Quantum graphity is a background-independent model for emergent macroscopic locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on N vertices. At high energy, the graph describing the system is highly connected and the physics is invariant under the full symmetric group acting on the vertices. We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low-dimensional and local. Consideration of the free energy associated with the dominant terms in the dynamics shows that this low-energy state is thermodynamically stable under local perturbations. The model can also give rise to an emergent U (1) gauge theory in the ground state by the string-net condensation mechanism of Levin and Wen. We also reformulate the model in graph-theoretic terms and compare its dynamics to some common graph processes.
We show that a class of background independent models of quantum spacetime have local excitations that can be mapped to the first generation fermions of the standard model of particle physics. These states propagate coherently as they can be shown to be noiseless subsystems of the microscopic quantum dynamics [3]. These are identified in terms of certain patterns of braiding of graphs, thus giving a quantum gravitational foundation for the topological preon model proposed in [1].These results apply to a large class of theories in which the Hilbert space has a basis of states given by ribbon graphs embedded in a three-dimensional manifold up to diffeomorphisms, and the dynamics is given by local moves on the graphs, such as arise in the representation theory of quantum groups. For such models, matter appears to be already included in the microscopic kinematics and dynamics.
The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU (2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group G q over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU (2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.
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