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.

arXiv comment70 pages, 3 figures; updated to version published in Classical and Quantum Gravity

References in corpus (17)

• The teleparallel equivalent of general relativity
• Representations of the holonomy algebras of gravity and non-Abelian gauge theories
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Phase space descriptions for simplicial 4d geometries
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity
• Hamiltonian formulation of teleparallel gravity
• Phase space structure of Chern-Simons theory with a non-standard puncture
• Discretization of 3d gravity in different polarizations
• 2+1D Loop Quantum Gravity on the Edge
• Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
• Lectures on Faster-than-Light Travel and Time Travel
• Dual 2+1D Loop Quantum Gravity on the Edge