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On representations of star product algebras over cotangent spaces on Hermitian line bundles. (English) Zbl 1038.53087
From the authors’ abstract: For every formal power series \(B\) of closed two-forms on a manifold \(Q\) and every value of an ordering parameter from \([0,1]\) we construct a concrete star product \(B \) on the cotangent bundle \(T^{*}Q\). The star product \(B\) is associated to the symplectic form on \(T^{*}Q\) given by the sum of the canonical symplectic form and the pull back of \(B\) to \(T^{*}Q\). Deligne’s characteristic class of \(B\) is calculated and shown to coincide with the formal de Rham cohomology class of \(^{*}B \) divided by \(i\). Therefore, every star product on \(T^{*}Q\) corresponding to the canonical Poisson bracket is equivalent to some \(B\). It turns out that every \(B\) 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.

MSC:
53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
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