We illustrate the relationship between spin networks and their dual representation by labelled triangulations of space in 2+1 and 3+1 dimensions. We apply this to the recent proposal for causal evolution of spin networks. The result is labelled spatial triangulations evolving with transition amplitudes given by labelled spacetime simplices. The formalism is very similar to simplicial gravity, however, the triangulations represent combinatorics and not an approximation to the spatial manifold. The distinction between future and past nodes which can be ordered in causal sets also exists here. Spacelike and timelike slices can be defined and the foliation is allowed to vary. We clarify the choice of the two rules in the causal spin network evolution, and the assumption of trivalent spin networks for 2+1 spacetime dimensions and four-valent for 3+1. As a direct application, the problem of the exponential growth of the causal model is remedied. The result is a clear and more rigid graphical...

Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of space-time. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we show that: • for Minkowski space-time, the deciding problem is as difficult as the Tarski’s problem of deciding elementary geometry, while • for a natural model of primordial space-time, the corresponding deciding problem is of the lowest possible complexity. 1

We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences - such as covariance - follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1

Isham’s topos-theoretic perspective on the logic of the consistent-histories theory [34] is extended in two ways. First, the presheaves of consistent sets of history propositions in their corresponding topos originally proposed in [34] are endowed with a Vietoristype of topology and subsequently they are sheafified with respect to it. The category resulting from this sheafification procedure is the topos of sheaves of sets varying continuously over the Vietoris-topologized base poset category of Boolean subalgebras of the universal orthoalgebra UP of quantum history propositions. The second extension of the topos in [34] consists in endowing the stalks of the aforementioned sheaves, which were originally inhabited by structureless sets, with further algebraic structure

Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as “directed ” topological quantum field theory. Two examples of such histories are described.

We propose a formulation of the holographic principle, suitable for a background independent quantum theory of cosmology. It is stated as a relationship between the flow of quantum information and the causal structure of a quantum spacetime. Screens are defined as sets of events at which the observables of a holographic cosmological theory may be measured, and such that information may flow across them in two directions. A discrete background independent holographic theory may be formulated in terms of information flowing in a causal network of such screens. Geometry is introduced by defining the area of a screen to be a measure of its capacity as a channel of quantum information from its null past to its null future. We call this a “weak ” form of the holographic principle, as no use is made of a bulk theory.