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 σ̃ ${}_{}{}^a{}_{}{}^{}$ ${\mathrm{}}_A$ ${\mathrm{}}^B$ and certain spin-connection one-forms $A_{\mathrm{aA}}$ ${\mathrm{}}^B$. Constraints of Einstein’s theory simply state that σ̃ ${}_{}{}^a{}_{}{}^{}$ 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 σ̃ ${}_{}{}^a{}_{}{}^{}$). In particular, the constraints are at worst quadratic in the new variables σ̃ ${}_{}{}^a{}_{}{}^{}$ 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.