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A example of amplitude in Relativistic Quantum Mechanics or specifically in QFT is the amplitude of a field configuration on a space-like hyper-surface of space-time to "lead" to another field configuration on another space-like hyper-surface of space-time. In the path-integral picture one simply integrates over all possible field configurations on the interior, giving each a weight in the normal way. Now if one wants to generalize this to finite closed boundaries, we would get an amplitude for each field configuration on a finite closed boundary of space-time. but how would we interpret this? This question relates to interpretations of quantum mechanics, has anyone investigated this line ?

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Actually in QFT (at least in particle physics) one usually calculates the amplitude between two spatial slices which are infinitely separated in time, i.e. the S-matrix. The infinite separation is needed to make the asymptotic state be Fock-like particles. Real time-dependent QFT (sometimes used in condensed matter) requires some very imaginative formulation of the path integral (see Keldysh formalism). I think what you're asking about has been pondered mostly by quantum gravity people, who have this problem in spades. See works by Rovelli. –  genneth Mar 27 '12 at 14:45
@genneth I think you should post your very informative comments as an answer. –  Slaviks Mar 27 '12 at 16:42
@genneth: This is not completely true--- it is true for S-matrix calculations, but the original pure field theory calculations of Schwinger, which were based on Feynman's path integral (in Schwinger's action principle reformulation) were between two finite time hypersurfaces, and this is still the cleanest way theoretically. The S-matrix thing was only in response to the quest for a pure S-matrix theory, which quantum field theory isn't. –  Ron Maimon Mar 28 '12 at 8:03
@RonMaimon: perhaps I'm misunderstanding the thrust of the question or your comment, but I think the OP wants to know whether it is possible and what it would mean to assign an amplitude to a field configuration defined on the boundary of a spacetime hypervolume, i.e. including time-like parts of the boundary. I wasn't aware that there was anything done on this formalism outside of quantum gravity circles? –  genneth Apr 1 '12 at 11:38
@RonMaimon: but I think that's the whole point of the OP's question --- what to do about the timelike (I assume that's what you meant to write in the last sentence) parts of the boundary, and what the resulting amplitude means. Mohamed should correct me if I has mis-understood. –  genneth Apr 1 '12 at 16:44

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