On Hamiltonian formulations of the Dirac system

TL;DR

This paper extends the Hamiltonian formulation of the Dirac system using the Dirac--Bergmann algorithm, deriving constraints, brackets, and quantization procedures for the Dirac field in various formalisms.

Contribution

It provides a detailed Hamiltonian analysis of the Dirac field, including constraints and brackets, and proposes a canonical second quantization method for different formulations.

Findings

Derivation of second class constraints for the Dirac field.

Construction of Dirac brackets on the full and reduced phase spaces.

Proposal of a second quantization recipe yielding correct anticommutators.

Abstract

We extend a previously successful discussion of the constrained Schr\"{o}dinger system through the Dirac--Bergmann algorithm to the case of the Dirac field. In order to follow the analogy, first we discuss the classical Dirac field as a spinorial variable, by introducing properly defined momenta and a suitably modified, factor ordered Poisson bracket. According to the Dirac--Bergmann algorithm two second class Hamiltonian constraints emerge, leading to a factor ordered Dirac bracket on the full phase space. This becomes the Poisson bracket on the reduced phase space in the canonical chart adapted to the shell. The Dirac equation is recovered both as consistency condition on the full phase space and as canonical equation on the reduced phase space. Alternatively, considering the Dirac field as odd Grassmann variable, we present the details of the Dirac--Bergmann algorithm (with either left and righ derivatives acting on Grassmann valued superfunctions and involving a different type of generalized Poisson and Dirac brackets). We propose a recipe for the canonical second quantization of all three versions of the generalized Dirac brackets, yielding the correct fundamental anticommutator.