Symplectic Quantization and General Constraint Structure of a Prototypical Second-Class System

TL;DR

This paper analyzes a general constrained Hamiltonian system with second-class constraints, deriving its algebraic structure and exploring applications in quantum field theory, cosmology, and Lorentz-violating models.

Contribution

It provides a comprehensive application of the Dirac-Bergmann and Faddeev-Jackiw formalisms to a prototypical second-class system, including explicit examples and potential cosmological implications.

Findings

Derived the Dirac brackets algebra for the system.

Applied the formalism to toroidal geometry and Lorentz-violating models.

Suggested cosmological applications based on the geometric analysis.

Abstract

We discuss a general prototypical constrained Hamiltonian system with a broad application in quantum field theory and similar contexts where dynamics is defined through a functional action obeying a stationarity principle. The prototypical model amounts to a Dirac-Bergmann singular system, whose constraints restrict the actual dynamics to occur within a differential submanifold, as is the case in the major part of field theoretical models with gauge symmetry. We apply the Dirac-Bergmann algorithm in its full generality unraveling a total of $4m$ second-class constraints and obtain the corresponding Dirac brackets algebra in phase space. We follow with the Faddeev-Jackiw-Barcelos-Wotzasek approach in which the geometric character of the mentioned submanifold is emphasized by means of an internal metric function encoding its symplectic properties. We consider two straightforward examples, applying our general results to constrained motion along a toroidal geometry and to a Lorentz violating toy model in field theory. Since toroidal geometry has been recently used in cosmological models, we suggest how our results could lead to different proposals for the shape of the universe in cosmology.