QUANTIZING CONSTRAINED SYSTEMS

We consider quantum mechanics on constrained surfaces which have non-E uclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order (h) over bar(2) multiplying the Gaussian curvature is addressed, We set ou t to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from the ir constrained values. We find additional ambiguous terms of order (h) over bar(2) involving freedom in the constraining potentials, demonst rating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizi ng a constrained system is inherently unresolvable. Hou;e-ver, there i s never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean const raint values, The issue is addressed from the perspectives of adiabati c approximations in quantum mechanics and Feynman path integrals, and semiclassically in terms of adiabatic actions.