Highly Irregular Quantum Constraints

Motivated by a recent paper of Louko and Molgado, we consider a simple system with a single classical constraint R(q) = 0. If ql denotes a generic solution to R(q) = 0, our examples include cases where R(ql) 6= 0 (regular constraint) and R(ql) = 0 (irregular constraint) of varying order as well as the case where R(q) = 0 for an interval, such as a ≤ q ≤ b. Quantization of irregular constraints is normally not considered; however, using the projection operator formalism we provide a satisfactory quantization which reduces to the constrained classical system when ~ → 0. It is noteworthy that irregular constraints change the observable aspects of a theory as compared to strictly regular constraints.