Classical Spheres, Division Algebras, and the Illusion of Quantum Non-locality:

Hi Rob,

Really glad to see you here. I appreciate that you're one who can always be counted on to wade into the foundational details.

The Fourier transform is popular I expect, because it makes a lot of calculation easier. In fact, the same applies to the complex plane in general -- I expect that some quantum theorists might take the Hilbert space and the quantum theory formalisms associated with it, as something special and perhaps even physical, but no mathematician is likely to make that mistake.

The algebraically closed property of the complex plane, however, *is* important to any spacetime geometry -- because we can get nonlinear functions from fewer assumptions, and still recover the real valued numbers that you demand for real physical results. Personally, I find the primary importance of analysis to physical applications is in the realization that all real functions of a real valued variable are continuous. This is critical to any constructive theory of complete measurement functions, such as Joy Christian's, whether the theory applies to physics or is only mathematical. There cannot be any representation of a probability space that creates a gap between everywhere simply connected points.

So I strongly agree with you that -- as you imply -- *if* probability measure is a *foundational assumption* of how nature works, then getting rid of complex numbers will leave only classical probability.

And that's what Lucien Hardy is getting at, too -- he's enumerated five axioms of which four incorporate both classical and quantum probability, and one which obviates continuous function classical physics at the foundational level.

You quote from Hardy's abstract:

"This work provides some insight into the reasons why quantum theory is the way it is. For example, it explains the need for complex numbers..."

And after arguing that quantum probability measures do not require Fourier transforms and therefore complex analysis, you say:

"Let me restate this more bluntly:

"Histograms are used to 'measure' probability distributions.

"Physicists have unwittingly fabricated a mathematical structure for quantum theory, that is identical to a histogram. Consequently, the theory only produces descriptions of probability distributions of measurements, rather than specific measurements, as in classical theories, which do not construct histograms."

I agree! It is only by the sum of histories and normalization that one recovers unitarity, in order to make quantum results coherent and mathematically compatible with observed outcomes.

In lecture notes on the first law of thermodynamics the author is careful to point out right away that "The value of a state function is independent of the history of the system." A continuous change of state (measurement function continuous from the initial condition) cannot be cumulative when there is no probability measure to normalize; the usefulness of a histogram in this case is limited to showing that unitary evolution is scale invariant -- that is, by both classical and quantum predictions, correlated values are independent of the time at which they were measured. The difference between the probabilistic measure and the continuous-function measure is that by assuming probability on a measure space and normalizing it, one gets only what one assumes true. The continuous measurement function (Joy's) gets the true result by a frequentist statistical analysis, independent of assuming some probability on the closed interval [0,1].

All best,

Tom

Joy,

Dang, it's hard enough to explain a measurement function continuous from the initial condition without having to invent new terms for probability! :-) Sigh.

I guess that until one understands topology enough to know the difference between simply connected, and disconnected and multiply connected spaces, it's going to be an uphill struggle. Heck, we haven't even gotten as far as critics' accepting that your framework contains no probabilistic measure space -- the statistical analysis is all based on a frequentist interpretation of aggregated random coin tosses. To me, that's what "classical probability" means, but I'm no expert in the literature.

Best,

Tom

Tom,

"In all previous simulations that were proposed to fail, that I saw -- there was no randomized input. The critics simply did not seem to understand that the arbitrary choice of vector by the programmer cannot be counted as the initial condition of a function continuous from the initial condition; they create a dependent condition based on the experimenter's choice -- and then when they don't get Joy's predicted correlations on their assumed probability space -- declare that something is wrong with the mathematics, rather than with their own assumptions about the initial condition and a probability space that isn't there in the first place. They don't grasp how Joy has left the choice of initial condition to nature and taken the experimenter out of it."

This was very illuminating. Thank you for posting it.

James Putnam

I love Vienna, and the university. My wife and I were there summer of 2002 for the Karl Popper centenary. My paper fell flat, though I was so dazzled by the famous scholars in attendance, that it didn't matter to me all that much.

There was a heat wave in Europe, and ice cold beers from the little taprooms lining the streets were a refreshing delight!

I wonder if Vienna will be hot again this year. :-)

Tom

Thanks, James. I find it suspect that free will is at the center of the observer-created reality of conventional quantum theory; the choice of the observer is not actually free, because the result that begs its own conclusion is a bound variable.

