Epr-bohm experiment and Bell’s inequality: Quantum physics meets probability theory

Our main aim in this paper is to inform the physics community (and especially experts in quantum information) about investigations of the problem of the probabilistic compatibility of a family of random variables: the possibility of realizing such a family based on a single probability measure (of constructing a single Kolmogorov probability space). These investigations were started a hundred years ago by Boole. The complete solution of the problem was obtained by the Soviet mathematician Vorobiev in the 1960s. It turns out that probabilists and statisticians obtained inequalities for probabilities and correlations that include the famous Bell’s inequality and its generalizations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 99–115, October, 2008.     

Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic–epistemic (implicate–explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical–quantum interrelations lose their sharpness; generally, simple dendrograms are “more quantum” than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data.     

Transport through a network of capillaries from ultrametric diffusion equation with quadratic nonlinearity

This paper is about a novel mathematical framework to model transport (of, e.g., fluid or gas) through networks of capillaries. This framework takes into account the tree structure of the networks of capillaries. (Roughly speaking, we use the tree-like system of coordinates.) As is well known, tree-geometry can be topologically described as the geometry of an ultrametric space, i.e., a metric space in which the metric satisfies the strong triangle inequality: in each triangle, the third side is less than or equal to the maximum of two other sides. Thus transport (e.g., of oil or emulsion of oil and water in porous media, or blood and air in biological organisms) through networks of capillaries can be mathematically modelled as ultrametric diffusion. Such modelling was performed in a series of recently published papers of the authors. However, the process of transport through capillaries can be only approximately described by the linear diffusion, because the concentration of, e.g., oil droplets, in a capillary can essentially modify the dynamics. Therefore nonlinear dynamical equations provide a more adequate model of transport in a network of capillaries. We consider a nonlinear ultrametric diffusion equation with quadratic nonlinearity - to model transport in such a network. Here, as in the linear case, we apply the theory of ultrametric wavelets. The paper also contains a simple introduction to theory of ultrametric spaces and analysis on them.     

Schrödinger representation in the bosonic string theory: Quantization without anomalous terms

Using the theory of infinite-dimensional pseudodifferential operators in superspace, a Schrödinger representation is introduced to the theory of bosonic strings (the superstucture emerges under BRST quantization). qp-Symbol quantization considered in this paper, possesses no anomalies (unlike quantization using normal ordering). In particular, under BRST quantization, the anomalous term vanishes in space of arbitrary dimension, so that there is no critical dimension in this theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 19–22, October, 1990.The author thanks V. S. Vladimirov, I. V. Volovich, and A. A. Slavnov for useful advice, and all the participants of the seminars in mathematical physics and quantum field theory of MIAN for useful discussions of mathematical aspects of field and string theory.     

Diffusion processes in linear spaces and pseudotopologies

The theory of diffusion processes with a nonnormable phase space (a nuclear Fréchet space) is developed and the Cauchy problem for parabolic equations relative to functions on this space is solved by probabilistic methods. A series of examples are given, demonstrating a significant difference between the theory of stochastic differential equations and parabolic equations in the case of locally convex spaces, on one hand, and the analogous theory in the case of Banach spaces, on the other hand. The difficulty which arises, when passing from a Banach space to a Fréchet space, involves basically a functional rather than a probabilistic character. There appears a sufficiently complex intertwinement of the theory of locally convex and pseudotopological spaces with probability theory.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 190–209, 1986.In conclusion, the author expresses his gratitude to O. G. Smolyanov for his constant interest in the paper and for useful advice.     

Trotter's formula for the heat conduction and Schrödinger equations in a non-Archimedean superspace

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 5, pp. 155–165, September–October, 1991.