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Journal of Physics A Mathematical and Theoretical

A generalized Wigner function on the space of irreducible representations of the Weyl–Heisenberg group and its transformation properties

Alberto Ibort, Vladimir I. Man’ko, Giuseppe Marmo, Alberto Simoni, F. Ventriglia

Published Mar 20, 2009• 14 citations
Publisher

Abstract

A natural extension of the Wigner function to the space of irreducible unitary representations of the Weyl–Heisenberg group is discussed. The action of the automorphisms group of the Weyl–Heisenberg group onto Wigner functions and their generalizations and onto symplectic tomograms is elucidated. Some examples of physical systems are considered to illustrate some aspects of the characterization of the Wigner functions as solutions of differential equations.

DOI
10.1088/1751-8113/42/15/155302

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Journal of Physics Conference Series•Mar 1, 2011•1 citation
A natural extension of the Wigner function to the space of irreducible unitary representations of the Weyl–Heisenberg group is discussed. The action of the automorphisms group of the Weyl–Heisenberg group onto Wigner functions and their generalizations and onto symplectic tomograms is elucidated. Some examples of physical systems are considered to illustrate some aspects of the characterization of the Wigner functions as solutions of differential equations.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
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 Heisenberg-Weyl group HW(d)HW(d)HW(d) related to a ddd-dimensional Hilbert space H(d)H(d)H(d), is enlarged into the Heisenberg-Weyl-parity group HWP(d)HWP(d)HWP(d) that incorporates parity transformations. It consists of 2d32d^32d3 elements, of which d3d^3d3 elements belong to the HW(d)HW(d)HW(d) subgroup, and extra d3d^3d3 elements which are related through a Fourier transform with the former ones. It is shown that HWP(d)HWP(d)HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. HWP(d)HWP(d)HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.2d22d^22d2 coherent states related to the HWP(d)HWP(d)HWP(d) group are introduced, which consist of d2d^2d2 coherent states related to the HW(d)HW(d)HW(d) subgroup, and extra d2d^2d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2d22d^22d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d2d^2d2 coherent states related to HW(d)HW(d)HW(d). One of the consequences of the HWP(d)HWP(d)HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.
The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.