Title:
Von Neumann, Gleason, and the Irreducibility of Probabilities in Hilbert Space Quantum Mechanics
Abstract:
The reducibility or fundamentality of chance in quantum physics has been a pressing issue since Born’s 1926 clarification of the probabilistic meaning of the wavefunction. That probabilities are irreducible in the standard formalism of quantum theory became, quite early, a central tenet in the Copenhagen orthodoxy, and a theorem formulated by von Neumann in 1927 was referred to as a proof of that statement. Furthermore, the theorem is usually known as an attempt of a refutation of the possibility of hidden variables, and Bell’s 1964 severe criticism of it became quite influential. In this talk—following a recent line of research developed by Jeffrey Bub, Guido Bacciagaluppi, Dennis Dieks, Michel Janssen & Tony Duncan, Chris Mitsch, and myself—I clarify the many misunderstandings that have surrounded von Neumann’s theorem in connection with hidden variables. Furthermore (and this is the work-in-progress part of the talk), I show that Bell’s criticism, rightly cashed out, amounts to the claim that von Neumann’s result is not enough to establish the irreducibility of probabilities in the Hilbert space formalism of quantum mechanics. Only in 1957, with the introduction of Gleason’s celebrated theorem, this central claim in quantum orthodoxy was firmly established.