General boundary quantum field theory: Timelike hypersurfaces in Klein-Gordon theory

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.