Augmented temporal logic formalism for histories-based generalized quantum mechanics

Abstract

An axiomatic development of dynamics of systems in the framework of histories is given which contains the history versions of classical and traditional quantum mechanics as special cases. We consider theories which admit a quasitemporal structure (a generalization of the concept of time using partial semigroups) and whose “single time” propositions have the mathematical structure of a logic; isomorphism of logics at different “instants of time” is not assumed. The concept of directed partial semigroup is introduced to incorporate the concept of direction of flow of time in quasitemporal theories. Starting with a few simple axioms, the space of history propositions is explicitly constructed and shown to be an orthoalgebra (as envisaged in the scheme of Isham and Linden [J. Math. Phys. 35, 5452–5476 (1994)]); its subspace consisting of history filters (“homogeneous histories”) is a meet semilattice. The partial semigroups employed allow semi-infinite irreducible decompositions.