Locality and Translations in Braided Ribbon Networks

An overview of microlocality in braided ribbon networks is presented. Following this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is followed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.Comment: 12 pages, 12 figure

Conserved Quantities in Background Independent Theories

We discuss the difficulties that background independent theories based on quantum geometry encounter in deriving general relativity as the low energy limit. We follow a geometrogenesis scenario of a phase transition from a pre-geometric theory to a geometric phase which suggests that a first step towards the low energy limit is searching for the effective collective excitations that will characterize it. Using the correspondence between the pre-geometric background independent theory and a quantum information processor, we are able to use the method of noiseless subsystems to extract such coherent collective excitations. We illustrate this in the case of locally evolving graphs.Comment: 11 pages, 3 figure

Network Codes for Real-Time Applications

We consider the scenario of broadcasting for real-time applications and loss recovery via instantly decodable network coding. Past work focused on minimizing the completion delay, which is not the right objective for real-time applications that have strict deadlines. In this work, we are interested in finding a code that is instantly decodable by the maximum number of users. First, we prove that this problem is NP-Hard in the general case. Then we consider the practical probabilistic scenario, where users have i.i.d. loss probability and the number of packets is linear or polynomial in the number of users. In this scenario, we provide a polynomial-time (in the number of users) algorithm that finds the optimal coded packet. The proposed algorithm is evaluated using both simulation and real network traces of a real-time Android application. Both results show that the proposed coding scheme significantly outperforms the state-of-the-art baselines: an optimal repetition code and a COPE-like greedy scheme.Comment: ToN 2013 Submission Versio

Evolution in Quantum Causal Histories

We provide a precise definition and analysis of quantum causal histories (QCH). A QCH consists of a discrete, locally finite, causal pre-spacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the Wheeler-DeWitt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.Comment: 20 pages. 8 figures. (v3: minor corrections, additional references [2,3]) to appear in CQ

Quantum gravity and the standard model

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. These are identified in terms of certain patterns of braiding of graphs, thus giving a quantum gravitational foundation for the topological preon model proposed by one of us. 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.Comment: 12 pages, 21 figures, improved presentation, results unchange

A discrete, unitary, causal theory of quantum gravity

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.Comment: 38 pages, 9 figures, v3 added to exposition and references, v4 expanded prospects sectio

A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime

We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system as a precursor to a quantum theory of gravity. The model has no a priori geometric properties, instead, locality is inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom. We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in this simple model.Comment: 23 pages, 6 figures; updated to published versio

Quantum Bose-Hubbard model with an evolving graph as a toy model for emergent spacetime

We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system as a precursor to a quantum theory of gravity. The model has no a priori geometric properties; instead, locality is inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom. We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in this simple model

Quantum Histories and Quantum Gravity

This paper reviews the histories approach to quantum mechanics. This discussion is then applied to theories of quantum gravity. It is argued that some of the quantum histories must approximate (in a suitable sense) to classical histories, if the correct classical regime is to be recovered. This observation has significance for the formulation of new theories (such as quantum gravity theories) as it puts a constraint on the kinematics, if the quantum/classical correspondence principle is to be preserved. Consequences for quantum gravity, particularly for Lorentz symmetry and the idea of "emergent geometry", are discussed.Comment: 35 pages (29 pages main body), two figure