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

To try and get a new technical discussion started ...

Suppose we contrast the Christian correlation function C_ab with the CHSH correlation function.

The former is deterministic, admitting random input; the latter is probabilistic, producing random output.

Could these functions be dual to each other? -- if so, it venerates Joy Christian's claim that entanglement is an illusion, without obviating the computability and mathematical validity of Bell-Aspect experimental results. Discrete output of Bell-Aspect -- on the assumption of entanglement -- would be identical to truncated output, from a continuously randomized input function (the ThrowDie function in Chantal Roth's programming terms) of Joy Christian's model.

See how the contrast makes duality possible? -- Bell-Aspect simply assumes entanglement, while the Christian framework follows Newton's prescription, "hypotheses non fingo." Entanglement is superfluous to deterministic, continuous, natural (and locally real) functions.

What makes Bell's theorem important to computer controlled applications, particularly security, is the assured pseudo-random integrity of the ouput, based on the assumption of quantum entanglement in which nonlocal results remain out of reach of computation by an adversary who has no knowledge of how the pseudo-random string was obtained.

Representative of such infomration technology is this 2010 paper by Stefano Pironio, et al:

"We quantify the Bell inequality violation through the CHSH correlation function [19]

I = SIGMAx,y (- 1)^xy [P(a = b|xy - P(a != b|xy]

where P(a = b|xy) is the probability that a = b given settings (x; y) and P(a != b|xy] is defined analogously. Systems that admit a local, hence deterministic [20], description satisfy I =< 2. Certain measurements performed on entangled states, however, can violate this inequality."

A measurement on an assumed entangled state, however, is an average of trials on a probabilistic space. A deterministic measure on a complete -- i.e., classically continuous -- space, is an arbitrary choice of boundary conditions on an arbitrary interval. In other words, does nonlinear input to a continuous function guarantee precisely correlated linear output?

To prove the above question in a positive way, we need an arithmetically continuous model computable from a discrete initial condition. I've been working on a proof, the strategy outlined here and here .

Dual quantum correlation functions that generate identical results -- one function assuming non-observable quantum entanglement; the other, a correlation measurement continuous from the initial condition -- tells us that entanglement is an artifact of an artificial probability space and not physically real.

What justifies the probability measure space? -- only the assumption that fundamental reality is probabilistic. This assumption is made apparent in Poronio et al's conclusion:

"Stepping back to the more conceptual level, note that Eq. (3) relates the random character of quantum theory to the violation of Bell inequalities. This bound can be modified for a situation where we assume only the no-signalling principle instead of the entire quantum formalism (see Figure 2 and 3 and Appedenix A.3). Such a bound lays the basis for addressing in a statistically significant way one of the most fundamental questions raised by quantum theory: whether our world is compatible with determinism (but then necessarily signalling), or inherently random (if signalling is deemed impossible)."

The no-signalling condition is a red herring; Joy's framework is completely compatible with the no-signalling condition -- it couldn't *not* be, in that it is fully relativistic. Non-relativistic quantum theory gets "inherently random" nonlocal results on the assumption that locality forbids signalling -- which only assumes what it means to prove. Joy's framework generates manifestly local quantum correlations that still forbid signalling, a la special relativity.

Tom

Another way to argue the point is to take Lucien Hardy's five reasonable axioms of quantum theory . Hardy uses the first four axioms to rule out classical probability. The fifth axiom (continuity) cannot live with the first axiom (probability).

We would instead drop the first axiom and keep the other four, to rule out quantum entangled probability.

Tom

Like Godfrey Hardy (don't know if there's any relation) Lucien Hardy is the kind of mathematician I trust, because he doesn't hedge his bets on observational (or on non-observed) outcomes. I argued with Gill in the past about the importance of theoretical independence from experiment; Gill claims conventional quantum theory *is* independent of experimental results -- and I think Hardy's paper shows clearly that it is *not*, when the assumption of a probability measure is excluded.

Hardy's axioms subheaded 1) Probabilities; 2) Simplicity; 3) Subspaces; 4) Composite systems; 5) Continuity are consistent with Joy Christian's framework when axiom (1) is dropped.

