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Mathematical aspects of the Feynman path integral

Daniel W. Boutros

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.

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Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
The Feynman path integral is defined over the space RT\mathbb{R}^TRT of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral ∫RT\int_{\mathbb{R}^T}∫RT​ and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.
We present a survey of the relations between infinite dimensional integrals, both of the probabilistic type (e.g. Wiener path integrals) and of oscillatory type (e.g. Feynman path integrals).Besides their mutual relations (analogies and differences) we also discuss their relations with certain types of partial differential equations (parabolic resp. hyperbolic), describing time evolution with or without stochastic terms.The connection of these worlds of deterministic and stochastic evolutions with the world of quantum phenomena is also briefly illustrated. The survey spans a bridge from basic concepts and methods in these areas to recent developments concerning their relations.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.