Strict deformation quantization of a particle in external gravitational and Yang-Mills fields (Q689438)

The author applies an adaptation of Rieffel's notion of ``strict deformation quantization'' 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 \({\mathcal A}_ 0 = C_ 0((T^*P)/H)\) is deformed into the \(C^*\)-algebra \({\mathcal A} = {\mathcal K}(L^ 2(P))^ H\) of \(H\)-invariant compact operators on \(L^ 2(P)\), which is isomorphic to \({\mathcal K}(L^ 2(Q)) \otimes C^*(H)\), involving the group algebra of \(H\). This deformation can be interpreted in terms of Lie groupoids and algebroids, as \({\mathcal A}_ 0\) is the \(C^*\)-algebra of the gauge groupoid of the bundle \((P,Q,H)\). There are also discussed -- from the point of view of the described formalism -- Wigner functions and the quantization of the Hamiltonian as well as position and momentum (including their domains).