arXiv subjects

Ziemowit Domański

Publications and source records attributed to Ziemowit Domański.

7 recordsLinked to original sources

arXiv2024

Time Evolution in Quantum Mechanics with a Minimal Time Scale

The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page-Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.

quant-ph↗

arXiv2022

Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. II. Isomonodromic Lax representation

This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this article we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the non-magnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlevé-type. We also present isomonodromic Lax representations for all one-, two- and three-dimensional Painlevé-type systems originating in our scheme. Based on these results we propose complete hierarchies of $P_{I}-P_{IV}$ that follow from our construction.

nlin.SI↗

arXiv2021

Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. I. Frobenius integrability

This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel-type and of Painlevé- type. In this article we present a deformation of autonomous Stäckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so called magnetic separable potentials to non-autonomous Frobenius integrable systems. We also provide a complete list of all $2$- and $3\,$-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, that originate in our construction. Further, we prove the equivalence between both classes of systems. Finally we show how Painlevé equations $P_{I}-P_{IV}$ can be derived from our scheme.

math-ph↗

arXiv2019

Lax Representations for Separable Systems from Benenti Class

In this paper we construct Lax pairs for Stäckel systems with separation curves from so-called Benenti class. For each system of considered family we present an infinite family of Lax representations, parameterized by smooth functions of spectral parameter.

math-ph↗

arXiv2019

Deformation quantization on the cotangent bundle of a Lie group

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Frechet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.

math-ph↗

arXiv2018

Deformation quantization with minimal length

We develop a complete theory of non-formal deformation quantization exhibiting a nonzero minimal uncertainty in position. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. Basic properties of the star-product are proved and the extension of the star-product to a certain Hilbert space and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, examples of position eigenvectors and states of maximal localization are given.

math-ph↗

arXiv2016

Coherence and squeezing along quantum trajectories

We perform a detailed analysis of the behavior of coherent and squeezed states undergoing time evolution. We calculate time dependence of expectation values of position and momentum in coherent and squeezed states (which can be interpreted as quantum trajectories in coherent and squeezed states) and examine how coherence and squeezing is affected during time development, calculating time dependence of position and momentum uncertainty. We focus our investigations on two quantum systems. First we consider quantum linear system with Hamiltonian quadratic in $q$ and $p$ variables. As the second system we consider the simplest quantum nonlinear system with Hamiltonian quartic in $q$ and $p$ variables. We calculate the explicit formulas for the time development of expectation values and uncertainties of position and momentum in an initial coherent state.

quant-ph↗