Reality conditions for (2+1)-dimensional gravity coupled with the Dirac field

TL;DR

This paper analyzes the canonical structure of (2+1)-dimensional gravity coupled with the Dirac field, introducing complex variables and examining reality conditions to ensure equivalence with the real theory.

Contribution

It presents a novel canonical formalism using complex variables and clarifies the reality conditions needed for consistent coupling of gravity and Dirac fields in three dimensions.

Findings

Simplified Dirac brackets using complex variables

Identified reality conditions for equivalence between complex and real theories

Analyzed the canonical structure of the coupled system

Abstract

The canonical formalism of three dimensional gravity coupled with the Dirac field is considered. We introduce complex variables to simplify the Dirac brackets of canonical variables and examine the canonical structure of the theory. We discuss the reality conditions which guarantee the equivalence between the complex and real theory.