Quantum Logic and the Histories Approach to Quantum Theory

TL;DR

This paper analyzes the Gell-Mann and Hartle axioms for a generalized histories approach to quantum theory, focusing on lattice structures and quasi-temporal theories relevant to quantum gravity and curved spacetime.

Contribution

It develops a framework for representing history propositions as projection operators, extending quantum logic to quasi-temporal and non-globally hyperbolic spacetimes.

Findings

Representation of history propositions as projection operators

Analysis of lattice structures in generalized histories

Application to quantum field theory in curved spacetime

Abstract

An extended analysis is made of the Gell-Mann and Hartle axioms for a generalised `histories' approach to quantum theory. Emphasis is placed on finding equivalents of the lattice structure that is employed in standard quantum logic. Particular attention is given to `quasi-temporal' theories in which the notion of time-evolution is less rigid than in conventional Hamiltonian physics; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space-time. The quasi-temporal structure is coded in a partial semi-group of `temporal supports' that underpins the lattice of history propositions. Non-trivial examples include quantum field theory on a non globally-hyperbolic spacetime, and a simple cobordism approach to a theory of quantum topology. It is shown how the set of history propositions in standard quantum theory can be realised in such a way that each history proposition is represented by a genuine projection operator. This provides valuable insight into the possible lattice structure in general history theories, and also provides a number of potential models for theories of this type.