New Hamiltonian formulation of general relativity
Abstract
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 Poissonbrackets relations, are the (densityvalued) soldering forms σ~ ^{a} _{A} ^{B} and certain spinconnection oneforms A_{aA} ^{B}. Constraints of Einstein's theory simply state that σ~ ^{a} satisfies the Gauss law constraint with respect to A_{a} and that the curvature tensor F_{abA} ^{B} and A_{a} satisfies certain purely algebraic conditions (involving σ~ ^{a}). In particular, the constraints are at worst quadratic in the new variables σ~ ^{a} and A_{a}. This is in striking contrast with the situation with traditional variables, where constraints contain nonpolynomial functions of the threemetric. Simplification occurs because A_{a} has information about both the threemetric and its conjugate momentum. In the fourdimensional spacetime picture, A_{a} turns out to be a potential for the selfdual 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 YangMills 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 (Poissonbracket) 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.
 Publication:

Physical Review D
 Pub Date:
 September 1987
 DOI:
 10.1103/PhysRevD.36.1587
 Bibcode:
 1987PhRvD..36.1587A
 Keywords:

 04.20.Fy;
 Canonical formalism Lagrangians and variational principles