Highly Irregular Quantum Constraints

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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)≠ 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 ℏ → 0. It is noteworthy that irregular constraints change the observable aspects of a theory as compared to strictly regular constraints.