 09122 Asao Arai
 Representations of a Quantum Phase Space with General Degrees of Freedom
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Jul 26, 09

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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 antisymmetric matrices, a quantum phase space (QPS) (or a noncommutative phase space) with $n$ degrees of freedom, denoted ${\rm QPS}_n(\Lambda)$,
is defined, where $\theta$ and $\eta$
are parameters measuring noncommutativity of the QPS.
Hilbert space representations of $\QPS_n(\Lambda)$ are considered.
A concept of quasiSchr\"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 quasiSchr\"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 quasiSchr\"odinger representation are described.
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