We elaborate on the proposed general boundary formulation as an
extension of standard quantum mechanics to arbitrary (or no) backgrounds.
Temporal transition amplitudes are generalized to amplitudes
for arbitrary space-time regions. State spaces are associated to general
(not necessarily spacelike) hypersurfaces.
¶ We give a detailed foundational exposition of this approach, including
its probability interpretation and a list of core axioms. We explain
how standard quantum mechanics arises as a special case. We include a
discussion of probability conservation and unitarity, showing how these
concepts are generalized in the present framework. We formulate vacuum
axioms and incorporate space-time symmetries into the framework. We
show how the Schrödinger–Feynman approach is a suitable starting point
for casting quantum field theories into the general boundary form. We
discuss the role of operators.