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-De Witt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L² functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model.

Abstract. Using numerical calculations, we compare three versions of the Barrett– Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model. 1.

This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper we discuss two different types of language that can be attached to a system, S. The first is a propositional language, PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our programme utilises the more powerful language L(S) and its representation in an appropriate topos.

We describe the spacetime reduction of the fields of a supergravity-Yang-Mills theory in generic curved spacetime background, and with large N flavor group, to linearized forms on an infinitesimal patch of local tangent space at a point in the spacetime manifold. Our novel prescription for spacetime reduction preserves all of the local symmetries of the continuum field theory Lagrangian in the resulting zero-dimensional matrix Lagrangian, thereby obviating difficulties encountered in previous matrix proposals for emergent spacetime in recovering the full nonlinear symmetries of Einstein gravity. It also obviates the challenges that must be faced by any proposal for a fundamental theory, holographic or topological, where gravity emerges instead as an induced interaction. We have conjectured in hep-th/0201129 that the zerodimensional matrix model obtained by the spacetime reduction of the circle-compactified type I-I ′-mIIA-IIB-heterotic supergravity-Yang-Mills theory with sixteen supercharges and large N flavor group, and inclusive of the full spectrum of Dpbrane charges, −2 ≤ p ≤ 9, offers a potentially complete framework for nonperturbative string/M theory. In this paper, we provide the details of such matrix Lagrangians, comparing with the results of simple planar reduction, and clarifying the emergence of the spacetime continuum in the large N limit of the zero dimensional matrix model. We explain the relationship of our conjecture for a fundamental theory of emergent local spacetime geometry to recent investigations of the hidden symmetry algebra of M theory, stressing insights that are to be gained from the algebraic perspective. We conclude with a list of open questions and directions for future work. 1

We investigate p-form electromagnetism—with the Maxwell and Kalb-Ramond fields as lowest-order cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose—a chain complex equipped with an inner product on (p + 1)-cochains—we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results—such as a ‘p-form Bohm–Aharonov effect’—that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show p-form electromagnetism in p + 1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d Yang-Mills theory as studied by Migdal and Witten. Our main result describes p-form electromagnetism as a ‘chain field theory’—a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.

An asymptotic formula for a certain 4d Euclidean spin foam 4-simplex amplitude is given for the limit of large spins. The analysis covers the model with Immirzi parameter less than one defined separately by Engle, Livine, Pereira and Rovelli (EPRL) and Freidel and Krasnov (FK). We are also able to analyse the EPRL model with Immirzi parameter greater than one. The asymptotic formula has one term which is proportional to the cosine of the Regge action for gravity, and it is shown that this term is present whenever the boundary data determines a non-degenerate Euclidean geometry for the 4-simplex. A scheme for resolving the phase ambiguity of the boundary data in these cases is also presented. 1

The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett-Crane model, as in statistical mechanics, even though this theory is based on a real-time (e iS ) rather than imaginary-time (e S ) path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative.

The 10j symbol is a spin network that appears in the partition function for the Barrett-Crane model of Riemannian quantum gravity. Elementary methods of calculating the 10j symbol require O(j 9 ) or more operations and O(j 2 ) or more space, where j is the average spin. We present an algorithm that computes the 10j symbol using O(j 5 ) operations and O(j 2 ) space, and a variant that uses O(j 6 ) operations and a constant amount of space. An implementation has been made available on the web.

Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as p-form electromagnetism, including the Kalb-Ramond field in string theory, and its nonabelian generalizations. It is desirable to discretize such ‘higher gauge theories ’ in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of p-form electromagnetism. We show that discrete p-form electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for p-form electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of p-form electromagnetism as a ‘chain field theory’ — a theory analogous to topological quantum field theory, but with chain complexes replacing