Journal of Mathematical Physics

A Continuous Representation of an Indefinite Metric Space

M. G. Gundzik

Published Apr 1, 1966• 6 citations

Abstract

An overcomplete family of states (OFS) is constructed for a countably infinite linear vector space with an indefinite metric for the case that the metric is diagonal with eigenvalues (−1)n, where n is an integer. A continuous representation is indicated and the properties of a semiclassical description of a quantum mechanical system (the pseudo oscillator whose creation and destruction operators ā and ā+ satisfy [ā, ā+] = −1) defined in this vector space are studied. It is found that a consistent OFS |z> can be constructed if the operator G(z) which generates the state |z> from the vacuum is unitary. Furthermore, with the statistical state of this system specified by a bounded pseudo-Hermitian density matrix ρ̄, the related semiclassical complex function ρA(z) for antinormal ordering of operators in the indefinite metric space is found to be bounded, with ρA(z) and [ρA(z)]2 integrable, continuous, and a boundary value of an entire analytic function of two complex variables. The semiclassical function ρN(z) for normal ordering is associated with a sequence of functions ρN(ν)(z) whose square is integrable and related to a sequence of tempered distributions ρN(ν) such that the corresponding sequence of density matrices ρ̄(ν) converges to ρ̄ in the norm.

DOI: 10.1063/1.1704977

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