Quantum gravitation

Abstract

This research proposal intends to analyze several aspects related to quantum gravity, ranging from noncommutative theories up to the study of higher derivative gravity theories. Noncommutative field theories arise as low energy limits of string theory in an antisymmetric tensor background. The main quantum property of these theories is the mixing of ultraviolet and infrared divergences which undermines renormalizability. We discovered that the only four dimensional renormalizable theory is the supersymmetric Wess-Zumino model. In three dimensions, we have also shown that the supersymmetric nonlinear sigma models are renormalizable in the 1/ N expansion. We intend to continue this research line analyzing supersymmetric gauge theories, mainly with extended supersymmetry, calculating the effective potential, and investigating its gauge invariance. Besides, we wish to study noncommutative gravity theories. The main attempts formulate gravity like a gauge theory. We intend to adopt the usual procedure of replacing the ordinary product by the Moyal one and to consider the noncommutative parameter as being a covariantly constant tensor. We are also planning to study exact non-solitonic solutions of several noncommutative theories, analyze their properties and the connection with tachyons in superstrings. In noncommutative theories the fields present exotic properties. The Seiberg- Witten map allows us to work with ordinary fields with their characteristics properties. In general, the formulation with commutative fields is incomplete since the dynamics is not known in explicit form. Besides, we have shown that the gauge field couples to matter, like a gravitational field. This shows a deep connection between noncommutative theories and gravitation which deserves to be better understood. This is another goal. Still using this formulation, we found that supersymmetry can not de realized off- shell, preventing a superspace formulation. We believe that supersymmetry becomes local in the gauge way as the gauge field, previously discussed, turns translations into a local symmetry. This aspect will also be explored. When it was understood that the nonperturbative regime of string theory included membranes and supergravity theories in eleven dimensions, several exceptional discoveries were made. One of them is the equivalence between certain gravity and gauge theories. The main one is known as the AdS/CFT correspondence and its major property is that it includes the holographic principle. We have done several contributions to this area, including a new quantization of the scalar field in anti De Sitter space and an improved version of the correspondence. We intend to follow this line analyzing the self-adjointness of the radial equation of motion, the passage from Euclidean to Lorentzian signature, the case of higher spins and the problem of selecting the relevant degrees of freedom in the bulk of anti De Sitter. General relativity is known to be perturbatively nonrenormalizable. However, gravity theories quadratic in the curvature tensor can be renormalizable and even present asymptotic freedom. Yet, they are sick since there are ghosts in the spectrum. As a rule, higher derivative theories have ghosts and are non unitary. We intend to analyze higher derivative theories introducing new ghosts so that a super selection rule can be generated in order to eliminate all ghosts from the spectrum to get a consistent theory. Duality has an essential role in string theory. In some models it suggests the existence of dimensions above eleven, including time like ones. Two or more times theories are troublesome since the dynamical evolution is ambiguous. However, if there exists a gauge symmetry which removes the extra time dimensions then it may turn the theory into a consistent one. We wish to analyze, using BRST-BFV techniques, a formulation of two times theories whose gauge group is Sp(2, R). We will build the path integral and show that several systems can be reached by different gauge choices. We also have some interest in beginning a research line based in a recent discovery that nonperturbative sectors of some supersymmetric gauge theories are related to matrix models. (AU)