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Journal of Physics A Mathematical and General

Combinatorial approach to Feynman path integration

Richard E. Crandall

Published Jul 21, 1993• 16 citations
Publisher

Abstract

Combinatorial relations can be used to convert the non-relativistic time-sliced Feynman path integral into perturbation expansions. These methods reveal that when the time interval is sliced into N increments, each order of perturbation theory sustains an error O(1/ square root N). In this way the author provides exact path integral results for the following potentials: delta-function comb, finite well, tunnelling barrier, and a generalized exponential cusp. For the tunnelling barrier it is seen how the celebrated (-1) reflection factor arises in the limit of infinite barrier height. The one-dimensional Coulomb problem is solved as a limiting case of the exponential cusp. In addition, for power potentials the author indicates how this path integral approach yields sometimes divergent, nevertheless asymptotic perturbation expansions.

DOI
10.1088/0305-4470/26/14/024

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Combinatorial relations can be used to convert the non-relativistic time-sliced Feynman path integral into perturbation expansions. These methods reveal that when the time interval is sliced into N increments, each order of perturbation theory sustains an error O(1/ square root N). In this way the author provides exact path integral results for the following potentials: delta-function comb, finite well, tunnelling barrier, and a generalized exponential cusp. For the tunnelling barrier it is seen how the celebrated (-1) reflection factor arises in the limit of infinite barrier height. The one-dimensional Coulomb problem is solved as a limiting case of the exponential cusp. In addition, for power potentials the author indicates how this path integral approach yields sometimes divergent, nevertheless asymptotic perturbation expansions.
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An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.
We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on Rn{\mathbb{R}^n}Rn , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in ℏ{\hbar}ℏ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the t→0{t \to 0}t→0 limit approaches the δ distribution. As such, our construction gives explicitly the full ℏ→0{\hbar\to 0}ℏ→0 asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.
It is proposed that each of several analytical techniques used to evaluate the path integral when a spinless particle encounters an infinite potential barrier relies on more constraints than is necessary. We review the image, forbidden-paths, and eigenfunction-expansion methods, and the infinitesimal-propagator method is developed herein. It is then shown that the latter two methods contain assumptions leading to extra terms in the path-integral integrand which serve only to aid in the analytical evaluation of the path integral but which do not affect the outcome of that evaluation. The analytical arguments are further confirmed by a recently developed numerical method that, for a limited number of cases, is a direct evaluation of the path integral.
This is the second, significantly expanded edition of the comprehensive textbook of 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular of the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's famous formula to include singular 1/r- and 1/r2-potentials. The second is a simple quantum equivalence principle governing the transformation of euclidean path integrals to spaces with curvature and torsion.The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation expansion. In contrast to ordinary perturbation expansions, divergencies are absent. Instead, there is a uniform convergence from weak to strong couplings, opening a way to precise approximate evaluations of analytically unsolvable path integrals.Tunneling processes are treated in detail. The results are used to determine the lifetime of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A new variational treatment extends the range of validity of previous tunneling theories from large to small barriers. A corresponding extension of large-order perturbation theory now also applies to small orders.Special attention is devoted to path integrals with topological restrictions. These are relevant to the understanding of the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.
Towards the Classification of Exactly Solvable Feynman Path Integrals: δδδ-Function Perturbations and Boundary-Problems as Miscellaneous Solvable Models
Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) Pöschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.