# Conic Constrained Particle Quantization within the DB, FJBW and BRST Approaches

## TL;DR

This paper investigates the quantization of a conic constrained particle, comparing Dirac-Bergmann and Faddeev-Jackiw methods, and constructs a gauge-invariant model with explicit BRST symmetry, revealing broader implications for constrained systems.

## Contribution

It provides a detailed comparison of Dirac-Bergmann and Faddeev-Jackiw quantization approaches for a conic constrained particle, and constructs a BRST-invariant gauge model generalizing known symmetries.

## Key findings

- FJBW reduces second class constraints from four to two.
- Constructed a gauge-invariant model with explicit BRST symmetry.
- Reproduced and generalized the BRST symmetry of the rigid rotor.

## Abstract

We consider a second degree algebraic curve describing a general conic constraint imposed on the motion of a massive spinless particle. The problem is trivial at classical level but becomes involved and interesting in its quantum counterpart with subtleties in its symplectic structure and symmetries. The model is used here to investigate quantization issues related to the Hamiltonian constraint structure, Dirac brackets, gauge symmetry and BRST transformations. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelos-Wotzasek iteration program to unravel the fine tuned and more relevant aspects of the constraint structure. A comparison with the longer usual Dirac-Bergmann algorithm, still more well established in the literature, is also presented. While in the standard DB approach there are four second class constraints, in the FJBW they reduce to two. By using the symplectic potential obtained in the last step of the FJBW iteration process we construct a gauge invariant model exhibiting explicitly its BRST symmetry. Our results reproduce and neatly generalize the known BRST symmetry of the rigid rotor showing that it constitutes a particular case of a broader class of theories.