scholarly journals Fock representations of Q-deformed commutation relations

2017 ◽  
Vol 58 (7) ◽  
pp. 073501 ◽  
Author(s):  
Marek Bożejko ◽  
Eugene Lytvynov ◽  
Janusz Wysoczański
Keyword(s):  
Commutation Relations ◽  
Deformed Commutation Relations ◽  
Fock Representations

scholarly journals Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

2019 ◽  
Vol 32 (05) ◽  
pp. 2030004
Author(s):  
Alexei Daletskii ◽  
Alexander Kalyuzhny ◽  
Eugene Lytvynov ◽  
Daniil Proskurin
Keyword(s):  
Linear Operator ◽  
Fock Space ◽  
Multicomponent System ◽  
Baxter Equation ◽  
Special Choice ◽  
Bounded Linear ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
System A ◽  
Fock Representations

Let [Formula: see text] be a separable Hilbert space and [Formula: see text] be a self-adjoint bounded linear operator on [Formula: see text] with norm [Formula: see text], satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator [Formula: see text] determines a [Formula: see text]-deformed Fock space [Formula: see text]. We start with reviewing and extending the known results about the structure of the [Formula: see text]-particle spaces [Formula: see text] and the commutation relations satisfied by the corresponding creation and annihilation operators acting on [Formula: see text]. We then choose [Formula: see text], the [Formula: see text]-space of [Formula: see text]-valued functions on [Formula: see text]. Here [Formula: see text] and [Formula: see text] with [Formula: see text]. Furthermore, we assume that the operator [Formula: see text] acting on [Formula: see text] is given by [Formula: see text]. Here, for a.a. [Formula: see text], [Formula: see text] is a linear operator on [Formula: see text] with norm [Formula: see text] that satisfies [Formula: see text] and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function [Formula: see text] in the case [Formula: see text] determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its [Formula: see text]-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

scholarly journals Kinetic energy properties and weak equivalence principle in a space with generalized uncertainty principle

2020 ◽  
Vol 35 (13) ◽  
pp. 2050096
Author(s):  
Kh. P. Gnatenko ◽  
V. M. Tkachuk
Keyword(s):  
Kinetic Energy ◽  
Energy Dependence ◽  
Uncertainty Principle ◽  
Equivalence Principle ◽  
Generalized Uncertainty Principle ◽  
Weak Equivalence ◽  
Weak Equivalence Principle ◽  
Deformation Function ◽  
Commutation Relations ◽  
Deformed Commutation Relations

A space with deformed commutation relations for coordinates and momenta leading to generalized uncertainty principle (GUP) is studied. We show that GUP causes great violation of the weak equivalence principle for macroscopic bodies, violation of additivity property of the kinetic energy, dependence of the kinetic energy on composition, great corrections to the kinetic energy of macroscopic bodies. We find that all these problems can be solved in the case of arbitrary deformation function depending on momentum if parameter of deformation is proportional inversely to squared mass.

scholarly journals A NOTE ON GAUSSIAN INTEGRALS OVER PARA-GRASSMANN VARIABLES

2004 ◽  
Vol 19 (11) ◽  
pp. 1705-1714 ◽  
Author(s):  
LETICIA F. CUGLIANDOLO ◽  
G. S. LOZANO ◽  
E. F. MORENO ◽  
F. A. SCHAPOSNIK
Keyword(s):  
Quadratic Form ◽  
Path Integral ◽  
Quadratic Forms ◽  
Disordered Systems ◽  
Commutation Relations ◽  
Deformed Commutation Relations ◽  
Grassmann Variables

We discuss the generalization of the connection between the determinant of an operator entering a quadratic form and the associated Gaussian path-integral valid for Grassmann variables to the para-Grassmann case [θp+1=0 with p=1(p>1) for Grassmann (para-Grassmann) variables]. We show that the q-deformed commutation relations of the para-Grassmann variables lead naturally to consider q-deformed quadratic forms related to multiparametric deformations of GL (n) and their corresponding q-determinants. We suggest a possible application to the study of disordered systems.

scholarly journals GENERALIZED DEFORMED PARA-BOSE OSCILLATOR AND NONLINEAR ALGEBRAS

1995 ◽  
Vol 10 (36) ◽  
pp. 2739-2748
Author(s):  
HA HUY BANG
Keyword(s):  
Fock Space ◽  
Single Mode ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
Special Cases ◽  
Deformed Commutation Relations

Generalized deformed commutation relations for a single mode para-Bose oscillator and for a system of two para-Bose oscillators are constructed. It turns out that generalized deformed para-Bose oscillators are not, in general, exactly independent. Furthermore, we also discuss about the Fock space corresponding to generalized deformed para-Bose oscillators. Finally, we show how SU(2) and SU(1, 1) generators can be constructed in terms of generalized deformed para-Bose creation and annihilation operators. The algebras SU(2) and SU(1, 1) of generalized deformed oscillators14,18 are the special cases of generalized deformed para-Bose oscillators algebras but, interestingly, they have the same form.

On Heisenberg's Uncertainty Principle and the CCR

1993 ◽  
Vol 48 (3) ◽  
pp. 447-451 ◽  
Author(s):  
Reinhard Honegger
Keyword(s):  
Uncertainty Principle ◽  
Heisenberg Uncertainty Principle ◽  
Commutation Relations ◽  
Integrable Functions ◽  
Heisenberg Uncertainty ◽  
Heisenberg’S Inequality ◽  
Fock Representations ◽  
Heisenberg's Uncertainty Principle ◽  
Square Integrable Functions ◽  
Rigorous Mathematical Analysis

Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.

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