Readers offer their own magic moments with John Bell
Robert B. Griffiths
Reinhold Bertlmann’s article “Magic moments with John Bell” was a pleasure to read. I add a note concerning some later developments that are significant for understanding the implications of Bell’s quantum foundations work.The crucial point is that under the Bell locality hypothesis for a joint measurement as given in the article,E(a,b)=∫dλρ(λ)A(a,λ)·B(b,λ)the quantity λ that represents the hidden variable(s) is a classical, not a quantum, object, and ρ(λ) is a classical probability distribution. Such expressions only make sense in the context of quantum theory when the operators of interest commute with each other, but that is not the case when this expression is used to derive a Bell inequality.Many years ago Arthur Fine pointed out that hidden variables and Bell inequalities were “imposing requirements to make well defined precisely those probability distributions for noncommuting observables whose rejection is the very essence of quantum mechanics.”11. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291 Thus the real issue is not one of locality but, instead, the proper use of probabilities in quantum theory.Fine’s point has been confirmed by the later development of a fully consistent way of introducing probabilities in quantum mechanics.22. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. That approach has shown that when quantum mechanics is interpreted using subspaces of the Hilbert space rather than classical hidden variables to represent physical properties, it is local: There are no ghostly nonlocal influences.We cannot know how Bell would have responded to those new developments, which only came to full fruition after his untimely death. The reader willing to dig into my somewhat tedious discussion33. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. of the Einstein-Podolsky-Rosen and Bohm situation will find an explicit proof that a measurement by Alice has not the slightest effect on the spin on Bob’s side. More recent is a general argument that quantum mechanics satisfies a principle of Einstein locality: What is done to system A has no effect on system B as long as there is no interaction between the two.44. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5Those proofs of quantum locality begin by taking very seriously John von Neumann’s proposal that quantum properties correspond to Hilbert subspaces, whose projectors do not, in general, commute with one another.55. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Bell, though, assumed that such properties can be represented by classical hidden variables, which always commute. Thus Bell’s theorem teaches that if one uses classical hidden variables in place of quantum properties, the result will contradict quantum theory and disagree with experiment. Indeed, while Bell started off by pointing out deficiencies in von Neumann’s argument against hidden variables, the end result of his work is an even better and much more conclusive argument, backed by experiment, for rejecting classical hidden variables.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291, Google ScholarCrossref2. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. Google Scholar3. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. Google Scholar4. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref5. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Google Scholar© 2015 American Institute of Physics.
Physics Today•Nov 30, 2015•0 citations