Reality Conditions for Lorentzian and Euclidean Gravity in the Ashtekar Formulation

TL;DR

This paper investigates the reality conditions in Lorentzian and Euclidean gravity using Ashtekar variables, showing their invariance under Wick rotation and how they lead to different quantum theories with distinct physical implications.

Contribution

It demonstrates the invariance of reality conditions under Wick rotation and reveals that Euclidean and Lorentzian formulations yield inequivalent quantum theories.

Findings

Reality conditions are invariant under Wick rotation.

Conformal factor is restricted to be real or purely imaginary.

Lorentzian and Euclidean quantizations are inequivalent.

Abstract

Using Ashtekar variables, we analyze Lorentzian and Euclidean gravity in vacuum up to a constant conformal transformation. We prove that the reality conditions are invariant under a Wick rotation of the time, and show that the compatibility of the algebra of commutators and constraints with the involution defined by the reality conditions restricts the possible values of the conformal factor to be either real or purely imaginary. In the first case, one recovers real Lorentzian general relativity. For purely imaginary conformal factors, the classical theory can be interpreted as real Euclidean gravity. The reality conditions associated with this Euclidean theory demand the hermiticity of the Ashtekar connection, but the densitized triad is represented by an anti-Hermitian operator. We also demonstrate that the Euclidean and Lorentzian sets of reality conditions lead to inequivalent quantizations of full general relativity. As a consequence, it seems impossible to obtain Lorentzian physical predictions from the quantum theory constructed with the Euclidean reality conditions.