QUANTUM GRAVITY OF A BRANE-LIKE UNIVERSE

Abstract

Quantum gravity of a brane-like Universe is formulated, and its Einstein limit is approached. Regge–Teitelboim embedding of Arnowitt–Deser–Misner formalism, parametrized by the coordinates yA(t,xi), is governed by some ρAB(y,y′,y″). Invoking a novel Lagrange multiplier λ, accompanying the lapse function N and the shift vector Ni, we derive the quadratic Hamiltonian [Formula: see text] The inclusion of mattter resembles minimal coupling. Setting PA=-i(δ/δyA), we derive a bifurcated Wheeler–Dewitt-like equation. Einstein gravity, associated with λ being a certain fourfold degenerate eigenvalue of ρAB, is characterized by a vanishing center-of-mass momentum ∫PAd3x=0. Troublesome (ρ-λI)-1 is then replaced by regular M-1, such that M-1(ρ-λI) defines a projection operator, modifying the Hamiltonian accordingly.