Quantum smooth boundary forces from constrained geometries

We implement the so-called Weyl-Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy boundary, a position-dependent mass, and an extra potential on the quantum level. The consistency of our quantization is discussed by analyzing the semi-classical phase space portrait of the derived quantum dynamics, which is obtained as a regularization of its original classical counterpart.