No-Go Theorem for Modeling with Von Neumann Observables

Vincenzo Capasso, Donato Fortunato, F. Selleri

Eftichios Bitsakis

Brian C. Hall

Reinhold A. Bertlmann, Nicolai Friis

Abstract In chapter 12 we give an overview of the historical development of hidden-variable theories aiming to attribute ‘hidden’ definite outcomes to any potential measurement on a quantum system. We start with a discussion of von Neumann’s assumption on hidden-variable models, which was later realized not to hold by Hermann, Kochen, Specker, and Bell, going into detail on Bell’s refutation of von Neumann’s arguments. We then turn to the notion of contextuality, which severely restricts hidden-variable models: we discuss Gleason’s theorem before stating and proving the famous Kochen-Specker theorem, as well as the simpler versions of Peres’ nonet and Mermin’s pentagram. These results show that quantum mechanics is not compatible with the assignment of definite measurement outcomes to all possible measurements, and such assignments can hence only be made once a restricted set of observables, a context, is selected

Alberto C. de la Torre

Miroslav Engliš

We want to point out the following strengthening of the classical theorem of Groenewold and van Hove: There exists no mapping Op from polynomial observables on the phase space into linear operators on which would map Poisson brackets into commutators, the position and momentum observables p and q into the usual (Schrödinger) position and momentum operators, and would obey the von Neumann rule for and . The point is that neither linearity, nor continuity etc. of Op are assumed.