edited by V S Vladimirov (Steklov Mathematical Institute), I V Volovich (Steklov Mathematical Institute), & E I Zelenov (Steklov Mathematical Institute)

p-adic numbers play a very important role in modern number theory, algebraic geometry and representation theory. Lately p-adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the non-Archimedean geometry of space-time at small distances.

This is the first book to deal with applications of p-adic numbers in theoretical and mathematical physics. It gives an elementary and thoroughly written introduction to p-adic numbers and p-adic analysis with great numbers of examples as well as applications of p-adic numbers in classical mechanics, dynamical systems, quantum mechanics, statistical physics, quantum field theory and string theory.

• Analysis on the Field p-Adic Numbers:
  • The Field of p-Adic Numbers
  • Analytic Functions
  • Additive and Multiplicative Characters
  • Integration Theory
  • The Gaussian Integrals
  • Generalized Functions
  • Convolution and the Fourier Transformation
  • Homogeneous Generalized Functions
• Pseudo-Differential Operators on the Field of p-Adic Numbers:
  • The Operator D?
  • p-Adic Schrodinger Operators
• p-Adic Quantum Theory:
  • p-Adic Quantum Mechanics
  • Spectral Theory in p-Adic Quantum Mechanics
  • Weyl Systems. Infinite Dimensional Case
  • p-Adic Strings
  • q-Analysis (Quantum Groups) and p-Adic Analysis
  • Stochastic Processes Over the Field of p-Adic Numbers