From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier

TL;DR

This paper introduces a positive complexifier that enables a Wick transform from Euclidean to Lorentzian loop quantum gravity, emphasizing the role of the diffeomorphism invariant Hilbert space for quantum analysis.

Contribution

It constructs a nearly everywhere differentiable positive complexifier for classical phase space and discusses its quantum implementation within the diffeomorphism invariant Hilbert space of LQG.

Findings

Proposes a positive complexifier for Wick rotation in LQG.

Highlights the importance of the diffeomorphism invariant Hilbert space.

Explores the potential to relate Lorentzian and Euclidean physical states via Wick rotation.

Abstract

We construct a positive complexifier, differentiable almost everywhere on the classical phase space of real triads and $SU(2)$ connections, which generates a Wick Transform from Euclidean to Lorentzian gravity everywhere except on a phase space set of measure zero. This Wick transform assigns an equal role to the self dual and anti-self dual Ashtekar variables in quantum theory. We argue that the appropriate quantum arena for an analysis of the properties of the Wick rotation is the diffeomorphism invariant Hilbert space of Loop Quantum Gravity (LQG) rather than its kinematic Hilbert space. We examine issues related to the construction, in quantum theory, of the positive complexifier as a positive operator on this diffeomorphism invariant Hilbert space. Assuming the existence of such an operator, we explore the possibility of identifying physical states in Lorentzian LQG as Wick rotated images of physical states in the Euclidean theory. Our considerations derive from Thiemann's remarkable proposal to define Lorentzian LQG from Euclidean LQG via the implementation in quantum theory of a phase space `Wick rotation' which maps real Ashtekar-Barbero variables to Ashtekar's complex, self dual variables.