Homogeneous Fedosov star products on cotangent bundles. II: GNS representations, the WKB expansion, traces, and applications. (English) Zbl 0989.53060

Summary: This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold \(Q\). For certain homogeneous star products of Weyl ordered type which we have obtained from a Fedosov type procedure in part I, see [Commun. Math. Phys. 198, 363-396 (1998; Zbl 0968.53056)] we construct differential operator representations via the formal GNS construction [see Martin Bordemann and Stefan Waldmann [Commun. Math. Phys. 195, No.3, 549-583 (1998; Zbl 0989.53057)]. The positive linear functional is integration over \(Q\) with respect to some fixed density and is shown to yield a reasonable version of the Schrödinger representation where a Weyl ordering prescription is incorporated. Furthermore we discuss simple examples like free particle Hamiltonians (defined by a Riemannian metric on \(Q)\) and the implementation of certain diffeomorphisms of \(Q\) to unitary transformations in the GNS (pre-)Hilbert space and of time reversal maps (involutive anti-symplectic diffeomorphisms of \(T^*Q)\) to anti-unitary transformations. We show that the fixed-point set of any involutive time reversal map is either empty or a Lagrangian submanifold. Moreover, we compare our approach to concepts using integral formulas of generalized Moyal-Weyl type. Furthermore we show that the usual WKB expansion with respect to a projectable Lagrangian submanifold can be formulated by a GNS construction. Finally we prove that any homogeneous star product on any cotangent bundle is strongly closed, i.e., the integral over \(T^*Q\) w.r.t. the symplectic volume vanishes on star-commutators. An alternative Fedosov type deduction of the star product of standard ordered type using a deformation of the algebra of symmetric contravariant tensor fields is given.


53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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