Quantum systems as classical systems

Abstract

A characteristical property of a classical physical theory is that the observables are real functions taking an exact outcome on every (pure) state; in a quantum theory, at the contrary, a given observable on a given state can take several values with only a predictable probability. However, even in the classical case, when an observer is intrinsically unable to distinguish between some distinct states he can convince himself that the measure of its ''observables'' can have several values in a random way with a statistical character. What kind of statistical theory is obtainable in this way? It is possible, for example, to obtain exactly the statistical previsions of quantum mechanics? Or, in other words, can a physical system showing a classical behaviour appear to be a quantum system to a confusing observer? We show that from a mathematical viewpoint it is not difficult to produce a theory with hidden variables having this property. We don't even try to justify in physical terms the artificial construction we propose; what we do is to give a general and rigorous argument showing how the interplay between the classical and quantum mechanics we offer is interpretable as the difference between an imaginary very expert observer and another nonexpert observer. This proves also that besides the well known theorems concerning the impossibility of hidden variables (cfr. Von Neumann [Neu] and Jauch-Piron [J-P]) there is also room for a result in favor of the possibility.