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

Richard Gill

Nick, can you give us links/references? "That's Diederik (Dirk) Aerts' iconic gedanken from the early 80s which I've noted in this very thread. Back in 2008 I dragooned him into a discussion about the two-vessel paper on the late Guardian Unlimited Talk. I'll send you the htmls if you're really interested. He discusses his friendships with Bell and Aspect in Geneva among other things ... some great material there. (And he was no fan of Joy's Bell papers at least at that time.) Posters were saying he was silly and crazy, which of course is the last thing he's ever been, and I used that as bait to inveagle (sp?) him and am certainly glad I did. The man is a prince."

Richard Gill

Tom said " 'The mathematical facts (tautologies) stand like a rock.' We agree, Richard. Here's the rock you can paint any color you wish: All real functions of a real valued variable are continuous."

Is that a physical fact or a mathematical fact? Or a computational fact? Please give references. I think I know a lot of real functions of real variables which are not continuous. For instance, the indicator function of the set of rational numbers. It sure exists in my *mathematical universe*. And in most other mathematicians' mathematical universe.

[deleted]

Richard,

I don't believe there are any online GUT archives. I have the discussion thread archived and will be happy to email it to you as a zip of several html files. Just be warned that because of the social format of the discussion there's an awful lot of sophomoric crap (some of it, alas, mine) along with the serious and relevant stuff. However you can just hit Ctrl-F and "Aerts" and catch the good material.

Nick Mann

Sophomoric c'est moi.

Thomas Ray

Richard wrote,

"Tom said ' 'The mathematical facts (tautologies) stand like a rock.' We agree, Richard. Here's the rock you can paint any color you wish: All real functions of a real valued variable are continuous.'

"Is that a physical fact or a mathematical fact? Or a computational fact? Please give references. I think I know a lot of real functions of real variables which are not continuous. For instance, the indicator function of the set of rational numbers. It sure exists in my *mathematical universe*. And in most other mathematicians' mathematical universe."

It's fundamental to topology that continuous functions graph unbroken over the manifold (Brouwer). However, before we discuss the details of how we get from real analysis to this theorem, to physical events, please tell me if you have any objections to Carl Mummert's commentary here.

Best,

Tom

Richard Gill

According to some definitions of the phrase "all functions", it can be a true theorem that "all real functions are continuous". In some intuitionistic systems, and some constructivist systems, this is true. As Zhen Liu wrote "There exists a Grothendieck topos in which the statement "every function from the Dedekind real numbers to the Dedekind real numbers is continuous" is true in the internal logic. To be a little more precise, ... "

In the mathematical universe usually inhabited by physicists, it is not a true theorem.

In other words, if your definition of "function" is rather restrictive, so that hardly anything is a function, you can make sure that all of the things which you deign to call functions are indeed continuous.

pjackson

Richard,

You believed 'you' couldn't pre-decide a setting sequence for my classical 'classroom' experiment reproducing the QM predictions without non local state-reduction. I'm not sure if you've missed my posts (a bit buried) or are struggling with prejudices, but I've taken up your challenge, so now challenge you. You said' wisely; "If we put nature into the picture as well, other possibilities arise." That is precisely what I'm proposing; No 'new science' just more consistently 'joined up' physics.

I offered one random 20 setting sequence, so please provide the second, or both sets if you prefer, and also more if you wish. For the moment they're simplified to 30 degree increments. I offered; 0,150,180,90,60,30,120,90,180,60,180,0,30,120,150,60,180,120,30,60.

I'll then let you run the experiment yourself if you wish as I walk through it. The interesting part may not be so much the derivation of local violations at each detector, that's already experimentally verified and only needs consistent interpretation, but more the intuitive way the local findings then resolve precisely to the weak measurement findings.

If anybody else would like to play Alice or Bob (or both if you're so inclined!)) and supply a set of 20 or more random numbers at increments between 0 and 180 degrees, I'd be delighted to run them, and explain the particle state they model.

Best wishes

Peter

Richard Gill

One setting is supplied at a time. One to Alice, one to Bob. I'll use the CHSH experiment settings for polarization of photons: Alice's will be either 0 or 45 degrees, Bob's will be either 22.5 or 67.5 degrees.

You can tell us what will happen in Alice's classroom when her single setting is delivered. She won't know what setting Bob is being given.

And what will happen in Bob's classroom when his single setting is delivered. He won't know what setting Alice is being given.

After that, anyone who likes can either do the math or write a computer simulation which will tell you the statistics after say 800 repetitions.

pjackson

Richard,

OK. I hope you're an Occam fan as it's simpler than imaginable. It also invokes 'nature' not QM's assumptions, so Bell's maths are fine starting from where he did.

