Complex Ashtekar Variables and Reality Conditions for Holst's Action

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 Ding 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 results perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity. © 2011 Springer Basel AG.