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arXiv:quant-ph/0004032 · Journal of Mathematical Physics

Phase space observables and isotypic spaces

Gianni Cassinelli, Ernesto De Vito, PEKKA J. LAHTI, Alberto Levrero

Published Sep 1, 2000• 9 citations
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Abstract

We give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observables to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.

DOI
10.1063/1.1286986

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We give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observables to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.
An orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group. From this, a series of lemmas is derived showing the informational completeness of natural covariant localization operators, as well as of the generalized Wigner distributions (matrix elements of the group). Some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators. The results are applied to special (phase space) representations of the Heisenberg, affine, and Galilei groups.
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
In the nonrelativistic setting with finitely many canonical degrees of freedom, a shift-covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for p = 1, 2, ∞ of a more general notion of p-regularity defined as the norm density of the span of translates of the operator in the Schatten-p class. We show that the relation between zero sets and p-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis.