On representations of star product algebras over cotangent spaces on Hermitian line bundles

Abstract

For every formal power series B of closed two-forms on a manifold Q and every value of an ordering parameter κ∈[0,1] we construct a concrete star product ★ κ B on the cotangent bundle T ∗Q . The star product ★ κ B is associated to the symplectic form on T ∗Q given by the sum of the canonical symplectic form ω and the pull back of B to T ∗Q . Deligne's characteristic class of ★ κ B is calculated and shown to coincide with the formal de Rham cohomology class of π ∗B divided by iλ. Therefore, every star product on T ∗Q corresponding to the canonical Poisson bracket is equivalent to some ★ κ B . It turns out that every ★ κ B 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.