On algebras of multidimensional probabilities

Abstract

Abstract The probability p(s) of the occurrence of an event pertaining to a physical system which is observed in different states s determines a function p from the set S of states of the system to [0, 1]. The function p is called a multidimensional probability or numerical event. Sets of numerical events which are structured either by partially ordering the functions p and considering orthocomplementation or by introducing operations + and · in order to generalize the notion of Boolean rings representing classical event fields are studied with the goal to relate the algebraic operations + and · to the sum and product of real functions and thus to distinguish between classical and quantum mechanical behaviour of the physical system. Necessary and sufficient conditions for this are derived, as well for the case that the functions p can assume any value between 0 and 1 as for the special cases that the values of p are restricted to two or three different outcomes.