General boundary quantum field theory: foundations and probability interpretation. (English) Zbl 1210.81039

Oeckl, Robert

Adv. Theor. Math. Phys. 12, No. 2, 319-352 (2008).

Summary: 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.

MSC:

81Q35	Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81T05	Axiomatic quantum field theory; operator algebras
81P05	General and philosophical questions in quantum theory
81T18	Feynman diagrams
81P16	Quantum state spaces, operational and probabilistic concepts