Third quantization and the Wheeler-DeWitt equation

Beginning with a proposal for the normalization of solutions to the Wheeler-DeWitt equation put forth by DeWitt we argue that the Wheeler-DeWitt equation naturally lends itself to a second quantization in analogy to the second quantization of the Klein-Gordon equation. We identify a conserved current, as well as DeWitt’s proposal for normalization, as coming from a Lagrangian which is the analog of a second-quantized string theory whose spatial coordinates parametrize the coset manifold SL(3,R)/SO(3). We derive a mode decomposition of the second-quantized Wheeler-DeWitt field in the linearized approximation to quantum gravity, the zero modes of which are given by the total three-volume as well as various anisotropy parameters. We discuss the possibility of adding topological interactions for the linearized theory and find a representation in terms of vertex operators. In a two-dimensional setting we discuss a connection between our formalism and a proposal by Green which may shed light on some of the interpretational problems of string theory.