Actions for Gravity, with Generalizations: A Review
Peter Peldan
TL;DR
This review surveys classical gravity actions across (2+1) and (3+1) dimensions, focusing on the Ashtekar reformulation and connected actions (EH, HP, ADM, CDJ, Plebanski). It analyzes constraint structures, canonical transformations, and the emergence of the Ashtekar Hamiltonian, CDJ-Lagrangian, and Plebanski framework, while outlining two broad generalizations: cosmological-constant neighbours and gauge-group extensions. The work also discusses the special status of 2+1 dimensions where certain complications disappear, and it assesses prospects for higher-dimensional or unified gravity–Yang-Mills theories, highlighting unresolved issues such as reality conditions and metric-signature choices. Overall, the paper clarifies interrelationships among actions and evaluates their potential to advance non-perturbative quantum gravity.
Abstract
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 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 Ashtekar- and 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.