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Title:
Strict deformation quantization of a particle in external gravitational and Yang-Mills fields
Authors:
Landsman, N. P.
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom
Publication:
Journal of Geometry and Physics, Volume 12, Issue 2, p. 93-132.
Publication Date:
08/1993
Origin:
ELSEVIER
Keywords:
deformation quantization, Poisson algebras, non-commutative geometry, 46 N 50, 81 S 05, 81 S 30, 02.40.+m, 03.65.Db.
Abstract Copyright:
(c) 1993 Elsevier Science B.V. All rights reserved.
DOI:
10.1016/0393-0440(93)90010-C
Bibliographic Code:
1993JGP....12...93L

Abstract

An adaptation of Rieffel's notion of ``strict deformation quantization'' is applied to a particle moving on an arbitrary Riemannian manifold Q in an external gauge field, that is, a connection on a principal H-bundle P over Q. Hence the Poisson algebra A0 = C0 ((T*P)/H) is deformed into the C*-algebra A = K (L2 (P))H of H-invariant compact operators on L2 (P), which is isomorphic to K(L2 (Q)) ⊗ C* (H), involving the group algebra of H. Planck's constant h¯ is a genuine number rather than a formal expansion parameter, and in the limit h¯ --> 0 commutators and anti-commutators converge to Poisson brackets and pointwise products, respectively, in a well-defined analytic sense. This deformation can be interpreted in terms of Lie groupoids and algebroids, as A0 is the Poisson algebra of the Lie algebroid (TP)/H, whereas A is the C*-algebra of the gauge groupoid of the bundle (P, Q, H. Other topics we discuss from the point of view of our formalism are Wigner functions, and the quantization of the Hamiltonian as well as position and momentum (including their domains).

Supported by a SERC Advanced Research Fellowship.

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