Can We Look at The Quantisation Rules as Constraints?

TL;DR

This paper investigates the possibility of interpreting Dirac quantisation conditions as constraints within phase-space, aiming to connect classical mechanics constraints with quantum evolution, but encounters challenges in implementation.

Contribution

It proposes a novel perspective of viewing quantisation rules as phase-space constraints and applies Dirac's theory to explore their implications in quantum mechanics.

Findings

Attempted to derive the Moyal operator as a total Hamiltonian failed.

Identified a conceptual gap in applying Dirac constraints to quantisation rules.

Suggested directions for future research to address the identified issues.

Abstract

In this paper we explore the idea of looking at the Dirac quantisation conditions as $\hbar$-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure.