Complex Ashtekar variables and reality conditions for Holst's action

TL;DR

The paper addresses unifying canonical loop quantum gravity with Lorentz-covariant spin-foam gravity by formulating Holst's action with complex Ashtekar variables and deriving reality conditions that mirror linear simplicity constraints. It develops a Hamiltonian framework, identifies a real-imposed constraint C_i{}^a=0 and related secondary constraints, and demonstrates that weakly implementing these reality conditions via the Dupuis-Livine projector recovers the standard LQG kinematical Hilbert space from SL(2,$\mathbb{C}$) data. This yields an isometry between the constrained complex-theory states and SU(2) Ashtekar variable states, offering a path to Lorentz-covariant descriptions compatible with spin-foam amplitudes, in line with Rovelli and Speziale’s Lorentz covariance results. Several open issues remain, notably the treatment of torsion, the strong vs weak imposition of all constraints, and a fully consistent dynamics that matches spin-foam evolution. The framework provides a concrete bridge between covariant spin-foam methods and canonical LQG, advancing prospects for a unified quantum gravity description.

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.

Figures (2)

• Figure 1: Phase space variables are smeared over links and faces.
• Figure 2: Links distinguishable only by their number of intermediate nodes should be identified.