Quantizing constrained systems

We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order ${ħ}^2$ multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order ${ħ}^2$ involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics and Feynman path integrals, and semiclassically in terms of adiabatic actions.