Quantum field logic

TL;DR

This paper develops a logical framework for algebraic quantum field theory, characterizing type III factors and providing a Hilbert-style calculus with completeness, advancing the understanding of quantum logic structures in relativistic quantum mechanics.

Contribution

It introduces an equational theory characterizing type III factors in AQFT and develops a Hilbert-style calculus with a completeness theorem.

Findings

Characterization of type III factors via equational logic.

Development of a Hilbert-style calculus for quantum logic.

Establishment of a completeness theorem for the calculus.

Abstract

Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of the structure of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras indexed by regions of a Lorentzian manifold. In several cases the mentioned net is represented by a family of von Neumann algebras, concretely, type III factors. Local quantum field logic arises as a logical system that captures the propositional structure encoded in the algebras of the net. In this framework, this work contributes to the solution of a family of open problems, emerged since the 30s, about the characterization of those logical systems which can be identified with the lattice of projectors arising from the Murray-von Neumann classification of factors. More precisely, based on physical requirements formally described in AQFT, an equational theory able to characterizethe type III condition in a factor is provided. This equational system motivates the study of a variety of algebras having an underlying orthomodular lattice structure. A Hilbert style calculus, algebraizable in the mentioned variety, is also introduced and a corresponding completeness theorem is established.