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

Still hoping that this thread will attract interactive exchange with those who know something about the foundations of quantum theory ...

Lucien Hardy has asked a poignant question: " ... could a 19th century theorist have developed quantum theory without access to the empirical data that later became available to his 20th century descendants?"

Hardy's answer is an axiomatic treatment of quantum theory that peels back the veil of Hilbert space formalism, to bare the skeleton of essential assumptions:

"Axiom 1 *Probabilities.* Relative frequencies (measured by taking the proportion of times a particular outcome is observed) tend to the same value (which we call the probability) for any case where a given measurement is performed on an ensemble of n systems prepared by some given preparation in the limit as n becomes infinite.

"Axiom 2 *Simplicity.* K is determined by a function of N (i.e. K = K(N)) where N = 1, 2, . . . and where, for each given N, K takes the minimum value consistent with the axioms.

"Axiom 3 *Subspaces.* A system whose state is constrained to belong to an M dimensional subspace (i.e. have support on only M of a set of N possible distinguishable states) behaves like a system of dimension M.

"Axiom 4 *Composite systems.* A composite system consisting of subsystems A and B satisfies N = N_A N_B and K = K_A K_B

"Axiom 5 *Continuity.* There exists a continuous reversible transformation on a system between any two pure states of that system."

One is compelled to see the beauty in this formalization, if one has a mathematical soul. For the consistency of calculation to meet the consistency of observation there need be continuity between the mathematical model and the physical result, without sacrificing the independence between the model's formal language and the physical manifestation. So it is with elegant understatement that Hardy adds:

"The first four axioms are consistent with classical probability theory but the fifth is not (unless the word 'continuous' is dropped). If the last axiom is dropped then, because of the simplicity axiom, we obtain classical probability theory (with K = N) instead of quantum theory (with K = N2). It is very striking that we have here a set of axioms for quantum theory which have the property that if a single word is removed -- namely the word 'continuous' in Axiom 5 -- then we obtain classical probability theory instead."

My personal investment in researching the mathematical physics of continuous functions goes back about 10 years. I think this recent preliminary result that I am in the process of formalizing, supports the case that reversible transformations of pure states is native to the plane. There can hardly be a purer quantum arithmetic state than a pair of distinct odd prime integers.

Tom

Some day I may get the link right on the first try, but don't count on it.

(My apologies. I wish there were some way to verify before posting. If this one doesn't work, then use the one in the post which started this thread.)

"Still hoping that this thread will attract interactive exchange with those who know something about the foundations of quantum theory ..."

OK Tom, I'll bite. Hardy's Abstract states that:

"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..."

Elsewhere in these discussions, I have pointed out that complex numbers, arising from the introduction of Fourier Transforms into QM, are not in fact NEEDED at all. They are merely sufficient. The unmeasurable phase of the Fourier Transform is what provides the continuous transform between "pure states" (Axiom 5). But that is also unnecessary, since the Fourier Transform itself is unnecessary.

The Fourier Transform, combined with the summation of the squared real and imaginary parts, is algebraically identical to a filter-bank that computes classical probabilities by merely counting discrete detections. In other words, it forms a histogram, which is why probabilities appear at all.

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.

Consider a purely classical analogy:

If you created a frequency modulated radio signal, with discrete states, (frequency shift keying) and then attempted to characterize those states via Fourier Transforms, you would encounter all the same problems that physicists have encountered in QM. That is why transforms are not used to "measure" such signals. Instead, a mathematical sturcture that actually measures the frequency, rather than the probability (histogram) of the frequency is used, to characterize the states, and thus demodulate the signal.

Rob McEachern

Hi Joy,

For sure there wouldn't be an exponential speedup as many are dreaming of. But there may be a little bit of a gain leveraging quantum-like probabilities based on your model. I guess due to your model, they shouldn't be called "quantum" probabilities any more. So what to call them?

Best,

Fred

Hi Fred, Joy,

What's wrong with "classical probability?" A quantum, whether it applies to a microscopic particle or a universe, is a probable result of a binary choice. It's only noncommutative following measurement. It's otherwise reversible in principle, as an element of a continuous function.

In other words, taking the classical case of a planetary orbit -- the trajectory is reversible before the initial condition prescribes a trajectory. That doesn't mean the two orbits are in superposition; the simple beauty of Joy's framework captured my imagination from the start, because it is dependent only on two fundamental principles: the topological space and the initial condition. The complete measurement function follows.

All best,

Tom

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.

Hi Joy,

Well... I don't think I'm gonna last another 50 years but I'm with you for as long as I last. :-) Yep, I have no reputation or career to protect other than what I said earlier about the path to truth. Seems like people now-a-days don't really care much about the truth. Cowards I guess.

Best,

Fred

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