dorsal/arxiv
View SchemaFormal GNS Construction and States in Deformation Quantization
| Authors | Martin Bordemann, Stefan Waldmann |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9607019 |
| URL | https://arxiv.org/abs/q-alg/9607019 |
Abstract
In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex $C^*$-algebras: in this approach the corresponding positive linear functionals (`states') take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations as the Bargmann and Schr\"odinger representation come out correctly, both formally and in a suitable convergence scheme. For certain Hamiltonian functions (contained in the Gel'fand ideal of the positive functional) a formal solution to the time-dependent Schr\"odinger equation is shown to exist. Moreover, we show that for every K\"ahler manifold equipped with the Fedosov star product of Wick type all the classical delta functionals are positive and give rise to some formal Bargmann representation of the deformed algebra.
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"abstract": "In this paper we develop a method of constructing Hilbert spaces and the\nrepresentation of the formal algebra of quantum observables in deformation\nquantization which is an analog of the well-known GNS construction for complex\n$C^*$-algebras: in this approach the corresponding positive linear functionals\n(`states\u0027) take their values not in the field of complex numbers, but in (a\nsuitable extension field of) the field of formal complex Laurent series in the\nformal parameter. By using the algebraic and topological properties of these\nfields we prove that this construction makes sense and show in physical\nexamples that standard representations as the Bargmann and Schr\\\"odinger\nrepresentation come out correctly, both formally and in a suitable convergence\nscheme. For certain Hamiltonian functions (contained in the Gel\u0027fand ideal of\nthe positive functional) a formal solution to the time-dependent Schr\\\"odinger\nequation is shown to exist. Moreover, we show that for every K\\\"ahler manifold\nequipped with the Fedosov star product of Wick type all the classical delta\nfunctionals are positive and give rise to some formal Bargmann representation\nof the deformed algebra.",
"arxiv_id": "q-alg/9607019",
"authors": [
"Martin Bordemann",
"Stefan Waldmann"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Formal GNS Construction and States in Deformation Quantization",
"url": "https://arxiv.org/abs/q-alg/9607019"
},
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