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.
"General boundary quantum field theory: Foundations and probability interpretation." Adv. Theor. Math. Phys. 12 (2) 319 - 352, April 2008.