In the subject of gauge theory a central aspect is the interaction form. These types of continuous homotopy or homotopy-like constructions involve curves that can be adjusted in certain ways so that an index is invariant or constant.

I am back home, but of course I have a lot of things to attend to here. I will try to expand on things in the future. The whole subject involves orbit spaces, or quotents of groups or spaces. The subject of four-manifolds is centered around the moduli, a 5-dim space that in a hyperbolic setting can be the AdS_5. Of course the hyperbolic setting is not Hausdorff and there are other problems. However, this is a form of orbit space that is mapped to the quantum SLOCC types of theory.

Cheers LC

Dear Lawrence,

You start with Goedel "no mathematical system can ever prove all possible atements as theorems about itself" and you propose HOTT (homotopy and type theory together) which of course fits the great categorization process at work in mathematics. I found a very recent preprint of Yuri Manin pointing the same direction http://xxx.lanl.gov/pdf/1501.00897.pdf

"Information is physical" and you seem to suggest that "mathematics is physical", and both are quantum (in your conclusion). I like your approach and thank you for a very original and readable text with non-trivial concepts.

This year, I am exploring the most discrete and anomalous/sporadic object ever found. I hope you can comment on it.

Best.

Michel

    This is in line with motives, categories and fundamental quantities as discrete elements from homotopy or varieties. This is as you say in line with category theory. In fact I think that ultimately the fundamental observables in the universe are topological categories, similar to Etale or Grothendieke theory.

    I see there being in a sense what I call the "body" of mathematics, which are those aspects of mathematics that can be, at least in principle, solved on a computer, and the "soul," which is the continuum mathematics of infinitesimals and infinities. My essay concentrates on the body, and not so much on the soul. I think for physical science the body is more directly associated with what is observed in the universe.

    The "body-soul" duality I tend to advocate is something one can "wear" as needed. I might by virtue of some argument want to invoke a mathematical objectivity of sets, continuous spaces and even to the point of Platonism. At other times I may put this entirely aside. In my essay I largely put this aside.

    Garrison Keillor has a feature on his show "Prairie Home Companion" called Guy Noir with the opening line, "On a dark night in a city that knows how to keep its secrets, one man seeks answers to life's persistent questions; Guy Noir, Private Eye." That is about my sense of the question about the relationship between physics and mathematics. We may never know for sure. Further, the universe may have a kernel of structure, symmetry and order to it that appears in a fractal-like form at different scales, but where nature also has this inherently chaotic or disordered nature to it as well, which I think is distilled down to the stochastic nature of quantum measurement.

    I will try to get to your essay in the near future. I just got back from some travelling.

    Cheers LC

    Dear Lawrence,

    Every scientist has his own way and velocity in going through the wonderful secrets of nature. At FQXi you already wrote many excellent essays like "Discrete time and Kleinian structures in Duality Between Spacetime and Particle Physics". I wonder if you already looked seriously at the concept of an orbifold? I see that it plays a role in the VOA associated to some sporadic groups. I also found http://arxiv.org/abs/math/0505431 for your topic of this year.

    I appreciate much the impetus you gave to my essay. After my first participation I learned how it works and don't take care to much of the lazzy inappropriate votes. You received from me the best endorsement. The goal is a continuing friendly discussion about the topics of mutual interest.

    Best.

    Michel

    Dear Michel,

    Of course I am aware of orbifolds with respect to superstring theory. The vertex operator algebra with partition function p(q) =tr q^N = Π_{N}1/(1 - q^n) is related to the Dedekind eta function. The trace results in the power [p(q)]^{24} In this there is a module or subalgebra of SL(2,Z), eg S(Z) ⊂SL(2,C), that forms a set of operators S(z)∂_z. This module or subgroup is then over certain primes, such as either Heegner primes or maybe primes in the sequence for the monster group. This is of course related to the Kleinian groups and the compactification of the AdS_5.

