We show that the real massive Klein-Gordon theory admits a description in
terms of states on various timelike hypersurfaces and amplitudes associated to
regions bounded by them. This realizes crucial elements of the general boundary
framework for quantum field theory. The hypersurfaces considered are
hyperplanes on the one hand and timelike hypercylinders on the other hand. The
latter lead to the first explicit examples of amplitudes associated with finite
regions of space, and admit no standard description in terms of ``initial'' and
``final'' states. We demonstrate a generalized probability interpretation in
this example, going beyond the applicability of standard quantum mechanics.Comment: 25 pages, LaTeX; typos correcte
