A moonshine dialogue in mathematical physics by Michel Planat

Dear Michel Planat,

I wonder if you could clarify your claim about Bell's inequality on p. 4 of your essay. You state that the sum of expectation values for the spin measurements you have in the diagram lead to the norm 2root(2), but of course one can only calculate an expectation value given the initial state of the pair of qubits. It is most usual to discuss these sorts of experiments done on singlet states, but for the singlet state the sum of expectation values for your experiments will be at most 2, which is within the bounds for a local theory. (This is because in the singlet state there is zero expected correlation between the outcome of x spin on one side and z spin on the other. As you have it written, the sum of expectation values is 0, but you can get 2 by relabeling.) So you evidently have some other state in mind, but you do not specify it.

This is important because any quantum state violating the inequality must be entangled. No product state will violate the inequality. I'm also not sure what you mean by "the proof of Bell's theorem does not mention entanglement". Bell's own proof, of course, does not mention entanglement, but predictions of violations of the inequality using the quantum formalism do require entangled states. And the derivation of the 2rt(2) bound, which is a property of quantum theory, will certainly require use of entangled states. So what is your comment about entanglement meant to say?

Any help on understanding what you have in mind here would be appreciated.

Regards,

Tim Maudlin

Michel,

I returned to see whom I rated as I tend to rate some and not others. I find that I did yours on 3/26 and I subsequently found your comments.

The cat map reminds me of machinations of Euler's Identity which I reference in my essay and simulations I used to do in aerospace.

Jim

Michel,

Maybe a stretch, but I was thinking of logarithmic spirals: In polar coordinates the logarithmic curve can be written as r = ae^b0. Logarithmic spirals occur everywhere in nature, from sea shells to galaxies, for example.

The dynamics of Euler's function. Then there are phase portraits of solutions: http://livetoad.org/Courses/Documents/ea8f/Notes/euler_formula.pdf

Jim

Dear Michel,

From Maudlin's subquestion 1) „Which mathematical concepts seem naturally suited to describe features of the physical world, and what does their suitability Imply about the physical world?"

I suggest three main candidates for the mathematical concept:

bit (it was the subject of the competition FQXi 2013);

exp(x) (You know the unique features of this function);

Euler's identity.

There are other useful functions, but less importance.

Suitable use of pervious can to describe features of the physical World.

What are your main candidates?

Best Regards,

Branko Zivlak

Dear Michel,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

Dear Michel,

I think that we both have the same goal: to understand quantum mechanics from a geometrical point of view. At the end, our approaches will be converge.

BTW, there is a new Springer journal Quatum Studies

(they send me an email). Maybe interesting for you?

Best

Torsten

Dear Michel,

Reading your essay is challenging, if one tries to reach beyond the pleasant dialogue effect that makes you a heir of Plato, Galileo, Lewis Caroll, Donald Knuth (Surreal numbers), or even Leslie Lamport [1] for instance, and actually see all these objects you are uncovering.

As for me, I have no other option than to trust you (as is usually the case in science --perfectly alright), with all these breathtaking connections. And though there is a lot of material I can hardly comprehend, the journey is pleasant.

So your essay proceeds by example, in the only really convincing manner, by showing new --i.e., unexpected-- occurrences of mathematics at work in uncovering new pieces of physics. (New is important, because once a connection is well known, we get acquainted to it, it vanishes into the background.)

Aren't these deep connections in fact within mathematics, since we are in a part of physics that is nearly completely mathematical? The concepts of physics we are considering are already extremely developed mathematically, and it is `natural' that they lend themselves to more mathematics (more connections, more structure). Of course, there is some selection to do because there are some experimental results that give some borders, and we naturally keep only the mathematics that speak to these facts.

Along this line you quote E. Frenkel, an if ``the Nobel prize in physics is really a Nobel prize in maths'', conversely Villani's decisive contribution that won him a Fields medal --Landau damping-- could count as a prize in physics too.

So, is it an implicit answer that you left to the reader to devise, that physics is by definition mathematical, therefore mathematics has to be efficient (reasonably or not being a matter of appreciation we don't quite have a standard to evaluate)?