Alice gets her setting and sets her (detector EM field angle) dial just before arrival of the 'heads tails' signal. Bob does exactly the same with his entirely discrete (independent) field setting at the same moment, far away. They could also use rotating filters and twin photomultipliers, but in either case we have an entirely 'discrete' and Local EM field model (DFM).

Her H/T signal negotiates the journey in the state sent, so opposite to Bob's. She records a binary output for collation, which we'll call Red or Green. However we now invoke nature, and indeed the experimental finding of quantum correlations of sequential arrivals in SINGLE particle/detector experiments. (I assume you're familiar, but citations available).

We first find we can 'say' more than just 'red' or 'green' of spin half quantum spin. But lets show the violations in the rough first to show they work, and then rationalise intuitively.

If you'd be so kind as to give me both the first settings and coin toss result i'll kick it off. Or if you're happy I'll just pick two of your above nominations.

Many thanks

Peter

Richard Gill

No Peter. I want an intelligible description of what happens in the two class-rooms. Each is delivered a closed envelope with one setting. I've told you what the settings could be.

The classrooms may have communicated in the past (obviously, they have) but now all communication is forbidden. Inside each classroom, the envelope is opened.

Each classroom has got to "output" a binary measurement outcome. We can call the two possible values "up" and "down". They can write it on paper and put it in another envelope.

Repeat 800 times and collect the statistics. We'll calculate four correlations and investigate CHSH.

pjackson

Richard,

That's fine. Alice and Bob haven't spoken for years. But we mustn't forget there IS a 'communication', one way only, from the very Stern Professor Gerlach, the tosser who sends each of them the Heads/Tails signal (spin 1/2 'up/down' quantum angular momentum). He has a 'position' in space so also dictates the opposing orientation of the propagation/arrival vector and spin axis.

I just need to re-jig the experimental protocol to change the 0,30,60,90,120,150,180 degree angles to your simper 4 cases. That means that while Alice (Bob) sets the detector field angle, the students can (one at a time) play at being the actual photomultipliers, so write down the binary result each time.

Frankly, including the odd 90 and 180 degree setting does help better explain the physics, as the DFM is primarily a physical rather than mathematical mechanism, but it can use whatever fair conditions you wish.

If that's all ok I'll give it a test run and revert with the full SP.

Thanks.

Peter

Richard Gill

So there are *five* parties.

There is a mysterious Source, Mr. Stern-Gerlach, who sends some kind of information or stuff or whatever to the two classrooms (bad name, since it is associated with measurement devices, not with the parametric down conversion laser stuff at the source).

There are two other External Setting Sources who quite independently send a request (setting) to the two classrooms. There are two Students in their two respective classrooms.

You want to call the students Alice and Bob. I would have preferred to give those names to the people who call the settings, who are outside the classrooms and far from the source, too. However we could call them Angus and Barbara, if that helps.

I am still waiting for the answer to this question: "I want an intelligible description of what happens in the two class-rooms. Each is delivered a closed envelope with one setting. I've told you what the settings could be."

Thomas Ray

I didn't know that physicists inhabit a mathematical world, Richard -- although I haven't yet read Max Tegmark's book, so perhaps I will learn differently. I also agree with Tegmark's premise, so I hope to learn as well that the mathematical universe includes all mathematics and not just the mathematics that anointed physicists and statisticians choose.

The dilemma for you, is that the constructive mathematics developed by Brouwer, Dedekind, Weierstrass and others is supported by constructive proofs. These proofs translate directly to physical reality by way of self-limitation; i.e., infinities don't appear completed and in need of normalization because -- guess what? -- "all real functions are continuous."

A function is a 1 to 1 relation between input and output. Self limiting and continuous, just like what? -- real life.

Quantum theory is nothing like real life, and the only way one will ever beat all "loopholes" in Bell-Aspect experiments is with a constructive proof of Bell's theorem.

Unfortunately for Bell loyalists, the Bell-Aspect program has a self-destruct mechanism built into it -- a non-reality that refuses a self-limiting function; conventional quantum theory will never be able to prove existence of a quantum mechanical boundary to the classical world that we experience.

Today or tomorrow, Bell's theorem will self destruct, in the physicists' world, not the mathematicians'. Mathematicians will continue to accept it, along with its domain boundary, just as we learned to accept the boundary of Euclidean geometry and the truth of the theorems that apply there, up to the fifth postulate.

Best,

Tom

Richard Gill

Perhaps it's more accurate to say that physicists *populate* a mathematical world. They have so far found little use for constructivism or intuitionism. Brouwer's own intuitionism makes his own fixed point theorem invalid.