    The AdS_5 compactification issue is something I started to return to. I gave up on this after the FQXi contest over this because it did not seem to gather much traction. The AdS_5 = SO(4,2)/SO(4,1) is a moduli space. The Euclidean form of this S^5 =~ SO(6)/SO(5) is the moduli space for the complex SU(2) or quaterion valued bundle in four dimensions. The AdS_5 is then a moduli space, and the conformal completion of this spacetime is dual to the structure of conformal fields on the boundary Einstein spacetime. This moduli is an orbit space, and this is the geometry of quantum entanglements.

    If this is the case then it seems we should be able to work out the geometry of 3 and 4 qubits according to cobordism or Morse theory. My idea is that the Kostant-Sekiguchi theorem has a Morse index interpretation. The nilpotent orbits N on an algebra g = h + k, according to Cartan's decomposition with [h,h] вЉ‚ h, [h,k] вЉ‚ k, [k,k] вЉ‚ h

    N∩G/g = N∩K/k.

    For map Ој:P(H) --- > k on P(H) the projective Hilbert space. The differential dОј = = П‰(V, V') is a symplectic form. The variation of ||Ој||^2 is given by a Hessian that is topologically a Morse index. The maximal entanglement corresponds to the ind(Ој).

    In general orbit spaces are group or algebraic quotients. Given C = G valued connections and A = automorphism of G the moduli or orbit space is B = C/A. The moduli space is the collection of self or anti-self dual orbits M = {∇ \in B: self (anti-self) dual}. The moduli space for gauge theory or a quaternion bundle in 4-dimensions is SO(5)/SO(4), or for the hyperbolic case SO(4,2)/SO(4,1) = AdS_5. The Uhlenbeck-Donaldson result for the hyperbolic case is essentially a form of the Maldacena duality between gravity and gauge field.

    With your presentation of the П€-problem and the connection between the Bell theorem and Grothendieck's construction, you push this into moonshine group О"^+_0(2). This leads to the conclusion or conjecture, I am not entirely clear which, that the moonshine for the baby monster group is coincident with the the Bell theorem. The connection to the modular discriminant is interesting. This then gets extended to О"^+_0(5). Your statement on page 7 that g(q) = П†(q)^24 is much the same with what I wrote above. There is a bit here that I do not entirely follow, but the ideas are intriguing. I would be interested in knowing if the hyperbolic tilings of О"^+_0(5) have a bearing on the discrete group structure of AdS_5.

    You may be familiar with Arkani-Hamed and Trka's amplitudhedron. The permutations arguments that you make give me some suspicion that this is related to that subject as well. This would be particularly the case is the О"^+_0(5) is related to the tiling and permutation of links on AdS_5 given that the isometry group of AdS_5 is SO(4,2) ~ SU(2,2) which can be called the twistor group. This is connected with Witten's so called "Twistor-string revolution."

    Thanks for the paper reference. That looks pretty challenging to read. I am not quite at the level of a serious mathematician, though I am fairly good at math and well versed in a number of areas.

    Cheers LC

      Dear Lawrence,

      I am really impressed by your knowkedge of so many things related to string theory. I propose that we start a collaboration because we have so many things to share and we are also quite complementary. I was very enthousiastic in writing the essay because new relations between several parts of maths and physics was taking place as by magic and also thanks to the computer. This is unreasonable in some sense!

      My best wishes,

      Michel

      Dear Michel,

      That might be interesting. I have been pondering how it might be that ﾎ"^+_0(5) is related to the tiling and permutation of links on AdS_5. The quotient SO(4,2)/SO(4,1) = AdS_5 is not an entanglement group, at least not as I know, but this might have some relationship to entanglement. This might be through the ﾎ"^+_0(5). Particularly if this is related to Langlands in some way.

      Cheers LC

      I want to give your paper some time Lawrence..

      But I want you to know that your essay is on my radar of important papers to read for detail (and I have skimmed it), while the contest is still underway. I see that you mention Bott-periodicity, which is a topic I would have touched on in my essay - had I allowed myself adequate time. My entry this year is briefer than I intended, because I did not.

