Joy Christian wrote:gill1109 wrote:Physical observables are represented by Hermitian [typo corrected] operators. But not every Hermitian operator represents a physical observable.
Wrong!
According to Hilbert space quantum mechanics --- which is universally accepted by all physicists --- the correspondence between Hermitian operators and observables is
one-to-one.
But that is not the main problem. The main problem is that Bell's theorem is based on the assumption that the sum of expectation values is equal to the expectation value of the sum.
This assumption is not valid for hidden variable theories. Therefore Bell's theorem against local theories is as invalid as von Neumann's theorem against general hidden variable theories.
Well, it is not universally agreed by physicists that the correspondence between Hermitian operators and observables is one-to-one. Only so-called "dynamical variables" are considered to be *observables*, and which Hermitian operators correspond to dynamical variables/observables depends on whether or not they satisfy transformation laws corresponding to changes of frames of reference of different observers. At least, that is what is written in the Wikipedia page on "Observable". If you disagree, find some reliable source which supports your point of view, and edit the article. It includes numerous references. You will have to rewrite most standard textbooks on quantum mechanics. Of course, disproving Bell's theorem should lead to the same result - all those books need to be rewritten.
But this is irrelevant to the main point in the OP. Let me mention an example. Let's take a look at sigma_x and sigma_z. They don't commute so can't be measured at the same time. Their sum is another 2x2 self-adjoint matrix. It is easily checked that it equals sqrt 2 times the matrix with orthonormal rows (1/sqrt 2, 1/sqrt2) and (1/sqrt 1, -1/sqrt 2). So it's another spin matrix! Let me call it sigma_u. Exercise to the reader: figure out what is the unit 3-vector u.
So we have sigma_x + sigma_z = sqrt 2 sigma_u and none of those three spin matrices can be measured at the same time. If the system is in the state rho (2x2 density matrix) then we find that trace(rho sigma_x) + trace(rho sigma_z) = sqrt 2 trace(rho sigma_u). The three expressions "trace rho sigma..." are the expectation values of the results of measuring the corresponding observable on a system in state rho. One could write <sigma_x> + <sigma_z> = sqrt 2 <sigma_u> and possibly add a subscript "rho" to the three "expectation operators" <...> to indicate the state of the system, whose dynamic variables (observables) one could consider observing, though not simultaneously. One can observe any one of the three as many times as one likes, but not simultaneously, on many new *preparations* of the same state.
The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is! Hidden variable theories reproduce exactly the predictions of quantum mechanics by the extension of the "universe" of quantum mechanics to a hidden universe of classical variables which underlie or explain in a classical statistical fashion, the probabilistic predictions of QM. QM tells us that when we observe sigma_x we get to observe +1 or -1 and we can compute the expected value of the measurement (ie the average of what we would see if we repeated the preparation of the state many, many times). So QM tells us the probabilities of the two outcomes of that measurement but does not explain, particle by particle, how it arises, from a deterministic evolution from the initial conditions of particle and measurement apparatus, which quite naturally vary from repetition to repetition. "Mere statistical variation" of "hidden", ie not directly observable, initial conditions were expected by Einstein to *explain* the random variation of outcomes of a measurement of the same observable on a system in the same "state". Einstein said "God does not toss dice" but actually, dice outcomes are deterministic, explained by classical physics, and Einstein did think that quantum dice were actually no different from classical dice. It's just that we can't (yet) directly see the true underlying dynamical variables of the system, nor (evidently) can we (yet) control them.
There is another problem. sigma_x and sigma_y when observed both produce outcomes +/- 1. One would expect measurement of their sum to produce outcomes which could be -2, 0 or +2. But when we measure their sum we get to observe +/- sqrt 2!
Sum of expectations equals expectation of sum is certainly true, but outcomes of sum are sums of outcomes of summands is not true!QM is definitely a bit weird.
This is a kind of toy version of the Kochen-Specker theorem.
By the way, take a look at the movie
https://www.youtube.com/watch?v=XDpurdHKpb8, Infinite Potential: The Life and Ideas of David Bohm, to catch a glimpse of how the physical establishment silenced David Bohm, who showed that there was a hidden variable theory which reproduced quantum theory. Einstein called Bohm his spiritual son. Oppenheimer, who had been David's mentor and supporter, got some top physicists together to find the mistakes in Bohm's theory. They failed. So they decided to shun Bohm. He fled the US and moved to Israel and later London. Nobody cited his papers. He was silenced.