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

TL;DR

This paper generalizes Dirac's quantization for systems with second-class constraints, applying it to quantum motion on a torus, revealing that intrinsic geometry leads to inconsistencies while extrinsic geometry provides a consistent framework.

Contribution

It introduces a self-consistent quantization approach for constrained systems and demonstrates its effectiveness for quantum motion on a torus using extrinsic geometry.

Findings

Intrinsic geometry approach conflicts with self-consistency.

Extrinsic geometry yields consistent momenta and Hamiltonian.

Intrinsic analysis is inadequate for quantum motion on a torus.

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 Schrodinger 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.