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Reports on Mathematical Physics

Quantum Logics Defined by Sets of Numerical Events

Dietmar W. Dorninger, Helmut Länger

Published Apr 1, 2019• 1 citation
Publisher
DOI
10.1016/s0034-4877(19)30041-2

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Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s in S. Such a function p from S to [0,1] is known as a numerical event or more accurately an S-probability. A set P of numerical events including the constant functions 0 and 1 and 1-p with every p in P becomes a poset when ordered by the order of real functions and can serve as a general setting for quantum logics. We call such a poset P a general set of events (GSE). The thoroughly investigated algebras of S-probabilities (including Hilbert logics), concrete logics and Boolean algebras can all be represented within this setting. In this paper we study various classes of GSEs, in particular those that are orthoposets and their interrelations and connections to known logics. Moreover, we characterize GSEs as posets by means of states and discuss the situation for GSEs to be lattices.
A class of ring-like structures of events (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 analyzed 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 [Formula: see text] from a set [Formula: see text] of states of a physical system to [Formula: see text] such that [Formula: see text] is the probability of the occurrence of an event when the system is in state [Formula: see text]. 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.