Finite Entropy Implies Finite Dimension in Quantum Gravity

Abstract

Quantum Field Theory (QFT) introduced us to the notion that a causal diamond in space-time corresponded to a subsystem of a quantum mechanical system defined on the global space-time. Work by Jacobson\cite{ted95}, Fischler and Susskind\cite{fs} and particularly Bousso\cite{bousso} suggested that in the quantum theory of gravity this subsystem should have a density matrix of finite entropy. These authors formalized older intuitive arguments based on black hole physics. Although mathematically, Type II von Neumann algebras admit finite entropy density matrices, the black hole arguments suggest that the number of physical states in these subsystems is finite. The conjecture that de Sitter (dS) space has a finite number of physical states was first made in\cite{tb}\cite{wf}. ``String theory constructions" of ``meta-stable dS states" always involve a model with an infinite number of states. The real challenge of such models is to define the asymptotic correlation functions of the space-time to which the dS space decays, and show how to isolate properties of the dS ``resonance" from those correlators. We argue that the theory of the resonance itself is adequately described, on time scales shorter than its lifetime, by a model with a finite number of states.