(t)a(t)+ b(t)y(t) = 0,dtdt(2.216)for which it is proportional to 1/a(t). The Wronskian serves to construct the Greenfunction for all such equations.12In terms of the Wronskian, Eq.
(2.214) has the general formDren (t) = −1[ξ(t)η(ta ) − ξ(ta )η(t)] .W(2.217)Inserting t = tb gives the desired determinantDren = −1[ξ(tb )η(ta ) − ξ(ta )η(tb )] .W(2.218)Note that the same functional determinant can be found from by evaluating thefunctionD̃ren (t) = −1[ξ(tb )η(t) − ξ(t)η(tb)]W(2.219)at ta . This also satisfies the homogenous differential equation (2.208), but with theinitial conditionsD̃ren (tb ) = 0,˙ (t ) = −1.D̃ren b(2.220)It will be useful to emphasize at which ends the Gelfand-Yaglom boundary conditions are satisfied by denoting Dren (t) and D̃ren (t) by Da (t) and Db (t), respectively,summarizing their symmetric properties as[∂t2 + Ω2 (t)]Da (t) = 0 ;[∂t2 + Ω2 (t)]Db (t) = 0 ;Da (ta ) = 0,Db (tb ) = 0,Ḋa (ta ) = 1,Ḋb (tb ) = −1,(2.221)(2.222)12For its typical use in classical electrodynamics, see J.D.
Jackson, Classical Electrodynamics,John Wiley & Sons, New York, 1975, Section 3.11.H. Kleinert, PATH INTEGRALS1252.4 Gelfand-Yaglom Formulawith the determinant being obtained from either function asDren = Da (tb ) = Db (ta ).(2.223)In contrast to this we see from the explicit equations (2.217) and (2.219) that thetime derivatives of two functions at opposite endpoints are in general not related.Only for frequencies Ω(t) with time reversal invariance, one hasḊa (tb ) = −Ḋb (ta ),for Ω(t) = Ω(−t).(2.224)For arbitrary Ω(t), one can derive a relationḊa (tb ) + Ḋb (ta ) = −2Ztbtadt Ω(t)Ω̇(t)Da (t)Db (t).(2.225)As an application of these formulas, consider once more the linear oscillator, forwhich two independent solutions areξ(t) = cos ωt,η(t) = sin ωt.(2.226)HenceW = ω,(2.227)and the fluctuation determinant becomes11Dren = − (cos ωtb sin ωta − cos ωta sin ωtb ) = sin ω(tb − ta ).ωω2.4.4(2.228)D’Alembert’s ConstructionIt is important to realize that the construction of the solutions of Eqs.
(2.221) and(2.222) requires only the knowledge of one solution of the homogenous differentialequation (2.211), say ξ(t). A second linearly independent solution η(t) can alwaysbe found with the help of a formula due to d’Alembert,η(t) = w ξ(t)Ztdt0,ξ 2 (t0 )(2.229)where w is some constant. Differentiation yieldsη̇ =˙ξηw+ ,ξξη̈ =¨ξη.ξ(2.230)The second equation shows that with ξ(t), also η(t) is a solution of the homogenous differential equation (2.211). From first equation we find that the Wronskideterminant of the two functions is equal to w:˙W = ξ(t)η̇(t) − ξ(t)η(t)= w.(2.231)1262 Path Integrals — Elementary Properties and Simple SolutionsInserting the solution (2.229) into the formulas (2.217) and (2.219), we obtainexplicit expressions for the Gelfand-Yaglom functions in terms of one arbitrary solution of the homogenous differential equation (2.211):Dren (t) = Da (t) = ξ(t)ξ(ta )Zttadt0,ξ 2 (t0 )D̃ren (t) = Db (t) = ξ(tb )ξ(t)Zttbdt0.
(2.232)ξ 2 (t0 )The desired functional determinant isDren = ξ(tb )ξ(ta )2.4.5Ztbtadt0.ξ 2 (t0 )(2.233)Another Simple FormulaThere exists yet another useful formula for the functional determinant. For this we solve thehomogenous differential equation (2.211) for an arbitrary initial position xa and initial velocityẋa at the time ta . The result may be expressed as the following linear combination of Da (t) andDb (t):x(xa , ẋa ; t) =i1 hDb (t) − Da (t)Ḋb (ta ) xa + Da (t)ẋa .Db (ta )(2.234)We then see that the Gelfand-Yaglom function Dren (t) = Da (t) can be obtained from the partialderivative∂x(xa , ẋa ; t).(2.235)Dren (t) =∂ ẋaThis function obviously satisfies the Gelfand-Yaglom initial conditions Dren (ta ) = 0 and Ḋren (ta ) =1 of (2.206) and (2.207), which are a direct consequence of the fact that xa and ẋa are independentvariables in the function x(xa , ẋa ; t), for which ∂xa /∂ ẋa = 0 and ∂ ẋa /∂ ẋa = 1.The fluctuation determinant Dren = Da (tb ) is then given byDren =∂xb,∂ ẋa(2.236)where xb abbreviates the function x(xa , ẋa ; tb ).
It is now obvious that the analogous equations(2.222) are satisfied by the partial derivative Db (t) = −∂x(t)/∂ ẋb , where x(t) is expressed in termsof the final position xb and velocity ẋb as x(t) = x(xb , ẋb ; t)x(xb , ẋb ; t) =i1 hDa (t) + Db (t)Ḋa (tb ) xb − Db (t)ẋb ,Da (tb )(2.237)so that we obtain the alternative formulaDren = −∂xa.∂ ẋb(2.238)These results can immediately be generalized to functional determinants of differential operators of the form −∂t2 δij −Ω2ij (t) where the time-dependent frequency is a D ×D-dimensional matrixΩ2ij (t), (i, j = 1, .
. . , D). Then the associated Gelfand-Yaglom function Da (t) becomes a matrixDij (t) satisfying the initial conditions Dij (ta ) = 0, Ḋij (tb ) = δij , and the desired functionaldeterminant Dren is equal to the ordinary determinant of Dij (tb ):Dren = Det −∂t2 δij − Ω2ij (t) = det Dij (tb ).(2.239)H.
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