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Journal of Geometry and Symmetry in Physics

Wigner Functions and Weyl Operators on the Euclidean Motion Group

Marin Drinov Academic Publishing House, Laarni B. Natividad, Job A. Nable

Published Jan 1, 2019• 2 citations
Publisher

Abstract

The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl–Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.

DOI
10.7546/jgsp-54-2019-37-54

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The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl–Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.
The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the ⋆\star⋆-product of phase space functions. In this article, the ⋆\star⋆-product of functions on the Euclidean motion group of rank three, E(3)\mathrm{E}(3)E(3), is constructed. C∗C^*C∗-algebra properties of ⋆s\star_s⋆s​ on E(3)\mathrm{E}(3)E(3) are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.
The objective of this work is to present some aspects of the phase space representation of quantum mechanics on the Euclidean Motion Group of rank three. In particular, the main contribution of this paper is the statement and proof of fundamental formula relating the Weyl transform for the group and the Wigner functions on the group, given by 〈F1|WkF2〉=∫R3∫so(3)k(c,C)WE(3)(F1,F2)dc dC. Here, Wk is the Weyl transform of the function k on E(3) and WE(3) is the Wigner transform. Thus, the various phase space objects such as the Wigner transform and the Weyl transform are also explicitly formulated on the Euclidean Motion Group.
This paper is devoted to the construction and analysis of the Wigner functions for noncommutative quantum mechanics, their marginal distributions, and star-products, following a technique developed earlier, viz, using the unitary irreducible representations of the group GNC, which is the three fold central extension of the Abelian group of ℝ4. These representations have been exhaustively studied in earlier papers. The group GNC is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom. The Wigner functions studied here reflect different levels of non-commutativity—both the operators of position and those of momentum not commuting, the position operators not commuting and finally, the case of standard quantum mechanics, obeying the canonical commutation relations only.