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Feynman Path Integral Using Operator Integration in Banach Space

Journal of Applied Mathematics and Physics 2024年第2期12卷 432-444页

作者： Reza R. Ahangar Erbil Cetin serife muge ege Mathematics Department Texas A&M University Kingsville USA Mathematics Department Ege University Bornova Türkiye

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.

关键词： Feynman Path Integral Lebesgue-Bochner-Stieltjes Integral Operator Integral Particle-Wave Function Operator Integration Position and Momentum Operators

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