The theory of non-Archimedean generalized functions and its applications to quantum mechanics and field theory
Khrennikov belongs to a group consisting mainly of Russian theoretical physicists who promote a \(p\)-adic model of quantum physics. They developed their ideas into a \(p\)-adic, and more generally into a non-Archimedean (n.a.) theory of physics and they created the mathematics necessary to do so. The paper starts with an excellent explanation of why the n.a. model is better adapted for the description of certain phenomena on the quantum level. The author then introduces n.a. versions of classical mathematical tools, while motivating the definitions and discussing the typical n.a. difficulties they represent. Each time some basic results about them are proved. The paper is too long to go into details here. It covers the following topics: -- The Gauss, Lebesgue and Feynman distributions, for finite- and infinite-dimensional spaces, over n.a. fields -- Integration theory for measures and functions that assume values in a n.a. field -- Quantum mechanics for n.a. wave functions -- Functional integrals and the quantization of n.a. models with an infinite number of degrees of freedom -- The \(p\)-adic probability distributions. Besides for theoretical physicists, this paper can also be considered as basic for pure mathematicians working in n.a. analysis. For them it provides indeed a source of inspiration for the development of new theories. Perhaps on stronger foundations and logically stricter than in the paper. And, as a reward, with applications in physics.
quantum mechanics for non Archimedean wave functions
functions that assume values in a n.a. field
scientific article; zbMATH DE number 859203
scientific article