1212.6373 (D. M. Xun et al.)

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

D. M. Xun, Q. H. Liu, X. M. Zhu

The present formalism of Dirac's theory for a system of second-class constraints can be well formulated either within purely intrinsic geometric framework or beyond, but whatever framework is taken, the results are not well compatible with those given by the Schr\"odinger's theory. A generalization of the Dirac's theory is made recently (Phys. Rev. A 84, 042101(2011)) to include the commutation relations [f,H], where f= position x_{i}, momentum p_{i} and Hamiltonian H, into the formalism as the second category of the fundamental ones. Through a careful analysis of the quantum motion on a torus, we demonstrate that the purely intrinsic geometry does not suffice for the Dirac's theory to be self-consistent, but an extrinsic examination of the torus in three dimensional flat space does. Thus the Dirac's theory turns out to be complementary to the Schr\"odinger's one. The former eliminates the intrinsic description, and the latter gives the unique form of the geometric potential, while both define the identical form of the geometric momentum.

View original: http://arxiv.org/abs/1212.6373