dorsal/arxiv
View SchemaHomogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications
| Authors | Martin Bordemann, Nikolai Neumaier, Stefan Waldmann |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9711016 |
| URL | https://arxiv.org/abs/q-alg/9711016 |
Abstract
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 Schr\"odinger 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.
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"abstract": "This paper is part II of a series of papers on the deformation quantization\non the cotangent bundle of an arbitrary manifold $Q$. For certain homogeneous\nstar products of Weyl ordered type (which we have obtained from a Fedosov type\nprocedure in part I, see q-alg/9707030) we construct differential operator\nrepresentations via the formal GNS construction (see q-alg/9607019). The\npositive linear functional is integration over $Q$ with respect to some fixed\ndensity and is shown to yield a reasonable version of the Schr\\\"odinger\nrepresentation where a Weyl ordering prescription is incorporated. Furthermore\nwe discuss simple examples like free particle Hamiltonians (defined by a\nRiemannian metric on $Q$) and the implementation of certain diffeomorphisms of\n$Q$ to unitary transformations in the GNS (pre-)Hilbert space and of time\nreversal maps (involutive anti-symplectic diffeomorphisms of $T^*Q$) to\nanti-unitary transformations. We show that the fixed-point set of any\ninvolutive time reversal map is either empty or a Lagrangean submanifold.\nMoreover, we compare our approach to concepts using integral formulas of\ngeneralized Moyal-Weyl type. Furthermore we show that the usual WKB expansion\nwith respect to a projectable Lagrangean submanifold can be formulated by a GNS\nconstruction. Finally we prove that any homogeneous star product on any\ncotangent bundle is strongly closed, i. e. the integral over $T^*Q$ w.r.t. the\nsymplectic volume vanishes on star-commutators. An alternative Fedosov type\ndeduction of the star product of standard ordered type using a deformation of\nthe algebra of symmetric contravariant tensor fields is given.",
"arxiv_id": "q-alg/9711016",
"authors": [
"Martin Bordemann",
"Nikolai Neumaier",
"Stefan Waldmann"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications",
"url": "https://arxiv.org/abs/q-alg/9711016"
},
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