Wigner–Weyl isomorphism for quantum mechanics on Lie groups

Abstract

The Wigner-Weyl isomorphism for quantum mechanics on a compact simple *

The Schwinger Representation of a Group: Concept and Applications

“…The WWI has been studied most extensively in the case of Cartesian quantum mechanics when, as mentioned in Section II, the configuration space is Q = R n and phase space is R 2n . It has been shown elsewhere that if we consider the configuration space to be a (compact simple) Lie group G, the kinematic structure of quantum mechanics shows striking new features absent in the Cartesian case, so the WWI also exhibits unexpected features [24]. Interestingly the SR of G plays a role in this context, and this will be outlined here.…”

Semiclassical dynamics of a rigid rotor: SO(3) covariant approach

“…The self-dual symbols, which arise when the same type of mapping is used both for the density operator and for the observables in order to compute average values by convoluting corresponding symbols, are usually called Wigner symbols 3 . The properties of such mappings essentially depend on the symmetry of the quantum system and on the requirements of covariance under a certain group of transformations [4][5][6][7][8][9][10][11][12][13][14]. In the simplest case of harmonic oscillators and spin-like systems, the transformation groups-Heisenberg-Weyl and SU (2), respectively-are easily identified.…”

Wigner function for the orientation state

“…However, in all these approaches the marginal distributions cannot be obtained from the Wigner function by integrating out the other variables. In [21,22] a phase space formulation is developed for quantum systems whose configuration space is a Lie group. This work takes into consideration the desired features of the standard Wigner function.…”