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Journal of Mathematical Physics

On the Fourier series representations of path integrals

Antoine Royer

Published Oct 1, 1984• 8 citations
Publisher

Abstract

The direct transformation from the polygonal to the Fourier series representation of Feynman path integrals, via a change of integration variables, is effected explicitly for cases where the Lagrangian is of the form L(x,ẋ,t)= 1/2 ẋ2−V(x,t). This transformation involves a ‘‘functional Jacobian’’ stemming solely from the velocity term in the Lagrangian; this is because N-segment polygonal paths of the type contributing significantly to the integral, and their N-term Fourier series approximants, coalesce together as N→∞, but not their derivatives. We also consider integrals over paths with fixed means T−1∫T0 dt x(t) =const. The usefulness of the Fourier representation is illustrated with the harmonic oscillator case V(x)= 1/2 ω2x2, in both the free and fixed means situations; in particular, the Fourier evaluation of the path integrals trivially determines the large time phases (Maslov indices), and the ranges of ω2 values for which the integrals are finite or infinite in the imaginary time cases.

DOI
10.1063/1.526058

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The direct transformation from the polygonal to the Fourier series representation of Feynman path integrals, via a change of integration variables, is effected explicitly for cases where the Lagrangian is of the form L(x,ẋ,t)= 1/2 ẋ2−V(x,t). This transformation involves a ‘‘functional Jacobian’’ stemming solely from the velocity term in the Lagrangian; this is because N-segment polygonal paths of the type contributing significantly to the integral, and their N-term Fourier series approximants, coalesce together as N→∞, but not their derivatives. We also consider integrals over paths with fixed means T−1∫T0 dt x(t) =const. The usefulness of the Fourier representation is illustrated with the harmonic oscillator case V(x)= 1/2 ω2x2, in both the free and fixed means situations; in particular, the Fourier evaluation of the path integrals trivially determines the large time phases (Maslov indices), and the ranges of ω2 values for which the integrals are finite or infinite in the imaginary time cases.
Several path integrals are exactly evaluated here using Gaussian path integration. A few general ideas are illustrated using the advantage of being able to exactly evaluate Gaussian path integrals. Path integrals defined over a particular collection of allowed indeterminate paths can sometimes be represented by a Fourier expansion of the paths. This leads to two important techniques for performing path integrations: Expanding the action about the classical solution of the Lagrangian; Expanding the degree of freedom in a Fourier expansion of the allowed paths. Various cases are considered to illustrate the usage of classical solutions and Fourier expansions, and these also provide a set of relatively simple examples to familiarize oneself with the nuts and bolts of the path integral. The Lagrangian of the simple harmonic oscillator is used for all of the following examples; all the computations are carried out explicitly and exactly. The following different cases are considered: • Correlators of exponential functions of the degree of freedom are discussed in Section 12.1. • The generating functional for periodic paths is evaluated in Section 12.2. • The path integral required for evaluating the normalization constant for the oscillator evolution kernel is discussed in Section 12.3. The path integral entails summing over all paths that start from and return to the same fixed position. • Section 12.4 discusses the evolution kernel for a particle starting at an initial position x i and, after time τ, having a final position that is indeterminate.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables N included. It is proved that when the cumulant expansion is truncated at order p, the asymptotic convergence rate of the density matrix behaves like N(-(2p+1)). The complex algebra associated with the proof is simplified by introducing a diagrammatic representation of the contributing terms along with an associated linked-cluster theorem. The cumulant terms at each order are expanded in a series such that the asymptotic convergence rate is maintained without the need to calculate the full cumulant at order p. Using this truncated expansion of each cumulant at order p, the numerical cost in developing Fourier path integral expressions having convergence order N(-(2p+1)) is shown to be approximately linear in the number of required potential energy evaluations making the method promising for actual numerical implementation.
Feynman's Lagrangian path integral was an outgrowth of Dirac's vague surmise that Lagrangians have a role in quantum mechanics. Lagrangians implicitly incorporate Hamilton's first equation of motion, so their use contravenes the uncertainty principle, but they are relevant to semiclassical approximations and relatedly to the ubiquitous case that the Hamiltonian is quadratic in the canonical momenta, which accounts for the Lagrangian path integral's "success". Feynman also invented the Hamiltonian phase-space path integral, which is fully compatible with the uncertainty principle. We recast this as an ordinary functional integral by changing direct integration over subpaths constrained to all have the same two endpoints into an equivalent integration over those subpaths' unconstrained second derivatives. Function expansion with generalized Legendre polynomials of time then enables the functional integral to be unambiguously evaluated through first order in the elapsed time, yielding the Schroedinger equation with a unique quantization of the classical Hamiltonian. Widespread disbelief in that uniqueness stemmed from the mistaken notion that no subpath can have its two endpoints arbitrarily far separated when its nonzero elapsed time is made arbitrarily short. We also obtain the quantum amplitude for any specified configuration or momentum path, which turns out to be an ordinary functional integral over, respectively, all momentum or all configuration paths. The first of these results is directly compared with Feynman's mistaken Lagrangian-action hypothesis for such a configuration path amplitude, with special heed to the case that the Hamiltonian is quadratic in the canonical momenta.