Actions for Gravity, with Generalizations: A Review

The search for a theory of quantum gravity has for a long time been almost fruitless. A few years ago, however, Ashtekar found a reformulation of Hamiltonian gravity, which thereafter has given rise to a new promising quantization project; the canonical Dirac quantization of Einstein gravity in terms of Ahtekar’s new variables. This project has already given interesting results, although many important ingredients are still missing before we can say that the quantization has been successful. Related to the classical Ashtekar Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1)and (3+1)-dimensions, and also generalizations of Einstein’s theory of gravity. In the first type of generalization, one introduces infinitely many new parameters, similar to the conventional Einstein cosmological constant, into the theory. These generalizations are called ”neighbours of Einstein’s theory” or ”cosmological constants generalizations”, and the theory has the same number of degrees of freedom, per point in spacetime, as the conventional Einstein theory. The second type is a gauge group generalization of Ashtekar’s Hamiltonian, and this theory has the correct number of degrees of freedom to function as a theory for a unification of gravity and Yang-Mills theory. In both types of generalizations, there are still important problems that are unresolved: e.g the reality conditions, the metric-signature condition, the interpretation, etc. In this review, I will try to clarify the relations between the new and old actions for gravity, and also give a short introduction to the new generalizations. The new results/treatments in this review are: 1. A more detailed constraint analysis of the Hamiltonian formulation of the Hilbert-Palatini Lagrangian in (3+1)-dimensions. 2. The canonical transformation relating the Ashtekarand the ADM-Hamiltonian in (2+1)-dimensions is given. 3. There is a discussion regarding the possibility of finding a higher dimensional Ashtekar formulation. There are also two clarifying figures (in the beginning of chapter 2 and 3, respectively) showing the relations between different action-formulations for Einstein gravity in (2+1)and (3+1)-dimensions. Email address: [email protected]