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

摘要

For every formal power series B of closed two-forms on a manifold Q and every value of an ordering parameter K is an element of [0, 1] we construct a concrete star product star(K)(B) on the cotangent bundle T*Q. The star product star(K)(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(K)(B) is calculated and shown to coincide with the formal de Rham cohomology class of pi*B divided by ilambda. Therefore, every star product on T*Q corresponding to the canonical Poisson bracket is equivalent to some star(K.)(B) It turns out that every star(K)(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. (C) 2003 Elsevier Science (USA). All rights reserved. [References: 28]