      I was happy to see that you mentioned the HOTT program, which I also find to be interesting and relevant. I especially like that their pursuit of univalent foundations is geometrically constructive, but it is tied to a rigorous analytic proof checking engine. I find this usage of constructivist Math as program code particularly elegant.

      More later,

      Jonathan

        In your Bio you wrote: "I think it is likely there is some subtle, and in some ways simple, physical principle that is not understood, or some current principle that is an obstruction."

        Einstein's constant-speed-of-light postulate is an obstruction. In a paper published in Science Miles Padgett showed that the speed of light (in a vacuum) is not a constant:

        "The speed of light is a limit, not a constant - that's what researchers in Glasgow, Scotland, say. A group of them just proved that light can be slowed down, permanently."

        Pentcho Valev

          Jonathan,

          I am working my way through reading these essays. I will try to get to yours before too long.

          The HOTT program does put mathematical foundations closer to algorithmic structures. It might be a way to address what I call the "body" of mathematics, which is that part of mathematics that is reduced to a computation. This can be computed in some way on a computer. The part of mathematics that involves infinitesimals and set theoretic infinities are what might be called the "soul." I don't deny the existence of this per se, but I don't think it has a direct connection to physics.

          I am working right now to find out how Bott periodicity applies with exceptional groups. The intention is to find a way that nilpotent sets can be mapped to max compact subsets as with the Kostant-Sekiguchi theorem.

          Cheers LC

          Pentcho,

          You spend a lot of time thumping this theme. Sadly, mostly this is just a demonstration that you don't know what you are talking about. I have no intention of getting into an argument over this, any more than I intend to argue for evolution to a committed creationist or global warming to a climate denialist.

          The speed of light is different in media, and some exotic media have been developed that can trap light. This does not falsify relativity.

          LC

          "The speed of light is different in media, and some exotic media have been developed that can trap light. This does not falsify relativity."

          They slowed down light IN A VACUUM:

          "Physicists manage to slow down light inside vacuum (...) ...even now the light is no longer in the mask, it's just the propagating in free space - the speed is still slow. (...) "This finding shows unambiguously that the propagation of light can be slowed below the commonly accepted figure of 299,792,458 metres per second, even when travelling in air or vacuum," co-author Romero explains in the University of Glasgow press release."

          Pentcho Valev

          This does not have a bearing on relativity, but is a quantum effect. One might say that the action of this mask that slows down photons can persist with a photon in much the same way as with the Wheeler Delayed Choice Experiment.

          LC

          Hi LC--

          I loved your essay. You covered an immense amount of ground--and did so in a cogent yet concise manner. Congratulations!

          I now turn to discuss some comments that you made in response to my essay. You raised the issues of super-Turing machines and the physics of super-tasking. I am not an expert on either. However, I have looked at several examples of physical super-tasking (e.g., carrying out an infinite number of physical operations within a finite time period). I did so because super-tasking appeared to be one place where physics might really need the concept of "physical infinity". As you know from my essay, I call into question the necessity and desirability of relying upon physical infinity.

          In fact, for me, super-tasking was the "tipping point" against physical infinity. In every example I looked at, I found that either: (a) the super-tasking scenario was unphysical and could not work realistically (e.g., because of friction, chaos, cannot propagate a signal faster than c, etc.); or (b) the underlying physics was so murky that I couldn't tell whether the scenario was physically realistic or not. I place super-tasking via Malament-Hogarth spacetimes in the latter category. Regarding super-tasking via M-H spacetimes, I strongly recommend Earman's book, "Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes". His Chapter 4 includes an excellent review of M-H super-tasking.

          Best regards,

          Bill.

            • [deleted]

            • Post #73

            I am glad you liked my essay. I threw in the subject the MH spacetimes and supertasking because that seems to be something that needs to be considered for a number of reasons. I think that non-eternal black holes can't be supertasking machines. The black hole decays by Hawking radiation and disappears before i^в€ћ, so there is no continuous stream of infinite amount of information that can approach an observer as they approach r^-. However, this probably means that NP-complete problems can be quickly solved for the internal observer and the exponential time is replaced with ~ r - r_- near r_-. This may mean that the NP-complete problem of compactifying all CY manifolds is computed by black holes. I do agree that it may be unlikely that superTuring computing is possible in a way that the output can be read by an exterior observer.

