Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications
| dc.creator | Bordemann, Martin | |
| dc.creator | Neumaier, Nikolai | |
| dc.creator | Waldmann, Stefan | |
| dc.date | 1997-11-19 | |
| dc.date.accessioned | 2026-07-07T05:57:42Z | |
| dc.date.available | 2026-07-07T05:57:42Z | |
| dc.description | 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 q-alg/9707030) we construct differential operator representations via the formal GNS construction (see q-alg/9607019). 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 Lagrangean 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 Lagrangean 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. | |
| dc.description | 29 Pages, LaTex2e | |
| dc.identifier | https://arxiv.org/abs/q-alg/9711016 | |
| dc.identifier | http://arxiv.org/abs/q-alg/9711016 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/88081 | |
| dc.subject | Quantum Algebra | |
| dc.title | Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications | |
| dc.type | text |