Minimal areas from entangled matrices

Abstract:
In holographic models of gravity, space emerges from the quantum mechanics of large N matrices. Perhaps the simplest instance of this phenomenon are states described by non commutative geometries. In these states diffeomorphisms arise as the N -> infinity limit of the U(N) gauge symmetry of the matrix theory. I will demonstrate that in such states there is a direct connection between microscopic (matrix) entanglement and geometry. I will define a U(N)-invariant notion of entanglement entropy in matrix quantum mechanics, associated essentially to all M x M sub-blocks of the matrices of fixed size M < N. I will show that this entanglement entropy is given by the area of a minimal surface. This formula has a strong resemblance to the Ryu-Takayanagi entropy in semiclassical gravitational theories.