Quantum tetrahedra and simplicial spin networks

A new link between tetrahedra and the group SU(2) is pointed out: by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close, the concept of quantum tetrahedron is seen to emerge. The Hilbert space of the quantum tetrahedron is introduced and it is shown that, due to an uncertainty relation, the ``geometry of the tetrahedron'' exists only in the sense of ``mean geometry''. A kinematical model of quantum gauge theory is also proposed, which shares the advantages of the Loop Representation approach in handling in a simple way gauge- and diff-invariances at a quantum level, but is completely combinatorial. The concept of quantum tetrahedron finds a natural application in this model, giving a possible intepretation of SU(2) spin networks in terms of geometrical objects.Comment: 11 pages, LaTeX, 1 figure.ps, typos corrected, references added and updated, a note added, E-mail and postal addresses change

The microscopic dynamics of quantum space as a group field theory

We provide a rather extended introduction to the group field theory approach to quantum gravity, and the main ideas behind it. We present in some detail the GFT quantization of 3d Riemannian gravity, and discuss briefly the current status of the 4-dimensional extensions of this construction. We also briefly report on recent results obtained in this approach and related open issues, concerning both the mathematical definition of GFT models, and possible avenues towards extracting interesting physics from them.Comment: 60 pages. Extensively revised version of the contribution to "Foundations of Space and Time: Reflections on Quantum Gravity", edited by G. Ellis, J. Murugan, A. Weltman, published by Cambridge University Pres

Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of ``dynamical triangulations'' is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian- and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde

Universe creation on a computer

The purpose of this paper is to provide an account of the epistemology and metaphysics of universe creation on a computer

A model of systems with modes and mode transitions

We propose a method of classifying the operation of a system into finitely many modes. Each mode has its own objectives for the system's behaviour and its own algorithms designed to accomplish its objectives. A central problem is deciding when to transition from one mode to some other mode, a decision that may be contested and involve partial or inconsistent information. We propose some general principles and model mathematically their conception of modes for a system. We derive a family of data types for analysing mode transitions; these are simplicial complexes, both abstract and concretely realised as geometric spaces in euclidean space . In the simplicial complex, a mode is represented by a simplex and each state of a system can be evaluated by mapping it into one or more simplices. This evaluation measures the extent to which different modes are appropriate for the state and can decide on a transition. To illustrate the general model in some detail, we work though a case study of an autonomous racing car