Para-Generalization of Peierls Bracket Quantization

TL;DR

This paper extends the Peierls quantization scheme to classical and quantum systems with parafermionic and parabosonic variables using a new parabracket formalism, unifying their relations.

Contribution

It introduces a parabracket formalism for parafermionic and parabosonic systems and demonstrates how Peierls quantization applies to these generalized variables.

Findings

Development of a parabracket formalism for parafermionic and parabosonic variables

Application of Peierls quantization to systems with paracommutation relations

Derivation of kinetic terms in the Lagrangian for such systems

Abstract

A convenient formalism is developed to treat classical dynamical systems involving $(p=2)$ parafermionic and parabosonic dynamical variables. This is achieved via the introduction of a parabracket which summarizes the paracommutation relations of the corresponding Green components in a unified manner. Furthermore, it is shown that Peierls quantization scheme may be applied to such systems provided that one uses the above mentioned parabracket to express the quantum paracommutation relations. Application of the Peierls scheme also provides the form of the parafermionic and parabosonic kinetic terms in the Lagrangian.