09-122 Asao Arai

Representations of a Quantum Phase Space with General Degrees of Freedom (239K, PDF) 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 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.

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