New Hamiltonian Formulation of General Relativity

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 \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$${\mathrm{}}_{A}$${\mathrm{}}^{B}$ and certain spin-connection one-forms ${A}_{\mathrm{aA}}$${\mathrm{}}^{B}$. Constraints of Einstein's theory simply state that \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$ satisfies the Gauss law constraint with respect to ${A}_{a}$ and that the curvature tensor ${F}_{\mathrm{abA}}$${\mathrm{}}^{B}$ and ${A}_{a}$ satisfies certain purely algebraic conditions (involving \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$). In particular, the constraints are at worst quadratic in the new variables \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$ and ${A}_{a}$. This is in striking contrast with the situation with traditional variables, where constraints contain nonpolynomial functions of the three-metric. Simplification occurs because ${A}_{a}$ has information about both the three-metric and its conjugate momentum. In the four-dimensional space-time picture, ${A}_{a}$ 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.