The experimenter is a free variable in Joy's framework, so the correlations that smooth the continuous measurement function of two other free variables guarantee the experimenter's free will by allowing the initial condition to be unbounded.

Just my unpolished opinion.

Best,

Tom

LOL! Joy, I couldn't separate my fortunes from yours now if I tried. I'm with you, brother -- I didn't think the question needed answering. :-)

And a lot can happen between now and next June.

Fred, I too think that it's fear as much as anything. Why else would those critics whose demands have now been fully satisfied stay away from a straightforward discussion? If they are secure in their knowledge of quantum foundations, Joy's model should be the easiest thing to refute.

They overplayed their hand.

Best,

Tom

Hi Joy, Tom,

That is just it. They are not secure in their knowledge of QM foundations. There is no one completely agreed upon interpretation of QM. But they all believe Bell for some unknown reason. The complete irony is that if Bell knew about Joy's model, he probably would be the first to say that it is correct and that he was in fact mistaken about how he modeled EPR-Bohm with linear probabilities. For me, it is natural that space would have spinor properties. That viewpoint should be much easier to accept now-a-days than in Bell's time.

Best,

Fred

Hi Fred,

"For me, it is natural that space would have spinor properties. That viewpoint should be much easier to accept now-a-days than in Bell's time."

It's only natural in Minkowski-Einstein continuous spacetime. Probabilistic quantum mechanics in the Hilbert space has no chance at all to unify relativity and quantum theory.

That's the rub. Bell's theorem proves that quantum configuration space cannot be mapped onto physical space without a nonlocal model. A fully relativistic framework of quantum correlations, however (Joy's), bridges the local-global distinction with topological features that obviate nonlocality. The measure space and the physical space are identical -- as they must be, in order to incorporate the continuum of time with space.

Among the many things detractors don't understand about this framework, is the difference between statistical functions that can determine measurement results, and probability measures that can't.

All best,

Tom

It is clear that Sascha Vongehr and his "peer reviewer" Richard Gill have an axe to grind in preserving probabilistic quantum theory against relativity, as shown in Vongehr's own abstract, which reads like a manifesto for a crusade against 300 years of well understood classical physics. That he can call it a campaign against "pseudo-science" is nothing more than psychological projection.

And it's not that I'm so concerned that Vongehr and Gill take themselves seriously. It's that more thoughtful researchers in quantum foundations also entertain the belief that reality is fundamentally probabilistic. Besides being a prescription for nihilism, it's demonstrably wrong. If probability were a true measurement function, the world would cohere only in a single binary outcome, instead of a statistical array of continuous measurement.

Tom

Joy, I agree that it's a travesty of no small proportion. One can expect a certain amount of snide comments, incredulity, hard feelings, in any case of professional rivalry in any field. However:

When I was introduced to what appeared to be an actual strategic campaign to smear and discredit -- with words like "crackpot," "charlatan," "academic fraudster," "pseudo-science" -- I was outraged. I cited and linked that sophomoric paper above, to make evident to reasonable people, the lengths that true believers will go. It's sickness, not science, that drives such efforts, no matter what academic trappings attend them.

Now, though -- the silence is deafening. Absolutely every demand the critics made to find your work acceptable has been met. It has become trite to say that evil triumphs only when good people do nothing. All I can do, is to publicly call on academics who still believe in honesty and justice -- particularly if they are in a position of influence -- to do whatever they can to remedy the wrong.

Not that long ago, Grigori Perelman quit mathematics for the very reason that many in the community allow such perfidy, when they could do something about it. The loss to science of talented people who are victimized by the jealousy and small-mindedness of a few, is immeasurable.

All best,

Tom

Obviously, last post was mine.

Hi Folks,

It's very sad to hear about Perelman. A definite loss to the world's intellectual advancement. But I don't think he will really stop thinking about Math; he'll just take the process underground.

A suggestion for the query in block above; how about 'strongly correlated probabilities'?

It is good to read that your detractors are not favored, at this point, Joy. But it is a travesty that such adolescent behavior is tolerated in academia today. I have been studying avidly - doing a lot of reading of Math books and papers lately - and I may have something intelligent to say before long. I'll check in from time to time, now that I see there is a discussion here.

All the Best,

Jonathan