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

TL;DR

This paper derives a boundary Schr"odinger equation and a new boundary constraint for quantum gravity on manifolds with boundaries, revealing boundary-specific relations among lapse and shift functions and introducing a previously overlooked boundary constraint.

Contribution

It introduces a novel boundary constraint equation in quantum gravity and analyzes the boundary conditions affecting lapse and shift functions in the Hamiltonian formalism.

Findings

Derived boundary Schr"odinger equation for quantum gravity.

Identified a new boundary constraint involving gravitational degrees of freedom.

Showed the boundary constraint's relevance in quantum evolution on manifolds with boundaries.

Abstract

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.