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

Abstract

A generalization of the Dirac's canonical quantization theory for a system with second-class constraints is proposed as the fundamental commutation relations that are constituted by all commutators between positions, momenta and Hamiltonian so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta so the theory either contains redundant freedoms or conflicts with experiments. The application of the generalized theory to quantum motion on a torus leads to two remarkable results: i) The theory formulated purely on the torus, i.e., based on the so-called the purely intrinsic geometry, conflicts with itself. So it provides an explanation why an intrinsic examination of quantum motion on torus within the Schrödinger formalism is improper. ii) An extrinsic examination of the torus as a submanifold in three dimensional flat space turns out to be self-consistent and the resultant momenta and Hamiltonian are satisfactory all around.

Generally covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere

“…So, for a Dirac fermion on S N −1 , the FQCs are set up and given by (13), which lead us to defining the generalized total angular momentum J in quantum mechanics. When quantizing a classical system, we put symmetries on the top priority: [4,5,7,20,21] Our philosophy is: The symmetry expressed by the Poisson or Dirac brackets in classical mechanics preserves in quantum mechanics; and so the Hamiltonian is determined by the symmetry. It can be considered a specific demonstration of the fundamental philosophical idea stating that symmetry dictates interactions in quantum mechanics [22].…”

Distribution of x ⋅ p for quantum states on a circle

“…In fact, the momenta p x and p y are special case of the so-called geometric momentum p = −i (∇ surf + M n/2) [6] on an N -dimensional surface which is embedded in (N + 1)-dimensional Euclidean space, where ∇ surf is the gradient operator on surface, and M is the mean curvature and n is the normal vector [6]. The geometric momentum is recently intensively studied [8][9][10][11][12][13][14][15][16].…”

The centripetal force law and the equation of motion for a particle on a curved hypersurface

“…It may not be a shortcoming, though. Instead, the over-description has the remarkable advantage to include the results predicted by the the confining potential technique [2,3,5,16,17,20,[23][24][25]. Thus, from the point of view of the operator algebra, a complete formulation of the quantization of the constrained motion is still an open problem [4,26].…”