Anyway, if the universe is ultimately discrete and finite then real numbers and real functions do not "exist" in reality, either. (Or: every topology is discrete and every function is continuous and simultaneously discontinuous. Both properties are empty). Discussions of abstract formalisms which either permit the formal existence or forbid the formal existence of continuous functions, has as much relevance to physics as medieval discussions of how many angels could dance on the head of a pin.

Thomas Ray

" ... if the universe is ultimately discrete and finite then real numbers and real functions do not 'exist' in reality, either. (Or: every topology is discrete and every function is continuous and simultaneously discontinuous. Both properties are empty)."

Real numbers and real functions don't have to exist in reality, if all real functions are continuous. Topologies are discrete only if m != n (invariance of domain, R^n = R^m IFF m = n). Since m = n in S^2 X S^2 (where R^3 is compactified by a point at infinity, accounting for m = n in the Euclidean limit of geometry), the topology of S^3 is continuous. (Poincare conjecture proven by Perelman.)

Real numbers and real functions are derived from the physical continuum, without having to assume physical reality. That makes the mathematical model completely independent of the physical result -- a principle you say you support yet have yet to demonstrate.

"Discussions of abstract formalisms which either permit the formal existence or forbid the formal existence of continuous functions, has as much relevance to physics as medieval discussions of how many angels could dance on the head of a pin.

Perhaps you will want to consult your alchemy texts to find out how many bosons can dance on the head of a pin.

John Cox

Richard,

If I may. I think what Tom distinguishes topologically of what characterizes the Reals is not only the criteria of both rational and irrational, but that they are immediate values of points inside the box obtained by multiplying all three orthogonal co-ordinate binaries at once, rather than a composite as the average of products of two binary multiples from adjacent orthogonal sides which leaves an ambiguity as to whether the point is on the sides of the box or absolutely interior. The Hamiltonions for multiplication and Octonions for division.

If your serious about radding vid cards with hardwired Hamiltonions, I don't Geek but if the cards don't carrying the Octonion matrix and what for some good reason I dunno is called the Fano Plane Mnemonic, you sound like you could html your own plug-in.

Play Nice kids, jrc

Richard Gill

Calculus was invented because integration and differentiation are easier than summation and differencing. We only ultimately need discrete mathematics, we only ultimately need the rational numbers (and we don't need all of them, only very few). We use "continuous" mathematics because it is easier, at least, as long as we only need good approximations. And we do only need good approximations. We use calculus to give us good approximations to solutions of discrete mathematical problems.

Similarly, continuous mathematical models are conceptually more easy than discrete models.

Thomas Ray

"Calculus was invented because integration and differentiation are easier than summation and differencing."

What a huge ration of baloney. The calculus was invented to explain the rate of change of the rate of change that characterizes continuous physical phenomena such as gravitation and optics. That infinite integration is dual to the differential reinforces one's suspicion that continuous functions are unbounded and independent of the particle phenomena derived from the continuum; i.e., foundationally free of singularities.

"We only ultimately need discrete mathematics, we only ultimately need the rational numbers (and we don't need all of them, only very few). We use 'continuous' mathematics because it is easier, at least, as long as we only need good approximations. And we do only need good approximations. We use calculus to give us good approximations to solutions of discrete mathematical problems."

Well, this explains your remarkable innocence of analysis and topology.

"Similarly, continuous mathematical models are conceptually more easy than discrete models."

So what?

Alan Lowey

Forgive me, but I've just read something from another blog discussion which I'm assuming is relevant to this one.

Someone posted this sentiment:

Quantum physics is common sense to anyone who understands the statistics of 2 coin flips and how those statistics are affected by observing the coins...

If I flipped 2 coins and at least 1 coin landed heads, then whats the chance both landed heads?

Its 1/3, not 1/4 or 1/2 like most people think, because there are 4 ways 2 coins can land and I only excluded "both tails" when I said "at least 1 coin landed heads" so that leaves 3 possibilities and I asked what is the chance of 1 of those 3 things which happen equally often. Its 1/3. If you still don't believe it, flip 2 coins many times and only ask the question when at least 1 of them lands heads and you will see that 1/3 of the time you ask the question they both land heads. The flaw in Human minds is the need to choose 1 of the coins and say it certainly landed heads, but I did not tell you any specific coin landed heads, and it does change the answer if you take that shortcut.

This kind of thinking is the problem. A real world event of tossing two coins is common sense. One coin has to land before you can claim "at least 1 coin landed heads". The rest is 'quantum quackery'..