Though I agree with Hardy (6.1, p 10) that the frequency interpretation of probability measure theory is to be preferred over Bayesian analysis (which I admittedly disdain in any form) -- I have to ask why the state (6.2) is assumed independent of its initial condition, as experimental preparation of the state implies (the experimenter is an element of the quantum system). A continuous function without a probability measure would not normalize the state vector; the function would give up information on the state of the system independent of state preparation.

(Correcting a misstatement in the previous post: I meant to say that Hardy rules out classical probability by adding the fifth axiom, while the first four support both classical and quantum probability.)

Tom

Thanks, Joy!

That is extremely good news, for a machine language that does not rely on a compiler to translate code to execution cannot be accused of rigging a result. The source code is executed on the spot and no information lost in translation.

So it cannot but be the case -- in the real time execution of random input to a function continuous from an initial condition, that correlation of two dichotomous variables of the function recorded on intervals that are indifferent to the outcome of a coin-toss probability -- is manifestly local.

I'm celebrating with you, here on the other side of the pond.

And all this from the simple assumption of the choice of topology! You might consider carving C_ab into a wooden bridge. :-)

You know, it occurred to me, the historical importance of the number 5 in a system of complete axioms -- Euclid's postulates; Dedekind-Peano arithmetic axioms; Lucien Hardy's five simple axioms of quantum mechanics -- and how replacing just one assumption with another changes the whole game. Such was the case of replacing Euclid's fifth postulate with one or the other axioms of non-Euclidean geometry, that made Riemannian geometry viable for general relativity. Replacing Hardy's probability axiom with a topological postulate would seem to form a complete axiomatic basis for your framework and put it a giant step closer to a coherent and falsifiable theory.

How would that postulate be worded?

And one more thing I'd like to know: where are the critics who said this couldn't done?

All best,

Tom

"...a machine language that does not rely on a compiler to translate code to execution cannot be accused of rigging a result. The source code is executed on the spot and no information lost in translation."

Did someone program the simulation in machine language?

James Putnam

James, all computer simulations are programmed by a set of instructions that are translated into a machine language native to the computer on which the instructions are executed.

The difference between a compiled set of instructions and an interpreted set is slight; both are what is called Turing-complete. Both are efficient. Instructions that use compilers take a comparatively longer time to write and a comparatively shorter time to execute; interpretative instructions take a comparatively shorter time to write and a comparatively longer time to execute. It all comes out in the wash; some methods are better adapted to specific tasks than others.

I suggest that to simulate a continuous function such as Joy's, with two randomly fluctuating (dichotomous) variables, the interpretative method reduces the chance that the random function (called ThrowDie in Chantal Roth's simulation) can be corrupted in the compilation of code. Though I can't speak as an expert -- perhaps one will show up.

No computer actually computes a continuous function -- computer code is digital. A differential equation, e.g., is converted to a difference equation before being executed in a program. The function can be made arbitrarily smooth.

(cf., the iteration of an n-sided-gon into the approximation of a circle. Or imagine that the randomly thrown straight lines of uniform length -- by which one can calculate curvature to arbitrary accuracy, by a Monte Carlo algorithm on a grid -- are not bound by grid lines, and nevertheless generates a beautiful regular curve, such as the sine wave of Joy's model, in a coordinate free manner. Are we looking at the wave function of the universe? If such a wave lives in the space of all possible correlated points of a parallelized 3-sphere, we need postulate neither wave function collapse nor nonlocality, to get the same strong quantum correlations as predicted by quantum theory.)

Tom

Tom,

Ok, I wasn't clear on the meaning of your statement. I looked at code that I was aware of being posted, and, didn't see rows of one's and zero's. I expected that such a task was far to monumental to undertake. Since I am not an expert, I asked anyway.

James Putnam

Joy, I've spent years now trying to understand why your critics' arithmetic differs from yours. And mine.

I concluded at some point that because they always assume the quantum probability measure, they deny the continuous function that completes the correlation measure. They never see it -- it never happens. Anything that might have happened is "nonlocal."

If one computes only on the basis of first order arithmetic, the probability is compelled to collapse to unity, as the wave function of a quantum observation is believed to collapse.

