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A generalization of the Dirac's canonical quantization theory for a systemwith second-class constraints is proposed as the fundamental commutationrelations that are constituted by all commutators between positions, momentaand Hamiltonian so they are simultaneously quantized in a self-consistentmanner, rather than by those between merely positions and momenta so the theoryeither contains redundant freedoms or conflicts with experiments. Theapplication of the generalized theory to quantum motion on a torus leads to tworemarkable results: i) The theory formulated purely on the torus, i.e., basedon the so-called the purely intrinsic geometry, conflicts with itself. So itprovides an explanation why an intrinsic examination of quantum motion on toruswithin the Schrodinger formalism is improper. ii) An extrinsic examination ofthe torus as a submanifold in three dimensional flat space turns out to beself-consistent and the resultant momenta and Hamiltonian are satisfactory allaround.

Quantum motion on a torus as a submanifold problem in a generalized Dirac's theory of second-class constraints