Quantum gravity of a brane-like Universe

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 y^A (t, xⁱ), is governed by some ρ_AB(y, y′, y″). Invoking a novel Lagrange multiplier λ, accompanying the lapse function N and the shift vector Nⁱ, we derive the quadratic Hamiltonian script H sign = 1/2 N [P_A((ρ - λI)^-1)^AB P_B + λ] + Nⁱy,^A_iP_A. The inclusion of matter resembles minimal coupling. Setting P_A = -i(δ/δy^A), we derive a bifurcated Wheeler-Dewitt-like equation. Einstein gravity, associated with λ being a certain fourfold degenerate eigenvalue of ρ_AB, is characterized by a vanishing center-of-mass momentum ∫ P_A d³x = 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.