Holographic Space-Time and Quantum Information

Tom Banks

doi:10.3389/fphy.2020.00111

Abstract

The formalism of Holographic Space-time (HST) is a translation of the principles of Lorentzian geometry into the language of quantum information. Intervals along time-like trajectories, and their associated causal diamonds, completely characterize a Lorentzian geometry. The Bekenstein-Hawking-'t Hooft-Jacobson-Fischler-Susskind-Bousso Covariant Entropy Principle, equates the logarithm of the dimension of the Hilbert space associated with a diamond to one quarter of the area of the diamond's holographic screen, measured in Planck units. The most convincing argument for this principle is Jacobson's derivation of Einstein's equations as the hydrodynamic expression of this entropy law. In that context, the null energy condition (NEC) is seen to be the analog of the local law of entropy increase. The quantum version of Einstein's relativity principle is a set of constraints on the mutual quantum information shared by causal diamonds along different time-like trajectories. The implementation of this constraint for trajectories in relative motion is the greatest unsolved problem in HST. The other key feature of HST is its claim that, the degrees of freedom localized in the bulk of a diamond are constrained states of variables defined on the holographic screen. This principle gives a simple explanation of otherwise puzzling features of BH entropy formulae, and resolves the firewall problem for black holes in Minkowski space. It motivates a covariant version of the CKN\cite{ckn} bound on the regime of validity of quantum field theory (QFT) and a detailed picture of the way in which QFT emerges as an approximation to the exact theory.

Highlights

Every known human or computer language has the notion of time hard wired into every sentence
In Banks and Kehayias [16] we proposed a different approach, inspired by Connes’ insight about the connection between the Dirac operator and Riemannian geometry
The fact that the fundamental variables are the fermionic generators in the fundamental representation of SU(K|L)8 follows from the finite dimension of the Hilbert space and the fact that dynamics is invariant under fuzzy area preserving maps of the holographic screen

Summary

INTRODUCTION

Every known human or computer language has the notion of time hard wired into every sentence. Despite the fact that the small mass is a very low entropy object, the final equilibrium state is a state of much higher entropy This indicates that before equilibration, the combined system lived in a much larger Hilbert space than that of the original black hole, but that the initial state had 2π RSm frozen degrees of freedom. The principle operating here is that a localized state in the causal diamond formed by the horizon of the black hole of mass m+M is a constrained state of the Hilbert space of that black hole Another important feature we learn from this discussion is that the constraints must have the property that they isolate the degrees of freedom of the small system, from that of the black hole. The same formalism can describe the production and decay of high entropy meta-stable excitations with all of the qualitative properties of black holes

THE HOLOGRAPHIC VARIABLES OF QUANTUM GRAVITY

TIME DEPENDENT HAMILTONIANS AND

Asymptotic Symmetries of HST

CONCLUSIONS