Abstract
Let S be a set of states of a physical system. The probabilities p(s) of the occurrence of an event when the system is in different states s ∈ S define a function from S to [0, 1] called a numerical event or, more precisely, an S-probability. If one orders a set P of S-probabilities in respect to the order of functions, further includes the constant functions 0 and 1 and defines p′ = 1 − p for every p ∈ P, then one obtains a bounded poset of S-probabilities with an antitone involution. We study these posets in respect to various conditions about the existence of the sum of certain functions within the posets and derive properties from these conditions. In particular, questions of relations between different classes of S-probabilities arising this way are settled, algebraic representations are provided and the property that two S-probabilities commute is characterized which is essential for recognizing a classical physical system.
Highlights
A characteristic feature of measurements in quantum mechanics is that one only deals with probabilities.Support of the research of the second author by the Austrian Science Fund (FWF), project I 1923-N25, and the Czech Science Foundation (GAC R), project 15-34697L, as well as by the project “Ordered structures for algebraic logic”, supported by AKTION Austria – Czech Republic, project 75p11, is gratefully acknowledged.Int J Theor Phys (2016) 55:4453–4461Let S be 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 ∈ S
Taking into account p(s) for all s ∈ S we obtain a function from S to [0, 1], which is called a multidimensional probability or, more precisely, an S-probability, or sometimes, more generally, a numerical event
If one fixes some (A, E) ∈ O × B one obtains a function p(A, ., E) : S → [0, 1] which assigns to each state s ∈ S the probability of the event that the measurement of A lies in E
Summary
A characteristic feature of measurements in quantum mechanics is that one only deals with probabilities. Taking into account further properties of quantum mechanical or classical physical systems, in this paper more features are added to a set P of S-probabilities resulting in various classes of multidimensional probabilities. Algebras of S-probabilities and orthomodular posets with a full set of states have been thoroughly studied in respect to algebraic properties and physical interpretations (cf [1,2,3,4] and [6,7,8,9,10]) and there are some results about a generalization of these structures (cf [5]), namely so-called generalized fields of events. In this paper the focus is on weakening and modifying the axioms of algebras of Sprobabilities motivated by possible outcomes of experimental data which would not fit into the forementioned concepts
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