Generalized Dirac bracket and the role of the Poincar\'e symmetry in the program of canonical quantization of fields 1

TL;DR

This paper discusses the canonical quantization of constraint systems with Fermi variables, emphasizing the construction of classical brackets compatible with quantum commutation relations, and illustrates the process with the Dirac field and spinor electrodynamics.

Contribution

It introduces a generalized Dirac bracket framework and explores Poincaré invariance's role in canonical quantization of fermionic fields.

Findings

Presented a consistent classical bracket for fermionic variables

Quantized the Dirac field with marginal Poincaré invariance

Extended the method to spinor electrodynamics

Abstract

An elementary presentation of the methods for the canonical quantization of constraint systems with Fermi variables is given. The emphasis is on the subtleties of the construction of an appropriate classical bracket that could be consistently replaced by commutators or anti--commutators of operators, as required by canonical quantization procedure for bosonic and fermionic degrees of freedom respectively. I present a consequent canonical quantization of the Dirac field, in which the role of Poincar\'e invariance is made marginal. This simple example provides an introduction to the Poincar\'e--free quantization of spinor electrodynamics in the second part of the paper.