Gauge theories and quantum gravity in a finite interval of time on a compact space manifold

We study gauge theories and quantum gravity in a finite interval of time tau, on a compact space manifold Omega. The initial, final and boundary conditions are formulated in gauge invariant and general covariant ways by means of purely virtual extensions of the theories, which allow us to "trivialize" the local symmetries and switch to invariant fields (the invariant metric tensor, invariant quark and gluon fields, etc.). The evolution operator U(t(f),t(i)) is worked out diagrammatically for arbitrary initial and final states, as well as boundary conditions on partial derivative Omega, and shown to be well defined and unitary for arbitrary tau=t(f)-t(i)