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

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

1998-11-09

arxiv.org/abs/math/9811055v2

Mathematics

Quantum Algebra

LaTeX2e, 38 pages, slight generalization of Theorem 4.4, minor typos corrected

Scientific paper

For every formal power series $B=B_0 + \lambda B_1 + O(\lambda^2)$ of closed two-forms on a manifold $Q$ and every value of an ordering parameter $\kappa\in [0,1]$ we construct a concrete star product $\star^B_\kappa$ on the cotangent bundle $\pi : T^*Q\to Q$. The star product $\star^B_\kappa$ is associated to the formal 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 $\star^B_\kappa$ is calculated and shown to coincide with the formal de Rham cohomology class of $\pi^*B$ divided by $\im\lambda$. Therefore, every star product on $T^*Q$ corresponding to the Poisson bracket induced by the symplectic form $\omega + \pi^*B_0$ is equivalent to some $\star^B_kappa$. It turns out that every $\star^B_kappa$ 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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