We sketch the so called convex, or operational, framework for statistical physical theories, based on the convex set formed by the states of the physical system. A notion of observable is introduced which encompasses the ones usually adopted in classical and in quantum mechanics. In this framework a structure of effect algebra naturally arises. We also discuss a notion of extension of a descriptive physical model, pointing at the possibility of constructing classical extension of quantum mechanics.
An observable on a quantum structure is any
σ-homomorphism of quantum structures from the Borel
σ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form
(−∞,t) is sufficient to determine uniquely the whole observable defined on quantum structures like
σ-MV-algebras,
σ-effect algebras, Boolean
σ-algebras, monotone
σ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.
This paper is the first of several parts introducing a new powerful algebra: the algebra of the pseudo-observables. This is a C*-algebra whose set is formed by formal expressions involving observables. The algebra is constructed by applying the Occam's razor principle, in order to obtain the minimal description of physical reality. Proceeding in such a manner, every aspect of quantum mechanics acquires a clear physical interpretation or a logical explanation, providing, for instance, in a natural way the reason for the structure of complex algebra and the matrix structure of the formulation of Werner Heisenberg of quantum mechanics. Last but not least, the very general hypotheses assumed, allow one to state that quantum mechanics is the unique minimal description of physical reality.