A holographic formulation of quantum general relativity

TL;DR

Smolin develops a holographic formulation of quantum general relativity in Lorentzian signature by encoding bulk gravitational dynamics on a finite boundary. The approach uses a constrained $Sp(4)$ topological field theory to yield a polynomial, left-right symmetric canonical structure, with boundary degrees of freedom described by a two-dimensional $SU(2)_L \oplus SU(2)_R$ WZW/Chern-Simons system and balanced spin networks that reproduce Barrett–Crane type amplitudes. The boundary Hilbert space consists of balanced intertwiners on punctured spheres and is finite-dimensional due to quantum-group deformation, satisfying the Bekenstein bound; dynamics are implemented by a boundary Hamiltonian that respects a preferred boundary time. The framework points toward extended supersymmetry via $Osp(4/N)$ and a holographic perspective on M-theory, unifying holography, boundary CFT, and spin-network quantization in a concrete, finite boundary setting.

Abstract

We show that there is a sector of quantum general relativity which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of SU(2)_L + SU(2)_R, WZW field theory on the n-punctured sphere, where n is related to the area of the boundary. The Bekenstein bound is explicitly satisfied. These results are based on a new lagrangian and hamiltonian formulation of general relativity based on a constrained Sp(4) topological field theory. The hamiltonian formalism is polynomial, and also left-right symmetric. The quantization uses balanced SU(2)_L + SU(2)_R spin networks and so justifies the state sum model of Barrett and Crane. By extending the formalism to Osp(4/N) a holographic formulation of extended supergravity is obtained, as will be described in detail in a subsequent paper.