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.

Using a form of modified dispersion relations derived in the context of quantum geometry, we investigate limits set by current observations on potential corrections to Lorentz invariance. We use a phenomological model in which there are separate parameters for photons, leptons, and hadrons. Constraints on these parameters are derived using thresholds for the processes of photon stability, photon absorption, vacuumČerenkov radiation, pion stability, and the GZK cutoff. Although the allowed region in parameter space is tightly constrained, non-vanishing corrections to Lorentz symmetry due to quantum geometry are consistent with current astrophysical observations.Date: January 2002. the authors find that, in an analysis of particle propagation, photon [17] and neutrino [16] dispersion relations are modified. In Section 3 we give a brief overview of threshold calculations before turning to a number of processes including: photon stability, photon non-absorption, vacuumČerenkov radiation for electrons and protons, proton non-absorption, and pion stability. We calculate constraints from the threshold calculations to investigate whether it is possible that observed effects may be accounted for by Lorentz symmetry corrections. In 3.2 we show that asymmetric momentum partitioning, first noticed by Liberati, Jacobson, and Mattingly [13], dramatically affects the constraints on the dispersion relation modifications. The results of Section 3 are summarized in Table 1. Finally in Section 4, we apply the constraints together with current observations to limit the extent of potential Lorentz symmetry corrections. We summarize the constraints in the final section and in Figures 2 and 3. We find that present day observations tightly constrainbut still leave open -the possibility of Lorentz symmetry corrections of this form.Particularly close to the present work is the paper by Jacobson, Liberati, and Mattingly [13] in which many of these results were summarized. For the most part the present work agrees with this paper where the subject overlaps, although this work also includes new threshold calculations and constraints for proton vacuum Cerenkov radiation, the GZK threshold, and pion stability.

The phase space of a classical particle in DSR contains de Sitter space as the space of momenta. We start from the standard relativistic particle in five dimensions with an extra constraint and reduce it to four dimensional DSR by imposing appropriate gauge fixing. We analyze some physical properties of the resulting theories like the equations of motion, the form of Lorentz transformations and the issue of velocity. We also address the problem of the origin and interpretation of different bases in DSR.

Graphity models are characterized by configuration spaces in which states correspond to graphs and Hamiltonians that depend on local properties of graphs such as the degrees of vertices and numbers of short cycles. As statistical systems, graphity models can be studied analytically by estimating their partition functions or numerically by Monte Carlo simulations. Results presented here are based on both of these approaches and give new information about the high-and low-temperature behavior of the models and the transitions between them. In particular, it is shown that matter degrees of freedom must play an important role in order for the low-temperature regime to be described by graphs resembling interesting extended geometries.