Canonical quantization of general relativity in discrete space-times

From the point of view of canonical quantum gravity, it has become imperative to find a framework for quantization which provides a {\em general} prescription to find the physical inner product, and is flexible enough to accommodate non-canonical variables. In this dissertation I consider an algebraic formulation of the Dirac approach to the quantization of constrained systems, due to A. Ashtekar. The Dirac quantization program is augmented by a general principle to find the ...

This note is to bring to the reader's attention the fact that general relativity and quantum mechanics differ from each other in one main aspect. General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is constructed such that its fundamental laws remain invariant to a change of topology. It is the goal of this paper to show that in order to obtain a complete description of quantum gravity one has to extend the pr...

We suggest to use "minimal" choice of quantum gravity theory, that is the quantum field theory, in which space-time is seen as Riemannian space and metric (or vierbein field) is the dynamical variable. We then suggest to use the simplest acceptable action, that is the squared curvature action. The correspondent model is renormalizable, has the correct classical limit without matter and can be explored using Euclidian path integral formalism. In order to get nonperturbative re...

This article describes the regularization of the generally relativistic gauge field representation of gravity on a piecewise linear lattice. It is a part of the program concerning the classical relativistic theory of fundamental interactions, represented by minimally coupled gauge vector field densities and half-densities. The correspondence between the local Darboux coordinates on phase space and the local structure of the links of the lattice, embedded in the spatial manifo...

A powerful strategy to treat quantum field theories beyond perturbation theory is by putting them on a lattice. However, the dynamical and symmetry structure of general relativity have for a long time stood in the way of a well-defined lattice formulation of quantum gravity. These issues are resolved by using Causal Dynamical Triangulations (CDT) to implement a nonperturbative, background-independent path integral for Lorentzian quantum gravity on dynamical lattices. We descr...

After many fruitless decades of trying to unify electromagnetism and gravitation, it is now being realized that this can be done only in discrete spacetime, as indeed the author had demonstrated. In this context, a unified description of gravitation and electromagnetism is provided within the framework of a gauge like formulation. Following the discrete spacetime structure, we then argue that the underpinning for the universe is an array of Planck scale oscillators.

We discuss in detail the uniform discretization approach to the quantization of totally constrained theories. This approach allows to construct the continuum theory of interest as a well defined, controlled, limit of well behaved discrete theories. We work out several finite dimensional examples that exhibit behaviors expected to be of importance in the quantization of gravity. We also work out the case of BF theory. At the time of quantization, one can take two points of vie...

We give an introduction to the canonical formalism of Einstein's theory of general relativity. This then serves as the starting point for one approach to quantum gravity called quantum geometrodynamics. The main features and applications of this approach are briefly summarized.

Lattice spinor gravity is a proposal for regularized quantum gravity based on fermionic degrees of freedom. In our lattice model the local Lorentz symmetry is generalized to complex transformation parameters. The difference between space and time is not put in a priori, and the euclidean and Minkowski quantum field theory are unified in one functional integral. The metric and its signature arise as a result of the dynamics, corresponding to a given ground state or cosmologica...

The basic features of the complex canonical formulation of general relativity and the recent developments in the quantum gravity program based on it are reviewed. The exposition is intended to be complementary to the review articles available already and some original arguments are included. In particular the conventional treatment of the Hamiltonian constraint and quantum states in the canonical approach to quantum gravity is criticized and a new formulation is proposed.