Canonical reduction for dilatonic gravity in 3+1 dimensions

TL;DR

This paper extends the 1+1-dimensional dilatonic gravity formalism to 3+1 dimensions, deriving a canonical reduction that facilitates potential quantum gravity analysis involving point particles.

Contribution

The authors develop a canonical reduction of dilatonic gravity in 3+1 dimensions using ADM formalism, generalizing previous 1+1-dimensional approaches.

Findings

Reduced Hamiltonian expressed in terms of particle variables

Dilaton field equation reduces to a logarithmic Schrödinger equation

Formalism retains key features enabling quantum gravity studies

Abstract

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 Schroedinger 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.