Quantum Gravity and Causal Structures: Second Quantization of Conformal Dirac Algebras

TL;DR

This paper develops a quantum gravity model based on causal structures and conformal Dirac algebras, extending previous models to include fermions through an osp(1|2) algebra, resulting in a Grassmann two-matrix Chern-Simons theory.

Contribution

It introduces a novel extension of quantum gravity models to include fermions via an osp(1|2) algebra, leading to a new Grassmann two-matrix Chern-Simons formulation.

Findings

Extended conformal algebra to include fermions using osp(1|2)

Formulated the quantum action as a Chern-Simons theory

Provided a framework for building physical observables in quantum gravity

Abstract

It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical observables. In particular, previous studies constructed quantum gravity models by quantizing the moduli of Laplace, weight and defining-function operators on Fefferman-Graham ambient spaces. The algebra of these operators underlies conformal geometries. We extend those results to include fermions by taking an osp(1|2) "Dirac square root" of these algebras. The theory is a simple, Grassmann, two-matrix model. Its quantum action is a Chern-Simons theory whose differential is a first-quantized, quantum mechanical BRST operator. The theory is a basic ingredient for building fundamental theories of physical observables.