Complex Ashtekar variables and reality conditions for Holst's action

TL;DR

This paper derives reality conditions for complex Ashtekar variables from Holst's action and demonstrates their implementation in quantum theory, aligning with loop quantum gravity and Lorentz covariance developments.

Contribution

It introduces reality conditions for complex Ashtekar variables in Holst's action and shows their weak implementation in quantum theory, connecting to loop quantum gravity.

Findings

Reality conditions mimic spin foam simplicity constraints

Weak implementation in quantum theory matches real Ashtekar connection

Aligns with Lorentz covariance in spin foam gravity

Abstract

From the Holst action in terms of complex valued Ashtekar variables additional reality conditions mimicking the linear simplicity constraints of spin foam gravity are found. In quantum theory with the results of You and Rovelli we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our result perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.