We generalize the $1+1$-dimensional gravity formalism of Ohta and Mann to $3+1$ dimensions by developing the canonical reduction of a proposed formalism applied to a system coupled with a set of point particles. This is done via the Arnowitt-Deser-Misner method and by eliminating the resulting constraints and imposing coordinate conditions. The reduced Hamiltonian is completely determined in terms of the particles’ canonical variables (coordinates, dilaton field and momenta). It is found that the equation governing the dilaton field under suitable gauge and coordinate conditions, including the absence of transverse-traceless metric components, is a logarithmic Schrödinger equation. Thus, although different, the $3+1$ formalism retains some essential features of the earlier $1+1$ formalism, in particular the means of obtaining a quantum theory for dilatonic gravity.

• Received 25 October 2015

DOI:https://doi.org/10.1103/PhysRevD.93.084017