Note on 't Hooft's Shock Wave Commutators From Near Horizon Conformal Field Theory

TL;DR

This paper develops a finite model of 't Hooft's shock wave commutation relations using a 1+1D conformal field theory framework, linking quantum degrees of freedom to the holographic screen's area, and addresses divergences in the relations.

Contribution

It constructs a finite, cutoff-based model of 't Hooft's shock wave relations derived from a conformal field theory approach inspired by holographic principles.

Findings

Model removes coincident point divergences in predictions.

Links CFT central charge to holographic screen area.

Provides a finite, consistent framework for shock wave commutators.

Abstract

We construct a finite model of 't Hooft's shock wave commutation relations from the ansatz\cite{Carlip}\cite{Solodukhin}\cite{BZ} that the quantum degrees of freedom in a causal diamond in a solution of Einstein's Equations are those of a (cut-off\cite{BZ}\cite{hilbertbundles} ) 1 + 1 dimensional conformal field theory (CFT) with central charge related to the area of the diamond's holographic screen by equating Cardy's formula with the generalized Bekenstein-Hawking formula\cite{ted95}\cite{fsb}. The particular CFT is an exactly marginal perturbation of free massless fermions, as motivated by the Holographic Space-Time\cite{hilbertbundles} (HST) program. The momentum cutoff in $1 + 1$ dimensions and the Dirac eigenvalue cutoff on the transverse geometry, modify the 't Hooft relations when the area of the diamond is finite and remove all coincident point divergences in predictions derived from these commutation relations.