A discrete, unitary, causal theory of quantum gravity

TL;DR

This paper introduces a background-free, discrete Lorentzian quantum gravity model where states evolve via unitary rules, enabling the construction of a physical Hilbert space with positive inner product under certain conditions.

Contribution

It presents a novel discrete, causal, and unitary framework for quantum gravity that is background independent and defines a consistent inner product for physical states.

Findings

Inner product is hermitian and gauge-invariant

Inner product is positive for states with finite past

Hilbert space of physical states can be constructed

Abstract

A discrete model of Lorentzian quantum gravity is proposed. The theory is completely background free, containing no reference to absolute space, time, or simultaneity. The states at one slice of time are networks in which each vertex is labelled with two arrows, which point along an adjacent edge, or to the vertex itself. The dynamics is specified by a set of unitary replacement rules, which causally propagate the local degrees of freedom. The inner product between any two states is given by a sum over histories. Assuming it converges (or can be Abel resummed), this inner product is proven to be hermitian and fully gauge-degenerate under spacetime diffeomorphisms. At least for states with a finite past, the inner product is also positive. This allows a Hilbert space of physical states to be constructed.