High-energy behavior and second class constraints in quantum gravity

Abstract

It is proposed that sensible high-energy behavior in a quantum theory of gravity may be achieved in a class of theories in which the connection and metric are independent and unconstrained and where the action is chosen so that no derivatives of the metric appear. This is because in these theories all ten of the metric field equations are realized as second class constraints. These can in principle be solved, expressing the operators g μν as functions of the operators for the components of the connection and their canonical momenta. Thus, the metric has no independent quantum fluctuations, and the instabilities resulting from the usual curvature squared terms are eliminated. Furthermore, there is no need to assume metric compatibility, as it is automatically restored in the low-energy limit by the dominance of dimension-two terms. In order to explore these ideas a toy model with two degrees of freedom, corresponding to a metric and a connection variable, is quantized and shown to have a sensible high energy limit, while a related model, in which a constraint analogous to metric compatibility is imposed, is found to be unstable at high energies.