Introduce second order arithmetic (analysis), and the game changes -- there is no collapse, no nonlocality.

The error -- just as you always claimed -- is built right into the assumption of what space one is working in. Where first order arithmetic applies, a many-sided die gives one real result with n-results in linear superposition; where second-order arithmetic applies, there is no probability for a linear outcome. The order relation (the primitive binary relation) in second order arithmetic will fluctuate (0,1) (1,0) continuously -- if one is judging this fluctuation by first order axioms, one reasons that because the statement, 0 < 1 is true and 1 < 0 is false on the real positive line R, what is less than 0 (the "distinguished member") is - 1 and mathematically illegal.

In the analytical case, however, because we are not confined to the space of the real line (topological space S^0), the distinguished member is a complex double zero {0,0} such that a measurement function continuous from the initial condition, assuming the primitive binary relation and nondegeneracy, is either [0, +/- 1] or [+/- 1, 0] in which the closed interval makes the difference between judging results probabilistically on the open interval (- oo, + oo) and finding true deterministic correlation of left and right independent variables on parallelized topological spheres. S^0 is trivial; S^1 has the complex {0,0} but still allows the open interval. Only at S^3 do we encounter a closed manifold suitable for linear independence of the random variables; we know by complex function analysis that the only allowable results on the S^3 equator are + 1, - 1 and i (sqrt -1). We don't even need the complete physical space of S^7 to make the case for this subset of the parallelizable spheres, S^1, S^3, S^7 (for those unfamiliar with topology, the notation means Euclidean spheres of two, four and eight dimensions, accommodating division algebras from the algebraically closed plane to quaternion algebra (S^3) and octonion algebra (S^7).

Let the critics come forward with counterarguments, if they have them. They certainly weren't shy of expressing their opinions when it wasn't clear if the continuous function simulation of nonlinear input could be programmed digitally (thanks again to courageous Chantal Roth) -- what now? Nothing to say?

All best,

Tom

Hi James,

I better understand what you mean, now. I also wondered whether writing a digital program of a continuous function with nonlinear input would be a "monumental task." The technical question is, "Is such a function algorithmically compressible?"

Precise numerical implementation of a continuous function isn't possible in principle; however, if we accept the arbitrarily close points of a continuous line (as mentioned, the conversion of a differential equation to a difference equation) is sufficiently smooth -- then the problem remains how to randomize the input such that we know that wherever and whenever we insert on the line a pair of randomly generated dichotomous variables fluctuating between +1 and -1, the discrete choices remain linearly independent of the continuous function.

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.

The second worry that I had apart from the algorithmic compressibility of the framework, is that if it should prove simulable, there remains the question -- "is the simulation of a continuous function itself a continuous function?" Only today has my last worry been laid to rest -- having heard of the algorithm being replicated in at least two other machine languages. This is important, because it answers "yes" to the question and obviates any doubt that the model is independent of experimenter bias and the variables are linearly independent.

Here is a prediction you can quote me on:

Because Gregory Chaitin has shown that linear arithmetic functions have a built in uncertainty (Chaitin's Omega number is sensitive to the machine language running it), quantum computing algorithms based on linear superposition and quantum entanglement will fail. Preparation of the state vector in the complex Hilbert space biases the function. Better start looking for nonlinear solutions and the arithmetic nonlinearity that powers them.

Best,

Tom

Joy,

I'm speaking the 3D+1 geometry of non-linear optics. We're observing the same volcano but from opposite sides so see the same but nothing the same!

I derive the QM prediction geometrically using helical dynamics in the essay, obeying Malus's Law. IQbit; The Intelligent Bit.

Most recent experiments use 'weak measurement' statistical methods which are 'blind' to the cosine curves as it can't correlate time-resolved individual photon pairs. Now single photon production is more a reality I've proposed a refined Aspect experiment which should reproduce the 'orbital asymmetry' present in

Tom, Joy,

You should find compressible continuous functions in helical paths. I've been discussing in an APS Quantum Physics blog. I reproduce a recent post below, and can probably dig out a thesis on the Gottfried-Jackson angle and helical frame if you can't find anything helpful.

