Quantum geometry of the null cone

TL;DR

This paper develops a non-perturbative quantum framework for gravitational null initial data, revealing a discrete spectrum for radiative modes and boundary areas, advancing understanding of quantum gravity on null surfaces.

Contribution

It introduces a novel quantization approach for null boundary data in tetradic gravity with a Holst term, including explicit solutions for constraints and spectra of observables.

Findings

Discrete spectra for boundary areas match previous results

Explicit solution of residual constraints in phase space

Quantization yields a straightforward physical Hilbert space

Abstract

We present a non-perturbative quantization of gravitational null initial data. Our starting point is the characteristic null initial problem for tetradic gravity with a parity-odd Holst term in the bulk. After a basic review about the resulting Carrollian boundary field theory, we introduce a specific class of impulsive radiative data. This class is defined for a specific choice of relational clock. The clock is chosen in such a way that the shear of the null boundary follows the profile of a step function. The angular dependence is arbitrary. Next, we solve the residual constraints, which are the Raychaudhuri equation and a Carrollian transport equation for an $SL(2,\mathbb{R})$ holonomy. We show that the resulting submanifold in phase space is symplectic. Along each null generator, we end up with a simple mechanical system. The quantization of this system is straightforward. Our basic strategy is to start from an auxiliary Hilbert space with constraints. The physical Hilbert space is the kernel of a constraint, which is a combination of ladder operators. The constraint and its hermitian conjugate are second-class. Solving the constraint amounts to imposing a simple recursion relation for physical states. On the resulting physical Hilbert space, the $SL(2,\mathbb{R})$ Casimir is a Dirac observable. This observable determines the spectrum of the two radiative modes. The area at the initial and final cross sections are Dirac observables as well. They have a discrete spectrum, which agrees with earlier results on this topic.