Spinor Representation for Loop Quantum Gravity

TL;DR

This paper introduces a spinor-based quantization approach for loop quantum gravity, establishing a unitary link to the standard Hilbert space and simplifying the analysis of gauge invariance and physical calculations.

Contribution

It develops a spinor representation of loop quantum gravity's phase space and proves its unitary equivalence to the traditional Hilbert space, facilitating easier computations.

Findings

Unitary equivalence between spinor and standard Hilbert spaces established.

Gauge invariant states characterized more easily in the spinor representation.

Calculations of physical quantities simplified via integrals over complex plane.

Abstract

We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.