POLYNOMIAL CONSTRAINTS FOR GENERAL-RELATIVITY USING REAL GEOMETRODYNAMICAL VARIABLES

The complex nature of Ashtekar's variables for general relativity has generated interest in Hamiltonian formulations in which the constraints continue to be polynomial but the canonical variables are real. In this paper, such a real, polynomial formulation is obtained using conformal geometrodynamical variables. However, like other geometrodynamical formulations, the constraints, although polynomial, are of a high order (ten). Nonetheless, this specific approach has the merit that it opens the door to obtaining a Hamiltonian framework in which the trace of the extrinsic curvature features as a configuration (rather than momentum) variable. Such a mixed representation may be suitable for isolating time in the quantum theory. Furthermore, as in the Ashtekar variables, the scalar constraint is now purely quadratic in the new momenta. This entire discussion sheds some light on the importance of polynomiality of constraints in the progress that has resulted from the Ashtekar variables.