I would like to try an example that (nearly) anyone can grasp: It has been shown that from any position, the Rubik's cube can be taken back to a reference position in 20 moves or less http://www.cube20.org/ (or 26, depending on what counts as a move --alas, not 24!). Given the huge size of the configuration set, its diameter being less than 20 means that the space is very highly connected. It is my best example to convey the shock of this connectivity to the layman. The many `coincidences' that we see, slowly and piece by piece, between mathematical structures initially seen as occurrences of distinct things appear to me in analogy (a sound analogy) with this high connectivity. Each time we are surprised by unexpected coincidences, hints us that we are far, far from seing the whole picture.

(Even simply when we keep on finding --or being shown-- more relationships between numbers, thus making some which look like nothing special, say 1729, very special indeed, we are in such a situation. In my comment on Wise's essay http://fqxi.org/community/forum/topic/2494, I have inserted a quotation on Lazare Carnot, to show how difficult it can be to see a structure, when you have not got properly acquainted to it, when you stick too much to initial vantage points. It is again such a simple example, I know of none which can be more convincing.)

We can also figure out that many, many things now look to us `naturally' the same, because we see groups behind them. The concept of group is so simple, so obvious, that it seems that for long, no one thought it could be of any use to take the pain to state it explicitly (certainly much like the concept of zero). But once it was there in clear, we started recognising it everywhere, with many cross-connections

So we slowly uncover `miracles', or mere connections. To begin with, we see many scattered, unorganised connections, that we begin to understand better when a structure shows, which links these connections into a whole.

Again, I cannot help but make the parallel with Bach-y-Rita experiment (see my essay), where from what at the beginning is confuse feelings in the back skin, objects, distinct, moving finally emerge, because their structure has been recognised, built up, abstracted.

So, congratulations on this moonshine explosion of new objects!

[1] Leslie Lamport, A simple approach to specifying concurrent systems, Commun. ACM, 32:1, January 1989, pp.聽32--45.

Dear Micheal,

I have been very busy with other work. It occurred to me that the dessin d'enfant for the simple y = x^3 is the same as the Dynkin diagram for the SO(8) group. It seems to me that one could easily build such constructions for the heterotic groups or for the sporadic groups with polytope realizations. These are in effect roots with Galois and abstract algebraic content. I was wondering if you had any insight into this.

Cheers LC

At it its core, math is about numbers. Natural number arises from counting orders and naming convention for the uniqueness of a places in a sequence. In that sense, the physical world is a book written in natural numbers.

Using an analogy, the English alphabet has 26 letters, and with the alphabet infinite books can be written. We examine the books and find that each book consists at least one of the five vowels, and each word is less than 100 letters long, and so forth.

We are puzzled by how a random book can be that way. But need not be so, if we realize that the rule of writing a book is quite simple although the end product is somewhat complex. We start with a letter, then a word, and then a passage, a chapter and so on. Each step has some simple but irreducible rules. This process masks the simple relationship between a book and the alphabet, if we simple look at them without the steps in between.

Dear Michael,

My point was not that there was an isomorphism, but that there might be some sort of relationship.

In addition I am wondering whether dessin d'enfant can be used to look at a general type of problem. You illustrate how Bell's theorem can be realized this way. I am interested in looking at whether this can be used to look at a general class of SLOCC groups. If they can't be found equivalent according to nilpotency on their Cartan subgroups then there is a polynomial invariant under that group which separates them.

Cheers LC

Dear Lawrence,

This is the type of application we can discuss. Until now, I focused on dessins due to their relationship to quantum geometries and contextuality as in my [12] and [17], now I mentioned in the essay the link to most sporadic groups, there are plenty of other applications, some have to be discovered. Cheers.

Michel

Dear Michel,

Thank you for a very enlightening and enjoyable essay! I really like how you combined rigorous analysis with a very entertaining narrative, and it's quite a thought provoking educational resource. It reminds me of genres along the lines of Flatterland. You presented a very intriguing discussion on Bell's theorem and the moonshine group, and your remark on self referencing was the perfect conclusion. I give this now the highest rating.

Thank you for your very stimulating comments and ideas on my essay (if you haven't yet rated my essay, I ask that you please take a moment to do that). I responded recently to your post and your comment inspires me to particularly revisit time entanglement and incompleteness from the perspective of gate dynamics. (And thank you for the time entanglement of connecting my past and present essays :) ) I look forward to seeing your further work and correspondence in the future. Take care,

Steve Sax