## von Neumann  and Gleason vs. Bell

Returning to physics topics today I want to talk about an important contention point between von Neuman and Gleason on one had, and Bell on the other. I had a series of posts about Bell in which I discussed his major achievement. However I do not subscribe to his ontic point of view and today I will attempt to explain why and perhaps persuade the reader with what what I consider to be a solid argument.

Before Bell wrote his famous paper he had another one in which he criticized von Neumann, Jauch and Piron, and Gleason. The key of the criticism was that additivity of orthogonal projection operators not necessarily implies the additivity of expectation values:

$$\langle P_u + P_v \rangle = \langle P_{u}\rangle + \langle P_{v}\rangle$$

The actual technical requirements in von Neumann and Gleason case were slightly different, but they can be reduced to the statement above and more importantly this requirement is the nontrivial one in a particular proof of Gleason's theorem

 Andrew Gleason

To Bell, additivity of expectation values is a non-natural requirement because he was able to construct hidden variable models violating this requirement. And this was the basis for his criticism of von Neumann and his theorem of the impossibility of hidden variables. But is this additivity requirement unnatural? What can happen when it is violated? I will show that violation on additivity of expectation values can allow instantaneous communication at a distance.

The experimental setting is simple and involves spin 1 particles. The example which I will present is given in late Asher Peres book: Quantum Theory: Concepts and Methods at page 191. (This book is one of my main sources of inspiration for how we should understand and interpret quantum mechanics. )

The mathematical identity we need is:

$$J_{z}^{2} = {(J_{x}^{2} - J_{y}^{2})}^2$$

and the experiment is as follows: a beam of spin 1 particles is sent through a beam splitter which sends to the left particles of eigenvalue zero for $$J_{z}^{2}$$ and to the right particles of eigenvalue one for $$J_{z}^{2}$$.

Now a lab on the right decides to measure either if $$J_z = 1$$ or if $$J_{x}^{2} - J_{y}^{2} = 1$$

For the laboratory on the right let's call the projectors in the first case $$P_u$$ and $$P_v$$ and in the second case $$P_x$$ and $$P_y$$

For the lab on the left let's call the projectors in the first case $$P_{w1}$$ and in the second case$$P_{w2}$$.

Because of the mathematical identity: $$P_u + P_v = P_x +P_y$$ the issues becomes: should the expectation value requirement hold as well?

$$\langle P_{u}\rangle + \langle P_{v}\rangle = \langle P_{x}\rangle + \langle P_{y}\rangle$$

For the punch line we have the following identities:

$$\langle P_{w1}\rangle = 1 - \langle P_{u}\rangle - \langle P_{v}\rangle$$
and
$$\langle P_{w2}\rangle = 1 - \langle P_{x}\rangle - \langle P_{y}\rangle$$

and as such if the additivity requirement is violated we have:

$$\langle P_{w1}\rangle \neq \langle P_{w2}\rangle$$

Therefore regardless of the actual spatial separation, the lab on the left can figure out which experiment the lab on the right decided to perform!!!

With this experimental setup, if additivity of expectation values is false, you can even violate causality!!!

Back to Bell: just because von Neumann and Gleason did not provide a justification for their requirements, this does not invalidate their arguments. The justification was found at a later time.

But what about the Bohmian interpretation of quantum mechanics? Although there are superluminal speeds in the theory, superluminal signaling is not possible in it. This is because Bohmian interpretation respects Born rule which is a consequence of Gleason't theorem and it respects the additivity of  expectation values as well. Bohmian interpretation suffers from other issues however.

1. Like others, this text is wrong at pretty much every level. First, the criticism of von Neumann's implication is rubbish. Additivity of operators surely implies additivity of their expectation values because the expectation value is a linear operation. Also, there is nothing "unnatural" about expectation values and their additivity.

That's why it's stupid to talk about "communication at a distance allowed by non-additivity". Indeed, additivity/linearity and locality are linked. But the point you're missing is that both of them are right - while both non-additivity and non-locality are wrong. So you're constantly stuck in the mode of thinking that 2+2=5 implies 20+20=50 or vice versa. You're obsessed with wrong statements and the would-be logical or causal relationships between them.

1. Again? No drinking and commenting at the same time. I am not criticizing von Neumann, I am criticizing Bell. I am defending von Neumannn. Read the text again.

2. I haven't drunk any alcoholic beverage for a very long time.

2. Are you proving that hidden variables models violating additivity requirement would violate Lorentz invariance also? I think previously it was established that these hidden variable models have to be non-local in the sense of non-factorizable. So is this a second strike against hidden variable models?
BTW, I never saw this (unnatural!!!) identity J(z)^2=(J(x)^2-J(y)^2)^2 before. I checked it by putting in matrices. It is probably true only for spin 1.It is certainly not true for spin 1/2. Does Peres give a general proof?

1. Dear Kashyap, Florin isn't answering. So at least ;-), here's what I would answer.

The violations of additivity and/or linearity in QM is a much more brutal deviation from proper physics than a Lorentz violation. When done incorrectly, any non-additivity or non-linearity implies violations of the unitarity or the rules of logic - probabilities don't add up to one etc.

When you restrict yourself to "fixed" theories where the probabilities are renormalized so that they do add up to one etc., then the logical inconsistency may be cured by construction but you always unavoidably create nonlocalities - something that can really propagate genuine information at spacelike separation. Exactly all the stories about nonlocalities that are bogus when we describe proper QM - something that has no nonlocality - become real nonlocalities. Results of experiments in the region A must unavoidably depend on experimenters' decisions in region B that is spacelike-separated etc.

And of course, when you have nonlocality, the Lorentz invariance is dead, too.

The identity you mention us surely correct for spin-one only. In that case, all the terms are diagonal 3x3 matrices. Both sides have two equal nonzero entries and the third is zero. The difference on the right hand side simply does (0,1,1) - (1,0,1) and cancels the third, z, component. So we get (1,-1,0) and by another squaring, we get (1,1,0) again.

It's obvious that the identity can't work for higher values of spin. It's not true even dimensionally. When the total spin is large, it behaves classically and Jx, Jy, Jz are three independent numbers. And it's surely not true that the identity works for three independent numbers/components like that.

2. Dear Lubos,
Thanks for the reply.The subject of hidden variables interests me greatly. I do not know if the issue is settled once for all. Probably it is. Let us wait for Florin's answer.
kashyap

3. Hello, I am back. Sorry for the delay, I was extremely busy and I neglected the blog for the week. The identity is for spin 1.

Let me prepare now the next post and if still interested I'll be back to this thread.