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Journal of Applied Mathematics and Physics

Feynman Path Integral Using Operator Integration in Banach Space

Reza Ahangar, Erbil Çetin, Şerife Müge Ege

Published Jan 1, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.4236/jamp.2024.122028

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•Jan 1, 2020•0 citations
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.
"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"--
The definition of Feynman path integrals (Feynman functional integrals) as integrals with respect to a generalized measure, called the Lebesgue–Feynman measure in the paper and being an infinite-dimensional analogue of the classical Lebesgue measure on finite-dimensional Euclidean space, is discussed. This definition, which is a formalization of Feynman’s original definition, is different from those used previously in the mathematical literature. It makes it possible to give a description of the origin of quantum anomaly which is a mathematically correct version of the description given in the book Path Integrals and Quantum Anomalies by K. Fujikawa and H. Suzuki (Oxford, 2004) (and erroneously qualified as wrong in the book Functional Integration: Action and Symmetries by P. Cartier and C. DeWitt-Morette (Cambridge Univ. Press, Cambridge, 2006)).
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.
Feynman's path integral formulation arose from his attempt to incorporate the Lagrangian framework into quantum mechanics, offering what he regarded as a more fundamental perspective than the Hamiltonian approach, particularly in the context of quantum electrodynamics. Inspired by an analogy proposed by P. A. M. Dirac, Feynman showed that Schrödinger's equation could be recovered by promoting this analogy to an equality, up to a constant factor. This insight laid the foundation for the path integral formalism. In this article, we first outline the historical development of dynamical frameworks in physics and then trace Feynman's reasoning as he constructed his formulation, beginning with the double-slit experiment and its interpretation. We subsequently derive the Schrödinger equation from the path integral and demonstrate the conservation of probability. These final sections aim to provide students with a clear link between this elegant yet less commonly presented approach and the more standard methods typically taught in undergraduate courses.
In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.
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.