Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 357

On representations of star product algebras over cotangent spaceson Hermitian line bundles

Martin Bordemann, Nikolai Neumaier, Markus Pflaum and Stefan Waldmann

For every formal power series $B$ of closed two-forms on a manifold $Q$ and every value of an ordering parameter $kappain [0,1]$ we construct a concrete star product $starBo$ on the cotangent bundle $T^*Q$. The star product $star_order^B$ is associated to the symplectic form on $T^*Q$ given by the sum of the canonical symplectic form $omega$ and the pull-back of $B$ to $T^*Q$. Deligne's characteristic class of $star_order^B$ is calculated and shown to coincide with the formal de Rham cohomology class of $B$ divided by $lambda$. Therefore, every star product on $T^*Q$ corresponding to the canonical Poisson bracket is equivalent to some $starBo$. It turns out that every $starBo$ is strongly closed. In this paper we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on $Q$. Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion.

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