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

TL;DR

This paper constructs and classifies star product representations on cotangent bundles over Hermitian line bundles, linking formal deformation quantization with pseudodifferential operator calculus and providing explicit formulas for WKB expansions.

Contribution

It introduces explicit constructions of star products on cotangent bundles associated with closed two-forms and classifies their representations using a global symbol calculus.

Findings

Calculated Deligne's characteristic class for the star products.

Showed equivalence of star products with certain Poisson brackets.

Derived a compact formula for the WKB expansion.

Abstract

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.