Quantum gravity in three dimensions, Witten spinors and the quantisation of length

TL;DR

This paper explores the quantisation of length in three-dimensional Euclidean quantum gravity, linking boundary spinor fields to a discrete length spectrum consistent with loop quantum gravity results.

Contribution

It introduces a boundary conformal field theory with SU(2) spinors, deriving a length operator with a discrete spectrum matching loop quantum gravity.

Findings

Length becomes a number operator in the quantum theory.

Spectrum of length operator is discrete.

Results align with loop quantum gravity spin network spectra.

Abstract

In this paper, I investigate the quantisation of length in euclidean quantum gravity in three dimensions. The starting point is the classical hamiltonian formalism in a cylinder of finite radius. At this finite boundary, a counter term is introduced that couples the gravitational field in the interior to a two-dimensional conformal field theory for an SU(2) boundary spinor, whose norm determines the conformal factor between the fiducial boundary metric and the physical metric in the bulk. The equations of motion for this boundary spinor are derived from the boundary action and turn out to be the two-dimensional analogue of the Witten equations appearing in Witten's proof of the positive mass theorem. The paper concludes with some comments on the resulting quantum theory. It is shown, in particular, that the length of a one-dimensional cross section of the boundary turns into a number operator on the Fock space of the theory. The spectrum of this operator is discrete and matches the results from loop quantum gravity in the spin network representation.