Canonical brackets of a toy model for the Hodge theory without its canonical conjugate momenta

TL;DR

This paper demonstrates deriving canonical brackets for a toy Hodge theory model, specifically a 1D rigid rotor, without using canonical conjugate momenta, relying solely on symmetry principles and basic quantum concepts.

Contribution

It introduces a novel method to obtain canonical brackets in Hodge theory models without defining conjugate momenta, expanding the toolkit for analyzing such theories.

Findings

Canonical brackets derived without conjugate momenta

Method relies on symmetry principles and basic quantum concepts

Applicable to a class of tractable Hodge theory models

Abstract

We consider the toy model of a rigid rotor as an example of the Hodge theory within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism and show that the internal symmetries of this theory lead to the derivation of canonical brackets amongst the creation and annihilation operators of the dynamical variables where the definition of the canonical conjugate momenta is not required. We invoke only the spin-statistics theorem, normal ordering and basic concepts of continuous symmetries (and their generators) to derive the canonical brackets for the model of a one (0 + 1)-dimensional (1D) rigid rotor without using the definition of the canonical conjugate momenta anywhere. Our present method of derivation of the basic brackets is conjectured to be true for a class of theories that provide a set of tractable physical examples for the Hodge theory.