GRAVITY AND BF THEORY DEFINED IN BOUNDED REGIONS

We study Einstein gravity in a finite spatial region. By requiring a w ell-defined variational principle, we identify all local boundary cond itions, derive surface observables, and compute their algebra. The obs ervables arise as induced surface terms, which contribute to a non-van ishing Hamiltonian. Unlike the asymptotically flat case, we find that there are an infinite number of surface observables. We give a similar analysis for SU(2) BF theory. (C) 1997 Elsevier Science B.V.