Strict deformation quantization of a particle in external gravitational and Yang-Mills fields

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 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-defined 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 ( 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).