When developing Quantum Field Theory, we usually refer to the Minkowski coordinates which cover the whole space in the case it is flat. I don't know how to set distribution theory and Hilbert space theory in this scenario.
I suppose the Hilbert space to be based on the whole Manifold, although in each hypersurface we are imposing canonical quantization (the so called Equal time canonical quantization).
Should be needed distribution theory on a Manifold at this point as we are dealing with something with more structure than $\mathbb{R}^3$? And I don't know how to relate Hilbert spaces on hypersurfaces and distrib. theory on hypersurfaces to the ones defined on the whole Manifold.
As at this point doing distribution theory (or defining a Hilbert space) on an hypersurface (or a submanifold) of a general manifold, or on a chart, it's non-sensical to me right now, as we are living on a semiriemannian Manifold.
