Deformation Quantization: Twenty Years After

Details Deformation Quantization: Twenty Years After Deformation Quantization: Twenty Years After

1998-09-10

arxiv.org/abs/math/9809056v1

AIP Conf.Proc.453:107-145,1998

Mathematics

Quantum Algebra

39 pages; to be published with AIP Press in Proceedings of the 1998 Lodz conference "Particles, Fields and Gravitation". LaTeX

Scientific paper

10.1063/1.57093

We first review the historical developments, both in physics and in mathematics, that preceded (and in some sense provided the background of) deformation quantization. Then we describe the birth of the latter theory and its evolution in the past twenty years, insisting on the main conceptual developments and keeping here as much as possible on the physical side. For the physical part the accent is put on its relations to, and relevance for, "conventional" physics. For the mathematical part we concentrate on the questions of existence and equivalence, including most recent developments for general Poisson manifolds; we touch also noncommutative geometry and index theorems, and relations with group theory, including quantum groups. An extensive (though very incomplete) bibliography is appended and includes background mathematical literature.

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