Local quantum field logic

Algebraic quantum field theory provides 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, this net is represented by a family of von Neumann algebras, specifically, Type III factors. Local quantum field logic emerges as a logical system that captures the propositional structure encoded in the algebras of the net and their respective locality conditions. Specifically, by considering an expanded language of orthomodular lattices that naturally arises from the Murray von Neumann dimension theory, we first provide equational conditions in the lattice of projectors of a von Neumann factor that uniquely characterize the Type III factor within the Murray-von Neumann classification. This equational system motivates the study of a variety of algebras with an underlying orthomodular lattice structure, which we shall refer to as LQF-algebras. A Hilbert-style calculus is also introduced, establishing a completeness theorem with respect to the variety of LQF-algebras.