            It is possible still that black holes are MH machines, even if they are finite in duration. This might be the case if black hole singularities are all the same thing. It could well be that black holes are all connected in a single quantum state that defines the singularity, and in a multiverse setting it could be that this is a great MH machine. The universe might then has underlying it a supertasking computer that is the ultimate quantum error correction code. I can go into this in detail if you want, though I will avoid that for now. Supertasking process in this setting is then associated with what were called shadow states. Shadow states are an old idea going back to the 1970s with S-matrix bootstrap physics. These are states which have T-matrix realizations, but they have no Born interpretation as associated with observables. The output of the MH spacetime machine can't be read!

            Cheers LC

            Hi Lawrence,

            Nice historical introduction. Interesting new maths. Most enjoyed philosophical concerns, which is more "down my street". Good that your essay is getting noticed. Good Luck, Georgina

            Hi Lawrence. Thank you for your reply to Georgina. Sorry I forgot to reply to you earlier.

            As I said it is not true that "a universal Turing machine is not able to enumerate all Turing machines". Turing machines can be automatically enumerated, but what is not possible is to find a general algorithm always correctly able to prove for any other algorithm, whether or not it will ever stop.

            You wrote that "Peano's number theory is incomplete, and so something funny does happen with N+1". It may be funny but it is not anything wrong with the existence of N+1. You seemed to mistake incompleteness with inconsistency, which are 2 very different things. All we need for a theory of arithmetic is that it is consistent, and indeed it is (even if we cannot have any formal proof for it). It is not a problem for a theory of arithmetic to be incomplete, anyway it remains a valid theory, and since it cannot be completed we must satisfy ourselves with this fact. There is no problem with the axioms, we only can never have enough axioms for all arithmetical truths to be deduced from them.

            I am not asking for symbolism, I consider the possibility to explain things with words as well. I only ask the words, whatever the details level, to stay in agreement with the logical structure of things as they actually are, a requirement which I found to be lacking in your essay ; and when one does not properly understand the logical structure of some issue, then better would be to avoid telling any story about it than telling a probably incorrect one. For example if you know a result but you are not familiar with the proof, it may be wiser to just tell the result but not try to give any sketch of proof that may not be the correct one, so as to better develop instead something else you would know better to do it correctly. Making things short to give the intuition of something can be good only if the intuition you provide is indeed a correct intuition, i.e. in coherence with the correct understanding.

            There is a concept of Polish set, of course, but what I meant is that this never aimed to constitute a "Polish set theory" as a candidate for the foundations of mathematics.

            Hi LC--

            I think that we are in agreement on the issue of super-tasking via M-H spacetimes. It's amazing the kinds of things that show up in our discussion threads! Thanks for taking the time to set out your position on this issue.

            Best regards,

            Bill.

            Some of my discussions were meant to illustrate something of the divide between computational mathematics and pure mathematics. With Peano arithmetic we know that Goedel's theorem indicates that something is not complete, even though much of it involves N ---> N 1. There are then numbers, such as some between 10^{10^{10^{10}}} and 10^{10^{10^{10^{10}}}} that have no description. There is a Berry paradox or self-referential form of incompleteness, based on the complexity or unnamable property of such numbers between these two, in not being able to describe numbers.

            I had a limited amount of space to describe this, and maybe I did not do the best at it. I tried to explain some of these ideas in physical terms without getting into depth on set theory or logic. It is also best I have found that keeping these essays on a level accessible to general readers to be a good strategy.

            LC

            There is not likely to be any way that supercomputing machines such as from MH spacetimes will produce readable output. This does not mean it is absent, but it may simply not involve quantum information that is directly read.

            LC