Finite states in 4 quantized dimensional gravity. The anisotropic minisuperspace Ashtekar–Klein–Gordon model (Part I).

Finite states in 4 quantized dimensional gravity. The anisotropic minisuperspace Ashtekar–Klein–Gordon model (Part I). Abstract In this paper we construct the generalized Kodama state for gravity in the homogeneous anisotropic case coupled to a Klein–Gordon scalar field with constant scalar potential. The basic method is to generate an asymptotic expansion about the pure Kodama state, viewed as a vacuum with respect to quantum fluctuations of the CDJ matrix. We compute propagation in the functional space of fields with respect to this vacuum via momentum space methods, finding that topologi-cal sectors remain preserved at the linearized level of the constraints. Nonlinear effects, the direct analogue of self-interactions in a quantum field theory, introduce higher topological sectors parametrized by the value of the scalar potential. Finally, we show that the generalized Kodama states form an orthogonal basis in this simple model, and we provide arguments for their normalizability based upon the functional form of the state.