p-Adic Valued Distributions in Mathematical Physics

• Andrei Khrennikov
Book

Part of the Mathematics and Its Applications book series (MAIA, volume 309)

1. Front Matter
Pages i-xvi
2. Andrei Khrennikov
Pages 1-29
3. Andrei Khrennikov
Pages 31-64
4. Andrei Khrennikov
Pages 65-83
5. Andrei Khrennikov
Pages 85-105
6. Andrei Khrennikov
Pages 107-114
7. Andrei Khrennikov
Pages 115-159
8. Andrei Khrennikov
Pages 161-191
9. Andrei Khrennikov
Pages 193-213
10. Andrei Khrennikov
Pages 215-234
11. Back Matter
Pages 235-264

Introduction

Numbers ... , natural, rational, real, complex, p-adic .... What do you know about p-adic numbers? Probably, you have never used any p-adic (nonrational) number before now. I was in the same situation few years ago. p-adic numbers were considered as an exotic part of pure mathematics without any application. I have also used only real and complex numbers in my investigations in functional analysis and its applications to the quantum field theory and I was sure that these number fields can be a basis of every physical model generated by nature. But recently new models of the quantum physics were proposed on the basis of p-adic numbers field Qp. What are p-adic numbers, p-adic analysis, p-adic physics, p-adic probability? p-adic numbers were introduced by K. Hensel (1904) in connection with problems of the pure theory of numbers. The construction of Qp is very similar to the construction of (p is a fixed prime number, p = 2,3,5, ... ,127, ... ). Both these number fields are completions of the field of rational numbers Q. But another valuation 1 . Ip is introduced on Q instead of the usual real valuation 1 . I· We get an infinite sequence of non isomorphic completions of Q : Q2, Q3, ... , Q127, ... , IR = Qoo· These fields are the only possibilities to com plete Q according to the famous theorem of Ostrowsky.

Keywords

Probability distribution Probability theory differential equation distribution functional analysis mathematical physics mechanics number theory statistical physics statistics thermodynamics

Authors and affiliations

• Andrei Khrennikov
• 1
1. 1.Moscow Institute of Electronic EngineeringZelenograd (Moscow)Russia

Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-8356-5