The theory of non-Archimedean generalized functions and its applications to quantum mechanics and field theory
type: Item
instance of: scholarly article
title: The theory of non-Archimedean generalized functions and its applications to quantum mechanics and field theory
zbMATH Open document ID: 0856.46052
DOI: 10.1007/BF02367241
published in: Journal of Mathematical Sciences (New York)
publication date: 1996-04-23T00:00:00Z
review text: 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.
reviewed by: N. de Grande-de Kimpe
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 46F99
Mathematics Subject Classification ID: 81S40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 859203
zbMATH Keywords: integration theory
zbMATH Keywords: quantum mechanics for non Archimedean wave functions
zbMATH Keywords: \(p\)-adic model of quantum physics
zbMATH Keywords: Feynman distributions
zbMATH Keywords: functions that assume values in a n.a. field
zbMATH Keywords: \(p\)-adic probability distributions
author: Andrei Khrennikov
MaRDI profile type: Publication
cites work: Q4002721
cites work: Q3261267
cites work: Q5591454
cites work: Q3696277
cites work: Q3261277
cites work: Q3662131
cites work: Q5521609
cites work: Q3349480
cites work: Superanalysis, I. Differential calculus
cites work: Q3729704
cites work: Q3469302
cites work: Q3491827
cites work: Q4731653
cites work: Q3995048
cites work: Q3487911
cites work: Q3317511
cites work: Q3781506
cites work: MEASURES ON LINEAR TOPOLOGICAL SPACES
cites work: Q3854176
cites work: Q3791690
cites work: Q3806089
cites work: Q3480632
cites work: Functional superanalysis
cites work: Q3488613
cites work: Q3970818
cites work: Q4005401
cites work: Mathematical methods of non-Archimedean physics
cites work: Q3986334
cites work: Q4007555
cites work: Q3362239
description: scientific article; zbMATH DE number 859203