Loop gravity in terms of spinors

TL;DR

This paper reformulates loop gravity using spinorial variables, providing a new mathematical framework that simplifies calculations and strengthens the connection between quantum states and discrete geometries.

Contribution

It introduces a unitarily equivalent spinor-based formulation of loop gravity on a generalized Bargmann space, replacing traditional group variable integrals.

Findings

Simplifies calculations in loop gravity

Establishes a direct link between spin network states and geometries

Provides a new mathematical framework for the theory

Abstract

We show that loop gravity can equally well be formulated in in terms of spinorial variables (instead of the group variables which are commonly used), which have recently been shown to provide a direct link between spin network states and discrete geometries. This results in a new, unitarily equivalent formulation of the theory on a generalized Bargmann space. Since integrals over the group are exchanged for straightforward integrals over the complex plane we expect this formalism to be useful to efficiently organize practical calculations.