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

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.