Spherically symmetric sector of self dual Ashtekar gravity coupled to matter: Anomaly-free algebra of constraints with holonomy corrections

TL;DR

This paper demonstrates that using self dual Ashtekar variables in spherically symmetric loop quantum gravity models results in an anomaly-free algebra of constraints with holonomy corrections, avoiding issues present in the real Ashtekar-Barbero formulation.

Contribution

It shows that self dual variables lead to an anomaly-free, undeformed algebra of constraints in spherically symmetric models with holonomy corrections, unlike the real formulation.

Findings

Algebra of constraints is anomaly free with self dual variables.

No signature change occurs in the quantum region.

Classical hypersurface deformation algebra is recovered.

Abstract

Using self dual Ashtekar variables, we investigate (at the effective level) the spherically symmetry reduced model of loop quantum gravity, both in vacuum and when coupled to a scalar field. Within the real Ashtekar-Barbero formulation, the system scalar field coupled to spherically symmetric gravity is known to possess a non closed (quantum) algebra of constraints once the holonomy corrections are introduced, which forbids the loop quantization of the model. Moreover, the vacuum case, while not anomalous, introduces modifications which are usually interpreted as an effective signature change of the metric in the deep quantum region. We show in this paper that both those complications disappear when working with self dual Ashtekar variables, both in the vacuum case and in the case of gravity minimally coupled to a scalar field. In this framework, the algebra of the holonomy corrected constraints is anomaly free and reproduces the classical hypersurface deformation algebra without any deformations. A possible path towards quantization of this model is briefly discussed.