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Modern Physics Letters A

ON THE REPRESENTATION OF THE WIGNER ALGEBRA AND WIGNER DEFORMATION OF THE FERMION ALGEBRA

Won Sang Chung

Published May 30, 2013• 2 citations
Publisher

Abstract

In this paper, we investigate the representation of the Wigner algebra by using the q-calculus like approach. For this algebra, the coherent state is constructed and the uncertainty relation for the ground state is discussed. Following Wigner's approach, the Wigner deformation of the fermion algebra is obtained and its Fock space is shown to have finite dimension.

DOI
10.1142/s0217732313500740

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In this paper, we investigate the representation of the Wigner algebra by using the q-calculus like approach. For this algebra, the coherent state is constructed and the uncertainty relation for the ground state is discussed. Following Wigner's approach, the Wigner deformation of the fermion algebra is obtained and its Fock space is shown to have finite dimension.
Abstract In this paper we introduce two types of q‐deformed Wigner algebras. One is q‐deformation of Wigner algebra with q‐reflection symmetry and another is q‐deformation of Wigner algebra with reflection symmetry. For two types of q‐deformed Wigner algebras, we investigate the representation and the eigenvalue of the position operator. Like q‐calculus, we introduce the (q; ν)‐ numbers, (q; ν)‐derivatives and (q; ν)‐Hermite polynomials for two algebras. For the deformation parameter q = 1 ‐ ϵ with small ϵ, we discuss the thermodynamics of the particle obeying the q‐deformed Wigner algebra.
Schwinger's finite ( D ) dimensional periodic Hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in is studied and, based on the admissibility of the underlying q -oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.