P-adic Mathematical Physics Research Articles

Free Choice in Quantum Theory: A p-adic View.

In this paper, it is rigorously proven that since observational data (i.e., numerical values of physical quantities) are rational numbers only due to inevitably nonzero measurements errors, the conclusion about whether Nature at the smallest scales is discrete or continuous, random and chaotic, or strictly deterministic, solely depends on experimentalist's free choice of the metrics (real or p-adic) he chooses to process the observational data. The main mathematical tools are p-adic 1-Lipschitz maps (which therefore are continuous with respect to the p-adic metric). The maps are exactly the ones defined by sequential Mealy machines (rather than by cellular automata) and therefore are causal functions over discrete time. A wide class of the maps can naturally be expanded to continuous real functions, so the maps may serve as mathematical models of open physical systems both over discrete and over continuous time. For these models, wave functions are constructed, entropic uncertainty relation is proven, and no hidden parameters are assumed. The paper is motivated by the ideas of I. Volovich on p-adic mathematical physics, by G. 't Hooft's cellular automaton interpretation of quantum mechanics, and to some extent, by recent papers on superdeterminism by J. Hance, S. Hossenfelder, and T. Palmer.

New Bases in the Space of Square Integrable Functions on the Field of p-Adic Numbers and Their Applications

In this paper we summarize the results obtained in some of our recent studies in the form of a series of theorems. We present new real bases of functions in L2(Br) that are eigenfunctions of the p-adic pseudodifferential Vladimirov operator defined on a compact set Br ⊂ ℚp of the field of p-adic numbers ℚp and on the whole ℚp. We demonstrate a relationship between the constructed basis of functions in L2(ℚp) and the basis of p-adic wavelets in L2(ℚp). A real orthonormal basis in the space L2(ℚp, u(x) dpx) of square integrable functions on ℚp with respect to the measure u(x) dpx is described. The functions of this basis are eigenfunctions of a pseudodifferential operator of general form with kernel depending on the p-adic norm and with measure u(x) dpx. As an application of this basis, we present a method for describing stationary Markov processes on the class of ultrametric spaces $$\mathbb{U}$$ that are isomorphic and isometric to a measurable subset of the field of p-adic numbers ℚp of nonzero measure. This method allows one to reduce the study of such processes to the study of similar processes on ℚp and thus to apply conventional methods of p-adic mathematical physics in order to calculate their characteristics. As another application, we present a method for finding a general solution to the equation of p-adic random walk with the Vladimirov operator with general modified measure u(∣x∣p) dpx and reaction source in ℤp.

Towards Three-Dimensional Conformal Probability

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and second-quantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion.We formulate this program in both the Archimedean and p-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where p-adic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a p-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author’s talk at the 6th International Conference on p-adicMathematical Physics and its Applications,Mexico 2017.

Holography and Local Fields

This paper, which is a summary (in which considerable creative license has been taken) of the author’s talk at the sixth international conference on p-adic mathematical physics and its applications (CINVESTAV, Mexico City, October 2017), reviews some recent work connecting field theories defined on the p-adic numbers and ideas from the AdS/CFT correspondence. Some results are included, along with general discussion of the utility and interest of p-adic analogues of Lagrangian field theories, at least from the author’s perspective. A few challenges, shortcomings, and ideas for future work are also discussed.

P-Adic mathematical physics: the first 30 years

$p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

P-Adic mathematical physics and B. Dragovich research

We present a brief review of some parts of p-adic mathematical physics related to the scientific work of Branko Dragovich on the occasion of his 70th birthday.

Application of p-Adic analysis methods in describing Markov processes on ultrametric spaces isometrically embedded into ℚ p

We propose a method for describing stationary Markov processes on the class of ultrametric spaces \(\mathbb{U}\) isometrically embedded in the field ℚp of p-adic numbers. This method is capable of reducing the study of such processes to the investigation of processes on ℚp. Thereby the traditional machinery of p-adic mathematical physics can be applied to calculate the characteristics of stationary Markov processes on such spaces. The Cauchy problem for the Kolmogorov-Feller equation of a stationary Markov process on such spaces is shown as being reducible to the Cauchy problem for a pseudo-differential equation on ℚp with non-translation-invariant measure m(x) dpx. The spectrum of the pseudo-differential operator of the Kolmogorov-Feller equation on ℚp with measure m(x) dpx is found. Orthonormal basis of real valued functions for L2 (ℚp,m(x) dpx) is constructed from the eigenfunctions of this operator.

On measurements, numbers and p-adic mathematical physics

In this short paper I consider relation between measurements, numbers and p-adic mathematical physics. p-Adic numbers are not result of measurements, but nevertheless they play significant role in description of some systems and phenomena. We illustrate their ability for applications referring to some sectors of p-adic mathematical physics and related topics, in particular, to string theory and the genetic code.

P-Adic valued quantization

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrodinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.

On p-adic mathematical physics

1. NUMBERS: RATIONAL, REAL, p-ADIC We present a brief review of some selected topics in p-adic mathematical physics. More details can be found in the references below and the other references are mainly contained therein. We hope that this brief introduction to some aspects of p-adic mathematical physics could be helpful for the readers of the first issue of the journal p-Adic Numbers, Ultrametric Analysis and Applications. The notion of numbers is basic not only in mathematics but also in physics and entire science. Most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. p-Adic numbers (see, e.g. [1]), introduced by Hensel, are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography.

GREEN FUNCTIONS ON THE p-ADIC VECTOR SPACE

Calculations of some integrals on the n-dimensional vector space over <TEX>$\mathbb{Q}_p$</TEX> are useful in getting some other formulations of quantum mechanics and the field theory of p-adic mathematical physics. For reasons of these, we estimate several integrals. As an application, we derive some properties for the p-adic Green functions.