On ring-like event systems in quantum logic

TL;DR

This paper explores a class of ring-like event systems that generalize Boolean rings, characterizing quantum logics within this framework and analyzing their structure to distinguish classical from quantum systems.

Contribution

It introduces and analyzes RLSEs, extending the correspondence between Boolean algebras and rings to orthomodular lattices, and applies this to physical systems.

Findings

Characterization of quantum logics within RLSEs

Extension of Boolean algebra-ring correspondence to orthomodular lattices

Criteria to distinguish classical and quantum systems based on numerical events

Abstract

A class of ring-like event systems (RLSEs) is studied that generalizes Boolean rings. Quantum logics represented by orthomodular lattices are characterized within this class and the correspondence between Boolean algebras and Boolean rings is enlarged to orthomodular lattices. The structure of RLSEs and various subclasses is analysed and classical logics are especially identified. Moreover, sets of numerical events within different contexts of physical problems are described. A numerical event is defined as a function p from a set S of states of a physical system to [0,1] such that p(s) is the probability of the occurrence of an event when the system is in state s\in S. In particular, the question is answered whether a given (small) set of numerical events will give rise to the assumption that one deals with a classical physical system or a quantum mechanical one.