Quantum Gravity on a Manifold with boundaries: Schr\"{o}dinger Evolution and Constraints

In this work, we derive the boundary Schr\"{o}dinger (functional) equation for the wave function of a quantum gravity system on a manifold with boundaries together with a new constraint equation defined on the timelike boundary. From a detailed analysis of the gravity boundary condition on the spatial boundary, we find that while the lapse and the shift functions are independent Lagrange multipliers on the bulk, on the spatial boundary, these two are related; namely, they are not independent. In the Hamiltonian ADM formalism, a new Lagrange multiplier, solving the boundary conditions involving the lapse and the shift functions evaluated on the spatial boundary, is introduced. The classical equation of motion associated with this Lagrange multiplier turns out to be an identity when evaluated on a classical solution of Einstein's equations. On the other hand, its quantum counterpart is a constraint equation involving the gravitational degrees of freedom defined only on the boundary. This constraint has not been taken into account before when studying the quantum gravity Schr\"{o}dinger evolution on manifolds with boundaries.