Alan

[deleted]

Richard,

I understand your points about the calculus and only needing some of the rational numbers for continuous functions, but I'm beginning to grasp the arguments of Joy Christian's framework of a measurement space which shows that quantum correlations are themselves not independent of continuous functions when space is measured topologically. Please bear with me, I came late to a higher learning and even later returning to it, so I approached this whole idea from a position of ignorance that precluded naivety, but which also spared me that peculiar prejudice which goes hand in glove of any long process of ingrained learned disciplines and practice. And having gotten to a point of starting to assimilate new knowledge, it feels great at this late stage of the entertainment to still be able to learn anything. Let me share this.

Topology does not require calculus which is what makes it so fast in computer graphics. And what drew my curiosity to this blog to begin with was that topology seemed to offer a possibility of accounting for the distribution of energy in a volume of electric charge in accord with inverse square law without some of the difficulties and consequences entailed by any calculus, not least of which is a mathematically inherent feedback at the boundary of minimum density limit. And it doesn't take calculus to get that feedback, it occurs in plane old solid geometry too. The multiplication and division of trivectors afforded by Hamiltonions and Octonions solves that problem with an absolute value interior of an orthogonal refererence (which could be taken as an octant of a sphere) obtained as an immediate result from all three co-ordinate binaries at once in a manner that should be near and dear to the heart of any committed Quantum Machinist; a natural pattern of results from pairs production which are plotted on a compact matrice with an operational norm of simple referencing. In the Hamiltonions that matrix is a four place by four place grid and the norm is to cross reference by rank and file which kind of obsures the recognition that you are multiplying two point binaries on the time metric inside the box instead of two point binaries on the outside of the box which would be one pair of adjacent sides.

There lies the distinction of real numbers topologically being associated with real space, and could use only some rational numbers but that does not and cannot discard the set of all irrational numbers from the set of Reals.

Apparently division of trivectors is a bit more tricky, but the pattern of pairs production among three binaries results in an eight place by eight place matrice and the Fano Plane Mnemonic is like a flow chart of the operating norm. I'm just starting to delve into how to do any math with it but the form of operating on trivectors to plot progressive points on the time metric interior of the orthogonal has an obvious utility. I'm guessing Fano is an individual who devised the simple flow chart mnemonic.

In my fourties I worked heavy construction on a (tough, good) crew of a small undercapitalized company competing for county and state highway bridge projects. Here is a macroworld example of Joy's framework. The surveyor contracted to set benchmark and stake work limits used a computer laser-rangefinder but we worked with old four post transit-levels and plumb bob from crossed string lines that could span up to 50 foot, more on one job. I'd always start over the bench and swing azimuth til the meniscus was dear to true then scope several hard points to make day benches and proceed then to sledging 2by4's into the dirt to build batter boards plumb up from limit stakes to pull the string lines from as the building lines for layout. In practice you want to keep the elavations of opposite batterboards about equal and level, and those of any crossing lines a little above the longer lines because the weight of the plumb bob will pull the upper line down a bit ( low stretch braided nylon line is used so under a stretch load of about 200 pounds it won't untwist ) and it's critical to locating a point on the ground underneath the lines that you can see that the plumb line is not causing a deflection of either horizontal line. This is analogous to what occurs if you measure space orthogonally without topology. Topological measurement space would be analogous to using the GPS computerized laser-rangefinder to locate a precise point in mid air and drop your plumb bob down from that. The string lines are like the imaginary lines pulled from a binary co-ordinate on one side of the box across to a binary co-ordinate on an adjacent side, but you have to do that three times to locate the point they all cross interior of the orthogonal. That means you have to allow associated points along those crossing lines to prove that none are deflecting the others and that obtains as a continuous function on any line. And the string-line example is macroscopically falsifiable and testable. Check for non-deflection with a red dot laser grazing any string line or in the case of the upper elevation line any deflection off the intersecting vertical plane.

Without a continuous function along the lines interior of the orthogonal, a point on the time metric cannot be proved by standard QM metrhodology as it can be by Dr. Joy Christian's topological metrhodolgy. Hence,classical to quantum correlation is established.

Sorry for the wordiness, jrc

John Cox

OOOPS!!!

jrc post 3/16 @17:56 GMT

bottom of paragraph 2 error, it should say:

...obscures the recognition that you are multiplying two trivector points on the time metric.... jrc

John Cox

Tom,

per your response to:

"Similarly, continuous mathematical models are conceptually more easy than discrete models."

of:

'So what?'

one might add, 'and so why not? KISS.

By Jove! I think I'm starting to get this! thanks much, jrc

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