Strict Deformation Quantization of a Particle in External Gravitational and Yang-mills Elds

An adaption of Rieeel's notion of`strict deformation quantization' is applied to a particle moving on an arbitrary Riemannian manifold Q in an external gauge eld, that is, a connection on a principal H-bundle P over Q. Hence the Poisson algebra A 0 = C 0 ((T P)=H) is deformed into the C-algebra A = K(L 2 (P)) H of H-invariant compact operators on L 2 (P), which is isomorphic to K(L 2 (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-deened analytic sense. This deformation can be interpreted in terms of Lie groupoids and algebroids, as A 0 is the Poisson algebra of the Lie algebroid (T P)=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).