arXiv subjects

Peter Prešnajder

Publications and source records attributed to Peter Prešnajder.

12 recordsLinked to original sources

arXiv2020

Some oscillatory representations of fuzzy conformal group SU(2,2) with positive energy

We construct the relativistic fuzzy space as a non-commutative algebra of functions with purely structural and abstract coordinates being the creaction and annihilation (C/A) operators acting on a Hilbert space $\mathcal{H}_F$. Using these oscillators, we represent the conformal algebra $su(2,2)$ (containing the operators describing physical observables, that generate boosts, rotations, spatial and conformal translations, and dilatation) by operators acting on such functions and reconstruct an auxiliary Hilbert space $\mathcal{H}_A$ to describe this action. We then analyze states on such space and prove them to be boost-invariant. Eventually, we construct two classes of irreducible representations of $su(2,2)$ algebra with \textit{half-integer} dimension $d$ ([1]): (i) the classical fuzzy massless fields as a doubleton representation of the $su(2,2)$ constructed from one set of C/A operators in fundamental or unitary inequivalent dual representation and (ii) classical fuzzy massive fields as a direct product of two doubleton representations constructed from two sets of C/A operators that are in the fundamental and dual representation of the algebra respectively.

math-ph↗

arXiv2018

Magnetic Monopoles in (Noncommutative) Quantum Mechanics

We utilize the close relation between the complex space $\textbf{C}^2$ and the real space $\textbf{R}^3$ to reformulate quantum mechanics in a manner which allows to, either or both, describe magnetic monopoles and quantize the underlying space, obtaining (noncommutative) quantum mechanics (with magnetic monopoles).

quant-ph↗

arXiv2018

Alternative Description of Magnetic Monopoles in Quantum Mechanics

We present an alternative description of magnetic monopoles by lifting quantum mechanics from 3-dimensional space into a one with 2 complex dimensions. Magnetic monopoles are realized as a generalization of the considered states. Usual algebraic relations and magnetic fields describing monopoles are reproduced, with the Dirac quantization condition satisfied naturally.

quant-ph↗

arXiv2018

Magnetic monopoles in noncommutative quantum mechanics 2

In this paper we extend the analysis of magnetic monopoles in quantum mechanics in three dimensional rotationally invariant noncommutative space $\textbf{R}^3_λ$. We construct the model step-by-step and observe that physical objects known from previous studies appear in a very natural way. Nonassociativity became a topic of great interest lately, often in a connection with magnetic monopoles. We show that this model does not possess this property.

math-ph↗

arXiv2017

Magnetic monopoles and symmetries in noncommutative space

In this paper, we review the progress in the analysis of magnetic monopoles as generalized states in quantum mechanics. We show that the considered model contains rich algebraic structure that generates symmetries which have been utilized in different physical contexts. Even though are we focused on quantum mechanics in noncommutative space $\textbf{R}_λ^3$, the results can be reconstructed in ordinary quantum mechanics in $\textbf{R}^3$ as well.

hep-th↗

arXiv2016

Magnetic monopoles in noncommutative quantum mechanics

We discuss certain generalization of the Hilbert space of states in noncommutaive quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).

physics.gen-ph↗

arXiv2016

NC plane waves, Casimir effect and flux tube potential with Lüscher terms

We analyze plane waves in a model of quantum mechanics in a three dimensional noncommutative (NC) space $R^3_λ$. Signature features of NC models are impossibility of probing distances smaller than a certain length scale λ and a presence of natural energetic cut-off at energy scale of order $1/λ^2$ (in convenient units). We analyze consequences of such restrictions on a 1 dimensional Casimir effect. The result shows resemblance to flux tube potential for quark-antiquark pairs and to effective bosonic string theories with Lüscher terms. Such behavior might effect the radius of possible compact (fuzzy) dimensions.

quant-ph↗

arXiv2015

Quantum Mechanics in Noncommutative space

This paper provides an examination of how are prediction of standard quantum mechanic (QM) affected by introducing a noncommutative (NC) structure into the configuration space of the considered system (electron in the Coulomb potential in the present case). The parameter controlling the extent of modification is denoted as λ. The coordinates in the NC space are realized via creation and annihilation operators acting in an auxiliary Fock space, this one being chosen in such a way that the rotational invariance of the system remains intact also in NCQM. Analog of Schrödinger equation for hydrogen atom is found and analytically solved, both for bound states and scattering. The exact formulas for NC corrections are given. None of the NC predictions contradicts experimentally verified QM results, since in the correspondence limit λ -> 0 both QM and NCQM coincide. Highly surprising feature of the NC version is the existence of bound states for repulsive potential at ultra-high energies. However, these disappear from the Hilbert space in the mentioned limit. The whole problem is solved also using a method analogous to that of Pauli. Besides rotational invariance, the dynamical symmetry related to the conservation of NC analog of Laplace-Runge-Lenz vector is being used and the results obtained this way are in the full agreement with those given by "Schrödinger-like" approach. The presented NC deformation of QM preserves all those mysterious properties of the Coulomb system that made it a distinguished key-stone of the modern physics.

math-ph↗

arXiv2011

Coulomb problem in non-commutative quantum mechanics - Exact solution

We investigate consequences of space non-commutativity in quantum mechanics of the hydrogen atom. We introduce rotationally invariant noncommutative space $\hat{\bf R}^3_0$ - an analog of the hydrogen atom ($H$-atom) configuration space ${\bf R}^3_0\,=\, {\bf R}^3\setminus \{0\}$. The space $\hat{\bf R}^3_0$ is generated by noncommutative coordinates realized as operators in an auxiliary (Fock) space ${\cal F}$. We introduce the Hilbert space $\hat{\cal{H}}$ of wave functions $\hatψ$ formed by properly weighted Hilbert-Schmidt operators in ${\cal F}$. Finally, we define an analog of the $H$-atom Hamiltonian in $\hat{\bf R}^3_0$ and explicitly determine the bound state energies $E^λ_n$ and the corresponding eigenstates $\hatψ^λ_{njm}$. The Coulomb scattering problem in $\hat{\bf R}^3_0$ is under study.

math-ph↗

arXiv2010

Quantum Field Theory on quantized Bergman domain

We present an oscillator realization of discrete series representations of group SU(2,2). We give formulas for the coherent state star-product quantization of a Bergman domain $D$. A formulation of a (regularized) non-commutative scalar field on a quantized $D$ is given.

math-ph↗

arXiv2008

Functional integral with $ϕ^4$ term in the action beyond standard perturbative methods II

To avoid problems with infinite measure, the functional integral for harmonic oscillator can be calculated by time - slicing method with continuum limit procedure proposed Gelfand and Yaglom. In previous article we proved by nonperturbative calculation the generalized Gelfand-Yaglom equation for anharmonic oscillator with positive or negative mass term. In this article we prove by step-by-step the calculation of the correction function to the Gelfand-Yaglom equation for an-harmonic oscillator.

hep-th↗

arXiv2007

Study of an Abelinization Transition in SU(2) Gluodynamics at Finite Temperature

We discuss the problem of an effective descriptions of the phase transition phenomena in the pure gluodynamics in SU(2) symmetric QCD. We choose the method of calculation following the conjecture that the infrared sector of the theory possesses the same confinement characteristic as the full theory. We show, that analytic descriptions of this phenomena is beyond the Gaussian method of evaluations of functional integrals. We propose a non-perturbative evaluation of functional integral, meanwhile for two dimensional Wiener integral for $ϕ^4$ theory.

hep-ph↗