Post; "I haven't found any hint that it may represent a longitudinal component, but studying related experimental results (grazing grounds I've always found useful), does give consistent hints about axial helicity consistent for instance with Jackson-Minkowski rotation viewable from the Gottfried-Jackson frame. The latter is nothing to do with me; but on the 'GJ angle' between the lab frame momentum of the Higgs and an emitted photon in the resonance rest frame. Rotating by the angle alpha on the y-axis gives the helicity rest frame. (alpha differentiates the z-axis in that frame and the approaching photons 3-momentum vector). I'll find some authoritative links on that if you want.

But translating the gobbldygook into possible physical models with a simplified description, it looks something like the (spin-0 but possibly spin 2!) Higgs may facilitate a marriage between particle and antiparticle into a coherent (massive) and conserved toroidal dynamic by perhaps adding the second supplemetary 'winding' spin of the body. So the primary spin may be considered as the 'ring' rotating (giving the primary helicity on axial translation[motion]), but the torus is held together by the poles (of the dipole) spinning, which produces the twin counter 'winding' of the torus.

The logical analogies of this are quite wide. For instance spin 1/2 and 1-1/2 can be physically represented by the charges ending up in the same place after only a half integer rotation of the ring. Sure, all circumstantial and highly speculative, but only in a similar way to jigsaw puzzles, and I've tried hard to falsify it and can't. I really thought Bells Theorem would destroy it, but, as you've seen, it avoids Bells (Bohr's 1020's) assumption of singles spin states because it has another couple hidden away!

The 'no-mirror symmetry' matter is also critical to the EPR case. Do take a look at that aspect and comment, as it seems to act as 'information pre-held' between the entangled particles." [PJ, 3 days ago]

best wishes

Peter

Let's take a trip back in time, to just a few months ago, when Richard Gill said, "We can be grateful for Christian that he has had the generosity to write his one page paper with a more or less complete derivation of his key result in a more or less completely explicit context, without distraction from the author's intended physical interpretation of the mathematics. The mathematics should stand on its own, the interpretation is 'free'. My finding is that in this case, the mathematics does not stand on its own."

Now that we know that that finding is wrong -- and instead that the mathematical interpretation of Bell's theorem is not free, and in fact imposes a dependent condition on the outcome -- where's Gill?

And where's Aaronson, who said, "I can't think of any better illustration than the comment thread below for my maxim that *computation is clarity.* In other words, if you can't explain how to simulate your theory on a computer, chances are excellent that the reason is that your theory makes no sense!"

The simulation of Joy's function looks pretty clear to me.

C'mon, guys. You dished it out.

Tom

Joy, I have tried my hardest to understand the sociology of this conflict, and I cannot. Even considering that the stakes are high, and that your idea is new, there are numerous high-stakes, new-idea propositions in theoretical physics that don't attract the level of disdain and uncollegial criticism that yours does.

Einstein was known to warn against the "cult of personality" in matters of scientific importance. He rejected all attempts to enshrine his own work into such a cult, although not altogether successfully.

If your research were such that it substituted techno-babble and mathematical esoterica for meaningful content (as so many papers these days do) -- I could understand the reaction. Such is not the case, however:

Nothing could be clearer than your eqn 3 in the above attached paper, and the conclusion, "Unless based on a prescription of this precise form, any Bell-type analysis simply does not get off the ground, because without completeness there can be no theorem."

Your argument is completely kosher and coherent -- from the precise prescription for a simply connected co-domain to the continuous measurement function that demonstrates the case.

That's exactly what science is suppposed to do: conjecture and predict.

All best,

Tom

Hi Joy,

Seeing that an old George Musser blog on quantum teleportation has attracted new comments, I did some surfing and found what I think is a meaningful discussion among Lawrence Crowell, Ray Munroe (rest in peace), Fred, me and you, over topological foundations, beginning with Lawrence's post on 22 December 2011, 17:40 GMT.

It would be nice if we could stimulate a new dialogue in that collegial manner -- Ray was wrong; however, he was wrong for the right reasons. One really does need to understand the topological principles that drive your result. And they need to understand that the topology isn't something that you made up to suit the case.

Best,

Tom