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Differentiable Path Integrals in Quantum Mechanics

Ignacio A. Reyes

Published Jan 1, 2014• 4 citations
OpenAlex

Abstract

A method is presented which restricts the space of paths entering the path integral of quantum mechanics to subspaces of C α , by only allowing paths which possess at least α derivatives. The method introduces two external parameters, and induces the appe

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A method is presented which restricts the space of paths entering the path integral of quantum mechanics to subspaces of Cα, by only allowing paths which possess at least α derivatives. The method introduces two external parameters, and induces the appearance of a particular time scale ϵD such that for time intervals longer than ϵD the model behaves as usual quantum mechanics. However, for time scales smaller than ϵD, modifications to standard formulation of quantum theory occur. This restriction renders convergent some quantities which are usually divergent in the time-continuum limit ϵ → 0. We illustrate the model by computing several meaningful physical quantities such as the mean square velocity 〈v2〉, the canonical commutator, the Schrödinger equation and the energy levels of the harmonic oscillator. It is shown that an adequate choice of the parameters introduced makes the evolution unitary.
These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum mechanics. No prior exposure to path integrals is assumed, however. The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. After an introduction including a very brief historical overview of the subject, we derive a path integral expression for the propagator in quantum mechanics, including the free particle and harmonic oscillator as examples. We then discuss a variety of applications, including path integrals in multiply-connected spaces, Euclidean path integrals and statistical mechanics, perturbation theory in quantum mechanics and in quantum field theory, and instantons via path integrals. For the most part, the emphasis is on explicit calculations in the familiar setting of quantum mechanics, with some discussion (often brief and schematic) of how these ideas can be applied to more complicated situations such as field theory.
Abstract This chapter discusses the Feymann path-integral approach to quantum mechanics. First, it derives a path integral expression for the evolution operator. Next, it shows that the classical equations of motion, that is, those obtained from the principle of least action, are obtained from this path integral formulation in the limit where the variation in the action of the problem at hand is small as compared to the Planck’s constant h. Last, it discusses how the WKB approximation is derived from the path integral formulation.
Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.