Asao Arai
Representations of a Quantum Phase Space with General Degrees of Freedom
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ABSTRACT. For each integer $n\geq 2$ and a parameter $\Lambda=(\theta,\eta)$ with $\theta$ and $\eta$ being
$n\times n$ real anti-symmetric matrices, a quantum phase space (QPS) (or a non-commutative phase space) with $n$ degrees of freedom, denoted ${\rm QPS}_n(\Lambda)$,
is defined, where $\theta$ and $\eta$
are parameters measuring non-commutativity of the QPS.
Hilbert space representations of $\QPS_n(\Lambda)$ are considered.
A concept of quasi-Schr\"odinger representation of $\QPS_n(\Lambda)$ is introduced. It is shown that there exists a general correspondence between representations of $\QPS_n(\Lambda) $ and those of the canonical commutation relations with $n$ degrees of freedom. Irreducibility of representations of $\QPS_n(\Lambda)$
are investigated. A concept of
Weyl representation of $\QPS_n(\Lambda)$ is defined.
It is proved that every Weyl representation of $\QPS_n(\Lambda)$
on a separable Hilbert space is unitarily equivalent to a direct sum of
a quasi-Schr\"odinger representation of the $\QPS_n(\Lambda)$ (a uniqueness theorem).
Finally representations of $\QPS_n(\Lambda)$ which are not unitarily equivalent to any
direct sum of a quasi-Schr\"odinger representation are described.