Tuesday, March 6, 2018

Bell tacitly assumed commuting observables

Physicist N. David Mermin just posted a revision of his 1993 paper, Hidden Variables and the Two Theorems of John Bell:
In all these cases, as Bell pointed out immediately after proving the Bell-KS theorem, we have “tacitly assumed that the measurement of an observable must yield the same value independently of what other measurements must be made simultaneously.” ...

This tacit assumption that a hidden-variables theory has to assign to an observable A the same value whether A is measured as part of the mutually commuting set A, B, C, ... or a second mutually commuting set A, L, M, ... even when some of the L, M, ... fail to commute with some of the B, C, ..., is called “non-contextuality” by the philosophers. Is non-contextuality, as Bell seemed to suggest, as silly a condition as von Neumann’s — a foolish disregard of “the impossibility of any sharp distinction between the behavior of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear,” as Bohr23 put it?
Yes, it is a silly condition, because it is contrary to everything we know about atomic physics. Many observables do not commute. If you want to measure position and momentum of a particle, then it makes a big difference which you measure first. That is the essence of the Heisenberg uncertainty principle.

Some people try to claim that Bell just made physically reasonable assumptions, but that is false.
To those for whom non-locality is anathema, Bell’s Theorem finally spells the death of the hidden-variables program.31 But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible. What Bell’s Theorem did suggest to Bell was the need to reexamine our understanding of Lorentz invariance, as he argues in his delightful essay on how to teach special relativity* and in Dennis Weaire’s transcription of Bell’s lecture on the Fitzgerald contraction.** “What is proved by impossibility proofs,” Bell declared, “is lack of imagination.”
Bell has his followers today, and they still refuse to accept impossibility proofs.

Update: A reader asks me to elaborate. Check the blog for past postings on this topic, for more detail.

A core tenet of quantum mechanics is that there are no hidden variables. Von Neumann was explicit about this in his textbook from about 1930. Bell, Einstein, and other dissenters have claimed that quantum mechanics is not a realistic theory, and that only a hidden variable theory achieves the sort of philosophical realism that they aspire too. Bell devised a clever way of testing his hidden variable ideas. Experiments have proved his hidden variable theories to be wrong.

I don't think Bell or Einstein ever stopped believing in hidden variable theories. Nearly everyone else, including myself, accepts that they have been proven wrong.

1 comment:

  1. Thank you!'

    Could you please elaborate on this?

    Did Bell think that hidden variable theories could still work?

    What is your personal take on hidden variable theories?

    Thank you! :0