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Physica A Statistical Mechanics and its Applications

Weyl's association, Wigner's function and affine geometry

Nandor L. Balazs

Published Jul 1, 1980• 31 citations
Publisher
DOI
10.1016/0378-4371(80)90134-x

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In the context of the Wigner–Weyl phase space formulation of quantum mechanics, a version of the uncertainty relations invariant under affine canonical transformations is derived. For a fixed Wigner distribution function possessing a finite covariance, ‘‘directions’’ of minimal uncertainty are found. The geometry of the Wigner ellipsoid and its Legendre transform, the dual Wigner ellipsoid, both of which are associated with the same covariance, is discussed. The results obtained are generalizations of the well-known fact that, for one degree of freedom, the area of the Wigner ellipse must be of order ℏ or larger. Instead of area, which is an invariant only when n=1, these results involve Poincaré invariants of certain curves and surfaces.
Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three-parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.
The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s . The representations of the central extension of the affine group denoted by G 0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.