Linearized quantum gravity in flat space with toroidal topology

Abstract

It is known that quadratic constraints must be imposed on linearized gravity in a spacetime with compact Cauchy surfaces and with Killing vectors. These constraints cannot be derived from the linearized Lagrangian. They require that the quantum states be invariant under the continuous isometry group of the background spacetime. This fact makes it non-trivial to construct a Hilbert space of linearized gravity in some spacetimes with compact Cauchy surfaces. A method for dealing with this problem has been proposed for the case of de Sitter spacetime. It is shown that this method can be used to construct a Hilbert space of linearized gravity in flat space with toroidal topology. The inner product is shown to be a modified Klein-Gordon inner product. Some speculations are made about applying this method to full quantum gravity.