We present a spacetime Hilbert space formulation of Feynman path integrals
(PIs). It relies on a tensor product structure in time which provides extended
representations of dynamical observables through a spacetime quantum action
operator. The “sum over histories” is identified with a spacetime quantum
trace, whose evaluation in different extended bases yields the distinct PI
representations. New insights naturally follow, including exact
discretizations, a nontrivial approach to the continuum limit, and a Hilbert
space treatment of spacetime symmetries. An equivalence between trace
expressions and expectation values in spacetime states is also exposed. The
relevance of the formalism in the development of general spacetime symmetric
Hilbert space extensions of quantum mechanics is also discussed.

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