Algebras of distributions suitable for phase‐space quantum mechanics. II. Topologies on the Moyal algebra
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Várilly Boyle, Joseph C.
Gracia Bondía, José M.
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Abstract
The topology of the Moyal *-algebra may be defined in three ways: the
algebra may be regarded as an operator algebra over the space of
smooth declining functions either on the configuration space or on the
phase space itself; or one may construct the *-algebra via a
filtration of Hilbert spaces (or other Banach spaces) of
distributions. We prove the equivalence of the three topologies
thereby obtained. As a consequence, by filtrating the space of
tempered distributions by Banach subspaces, we give new sufficient
conditions for a phase-space function to correspond to a trace-class
operator via the Weyl correspondence rule.
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Keywords
Quantum mechanics in phase space, Tempered distributions, Locally convex spaces
Citation
https://aip.scitation.org/doi/10.1063/1.527984