Spin networks and cosmic strings in 3+1 dimensions

Abstract

Spin networks, the quantum states of discrete geometry in loop quantum gravity, are directed graphs whose links are labeled by irreducible representations of SU(2), or spins. Cosmic strings are 1-dimensional topological defects carrying distributional curvature in an otherwise flat spacetime. In this paper we prove that the classical phase space of spin networks coupled to cosmic strings may obtained as a straightforward discretization of general relativity in 3+1 spacetime dimensions. We decompose the continuous spatial geometry into 3-dimensional cells, which are dual to a spin network graph in a unique and well-defined way. Assuming that the geometry may only be probed by holonomies (or Wilson loops) located on the spin network, we truncate the geometry such that the cells become flat and the curvature is concentrated at the edges of the cells, which we then interpret as a network of cosmic strings. The discrete phase space thus describes a spin network coupled to cosmic strings. This work proves that the relation between gravity and spin networks exists not only at the quantum level, but already at the classical level. Two appendices provide detailed derivations of the Ashtekar formulation of gravity as a Yang–Mills theory and the distributional geometry of cosmic strings in this formulation.