Pseudoclassical system with gauge and time-reparametrization invariance

TL;DR

This paper introduces a pseudoclassical model with gauge and reparametrization invariance, analyzing its constraints, gauge fixing, and quantization, revealing that the secondary first-class constraint is not a symmetry generator.

Contribution

It demonstrates that the Dirac conjecture does not hold in this model and provides a detailed quantization approach using deformation and Schrödinger methods.

Findings

Secondary first-class constraint is not a symmetry generator

Gauge fixing removes gauge ambiguities with only primary constraint

Quantization yields a positive energy state

Abstract

We present a pseudoclassical mechanics model which exhibits gauge symmetry and time-reparametrization invariance. As such, first- and second-class constraints restrict the phase space, and the Hamiltonian weakly vanishes. We show that the Dirac conjecture does not hold -- the secondary first-class constraint is not a symmetry generator -- and only the gauge fixing condition associated with the primary first-class constraint is needed to remove the gauge ambiguities. The gauge fixed theory is equivalent to the Fermi harmonic oscillator extended by a boundary term. We quantize in the deformation quantization and in the Schrodinger representation approaches and observe that the boundary term prepares the system in the state of positive energy.