Toward a Unique Filter for the Gravitational Path Integral

Abstract: In recent work, McNamara and Wang demonstrated that the failure of factorization for a gravitational path integral (GPI) can be understood as an incompleteness of its spectrum of states by constructing a unitary quantum field theory (QFT) into which the GPI embeds. We examine this construction from an algebraic point of view, and use it to verify a conjecture of the author concerning the existence and uniqueness of a unitary completion for the GPI by means of the theory of conditional expectations. We explore how the unitary QFT can be regarded as a single coherent theory in which the quantitative signatures of ensemble averaging emerge purely from projecting to the underlying, smooth GPI. The conditional expectation relating the two theories may therefore be interpreted as a filter in the sense of Liu. The factorizing theory differs from a more standard QFT in the sense that it is reducible, meaning the Hilbert space associated with the empty set is non-trivial. We explain how the structure of reducible QFTs are generically encoded in weak Hopf algebraic symmetries and demonstrate how the weak structure allows for generalized $α$-sectors and half wormholes to coexist within a single Hilbert space.