A. Yu. Khrennikov

On p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.

Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases.

Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis.

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions.In this thesis we restrict our study to the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions.The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball.Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.

Generalized probabilities taking values in non-Archimedean fields and in topological groups.

We develop an analog of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov’s method of axiomatization of probability theory, and the main distinguishing features of frequency probabilities are taken as axioms in the measure-theoretic approach. We also present a survey of non-Kolmogorovian probabilistic models, including models with negative-, complex-, and p-adic-valued probabilities. The last model is discussed in detail. The introduction of probabilities with p-adic values (as well as with more general non-Archimedean values) is one of the main motivations to consider generalized probabilities with values in more general topological groups than the additive group of real numbers. We also discuss applications of non-Kolmogorovian models in physics and cognitive sciences. A part of the paper is devoted to statistical interpretation of probabilities with values in topological groups (in particular, in non-Archimedean fields).

p-Adic probability theory and its applications. The principle of statistical stabilization of frequencies

The development ofp-adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events arep-adic numbers. The foundations of this theory are developed here. The frequency definition of probability is used. A general principle of statistical stabilization of relative frequencies is formulated. By virtue of this principle, statistical stabilization of relative frequencies, which are, like all experimental data, rational numbers, can be considered not only in the real topology but also inp-adic topologies.

p-adic orthogonal wavelet bases

We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L2(ℚp).

Second quantization and pseudodifferential operators

A theory of distributions on an infinite-dimensional phase space is proposed. Infinite-dimensional pseudodifferential operators are introduced on the basis of this theory.

Representation of a quantum field Hamiltonian inp-adic Hilbert space

Gaussian measures on infinite-dimensional p-adic spaces are defined and the corresponding L2-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in such spaces and the formal analogy with the usual Segal representation is discussed. It is found that the parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls. In p-adic Hilbert space, representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. The Hamiltonians with singular potentials are realized as bounded symmetric operators in L2-space with respect to a p-adic Gaussian measure.

Superanalysis: Theory of generalized functions and pseudodifferential operators

A theory of distributions on an infinite-dimensional superspace is constructed. Feynman and Gaussian supermeasures are defined on the basis of this theory and superpseudodifferential operators are introduced. We obtain the basic formulas in the theory of superpseudodifferential operators and the Feynman-Kac formula for the symbol of the evolution operator (here, in contrast to the algebraic approach, the functional integral is an integral over a space of actual trajectories in the phase superspace). The differential calculus in graded /Lambda/ modules proposed in this paper is needed for the introduction of infinite-dimensional superpseudodifferential operators and also for nonsequential definition of the Feynman integral.

Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups

A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group Rn, the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrödinger operator based on Schwingers approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621–648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrödinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used.