Related papers: Ona relation between classical and free infinitely…
Finding classical canonical observables consists of taking a function space over phase space. For constrained theories, these functions must form zero brackets with a closed algebraic structure of first-class constraints. This brackets…
Two examples of the situation when the classical observables should be described by a noncommutative probability space are investigated. Possible experimental approach to find quantum-like correlations for classical disordered systems is…
We compare two henselisations of a residually discrete valuation domain. Our constructive proof that a certain natural morphism is an isomorphism is also a proof in classical mathematics. Although this isomorphism is implicitly accepted as…
The hidden-variable question is whether or not various properties --- randomness or correlation, for example --- that are observed in the outcomes of an experiment can be explained via introduction of extra (hidden) variables which are…
Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n-1 symmetries polynomial in the canonical momenta, so that they are in…
We consider non-stationary dynamical systems with one-and-a-half degrees of freedom. We are interested in algorithmic construction of rich classes of Hamilton's equations with the Hamiltonian H=p^2/2+V(x,t) which are Liouville integrable.…
In this paper we deal with free functional inequalities on the circle. There are some interesting changes as opposed to the classical case. For example, the free Poincar\'e inequality has a slight change which seems to account for the lack…
This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are…
We study the role of continuous measurement in the quantum to classical transition for a system with coupled internal (spin) and external (motional) degrees of freedom. Even when the measured motional degree of freedom can be treated…
Classical entanglement is a powerful tool which provides a neat numerical estimate for the study of classical correlations. Its experimental investigation, however, has been limited to special cases. Here, we demonstrate that the…
We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies…
We formulate the problem of finding self-dual Hamiltonians (associated with integrable systems) as deformations of free systems given on various symplectic manifolds and discuss several known explicit examples, including recently found…
In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $\lambda \in [0,1]$, the rectangular free convolution with ratio $\lambda$. Here, we investigate the related notion of infinite divisiblity,…
Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity…
An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear…
In this paper we study some analytic properties of bi-free additive convolution, both scalar and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of…
In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are…
Following a review of quantum-classical hybrid dynamics, we discuss the ensuing proliferation of observables and relate it to measurements of (would-be) quantum mechanical degrees of freedom performed by (would-be) classical ones (if they…
In this work simple and effective quantization procedure of classical dynamical systems is proposed and illustrated by a number of examples. The procedure is based entirely on differential equations which describe time evolution of systems.
A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and…