On the time evolution in totally constrained systems with weakly vanishing Hamiltonian

TL;DR

This paper compares the Dirac and canonical methods for describing evolution in constrained systems with weakly vanishing Hamiltonian, showing the canonical approach can describe evolution in physical time without gauge fixing, and extends the analysis to infinite dimensions.

Contribution

It demonstrates that the canonical method provides a gauge-independent description of evolution in constrained systems and extends the formalism to infinite-dimensional cases.

Findings

Canonical method describes evolution in physical time without gauge fixing.

Operator quantization leads to Schrödinger equation with Hamiltonian operator.

Extension of the formalism to infinite-dimensional systems is achieved.

Abstract

The Dirac method treatment for finite dimensional singular systems with weakly vanishing Hamiltonian leads to obtain the equations of motion in terms of parameter $\tau$. To obtain the correct equations of motion one should use gauge fixing of the form $\tau - f(t)=0$. It is shown that the canonical method leads to describe the evolution in both standard and constrained finite dimensional systems with weakly vanishing Hamiltonian in terms of the physical time $t$, without using any gauge fixing conditions. Besides the operator quantization of the these systems is investigated using the canonical method and it is shown that the evolution of the state $\Psi$ with the time $t$ is described by the Schr/"odinger equation $i\frac{\partial \Psi}{\Partial t} = {\hat H}\Psi$. The extension of this treatment to infinite dimensional systems is given.