New Hamiltonian formulation of general relativity

Abhay Ashtekar

Research output: Contribution to journalArticlepeer-review

976 Scopus citations


The phase space of general relativity is first extended in a standard manner to incorporate spinors. New coordinates are then introduced on this enlarged phase space to simplify the structure of constraint equations. Now, the basic variables, satisfying the canonical Poisson-brackets relations, are the (density-valued) soldering forms σ a A B and certain spin-connection one-forms AaA B. Constraints of Einstein's theory simply state that σ a satisfies the Gauss law constraint with respect to Aa and that the curvature tensor FabA B and Aa satisfies certain purely algebraic conditions (involving σ a). In particular, the constraints are at worst quadratic in the new variables σ a and Aa. This is in striking contrast with the situation with traditional variables, where constraints contain nonpolynomial functions of the three-metric. Simplification occurs because Aa has information about both the three-metric and its conjugate momentum. In the four-dimensional space-time picture, Aa turns out to be a potential for the self-dual part of Weyl curvature. An important feature of the new form of constraints is that it provides a natural embedding of the constraint surface of the Einstein phase space into that of Yang-Mills phase space. This embedding provides new tools to analyze a number of issues in both classical and quantum gravity. Some illustrative applications are discussed. Finally, the (Poisson-bracket) algebra of new constraints is computed. The framework sets the stage for another approach to canonical quantum gravity, discussed in forthcoming papers also by Jacobson, Lee, Renteln, and Smolin.

Original languageEnglish (US)
Pages (from-to)1587-1602
Number of pages16
JournalPhysical Review D
Issue number6
StatePublished - 1987

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy (miscellaneous)


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