Real Ashtekar Variables for Lorentzian Signature Space-times

TL;DR

This paper proposes a new formulation of Ashtekar variables that describes Lorentzian space-times without complex variables, simplifying the reality conditions in quantum gravity.

Contribution

It introduces a modified Hamiltonian constraint within the $SO(3)$ Yang-Mills phase space to handle Lorentzian signatures without complex variables.

Findings

Retains geometrical features of Ashtekar formalism

Allows use of loop variables in quantum theory

Avoids complex variables in Lorentzian space-times

Abstract

I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual $SO(3)$ Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum theory. The key issue in the new formulation is how to deal with the more complicated Hamiltonian constraint that must be used in order to avoid the introduction of complex fields.