Dear Jeff,

Thanks for letting me know about your essay. I am familiar with the Nordstrom theory of course, but it is always good to learn more.

Sean.

Dear Richard,

I'm glad that we share the same kinds of intuitions about shapes. The idea of approaching scale from the point of view of entropy is an incredibly interesting one. I agree that there could be a lot of value in doing so. I'm interested to see what you say about it in your essay.

Sean.

Dear Hugh,

Observational evidence is always important. Thanks for pointing out this work.

Sean.

Dear Peter,

Thanks for you flattering remarks and the detailed comments. I'm glad you enjoyed the explanations and are happy for the critiques on the content.

Your approach sounds interesting, but I will have to look more closely at the details to be able to make an educated opinion.

Good luck in the competition!

Sean.

Dear Ojo,

I'm curious to hear why you think spacetime is countable. However, I hope that you judge the essays by their content alone.

Sean.

Dear Jim,

Agreed.

Best of luck in the competition,

Sean.

Dear Don,

Thanks for the rating.

That sounds like an interesting idea. Certainly discretizing the momentum could be a path to helping solve some issues. That you for pointing this out.

Best of luck in the competition,

Sean.

Dear Hector,

My apologies. I am travelling extensively right now and haven't had much time to respond to posts. I did read your post though. Thanks for the description.

Cheers,

Sean.

Dear Patrick,

Sounds interesting, but I will have to look more closely at the details to judge.

Good luck in the competition.

Sean.

Dear Willard,

I appreciate the reflective comments and will have a look at the work you recommend.

I would not deny that it is impossible to have a form of discreteness while still having scale invariance. For example, it could be possible that there is a kind of "minimum" ratio instead of an absolute minimum length or it may be that the scale invariant degrees of freedom come in countable discrete packets. I wouldn't be surprised if there exists models that exhibit this kind of behaviour so I do take your point.

The point I was trying to make was that scale invariance could provide an interesting alternative to fundamental discreteness as a resolution of the non-renormalizability of General Relativity. Postulating a fundamental minimum length is a quite a traumatic thing to do to spacetime. I think it's valuable to explore alternatives.

Re: your last comment, I would mainly object to the use of the word *fundamental*, which I think is widely and deceptively overused in physics - especially in this case.

All the best in the essay. I am eager to have a look at your paper, but please have patience: I am travelling and am very busy with the Loop conference right now, so it might take me a bit of time to get to it.

Best of luck with the competition!

Sean.

Dear Ken,

Thanks so much for your very generous comments on the essay! I really enjoy this kind of writing and would be happy to do more of it (do you know how I could look into doing this??).

Yes, conformal invariance is a really important thing to look for in a quantum field theory because this is the definition of a fixed-point of the RG flow. I'm not sure why this point is not emphasised more. Perhaps because anomalies tend to mess everything up? Case in point: the conformal invariance of the string worldsheet is one of its main features, but conformal anomalies are the reason why string theories are only consistent in 10d with SUSY. In any case, conformal anomalies are probably the most important challenge Shape Dynamics faces in the quantum theory.

You're right about the Higgs in the standard model. This is a topic that we will have to think about very carefully eventually. For the moment, we are just focusing on the gravity sector. However, we can always "fix" the problem by trying to replace the Higgs with a more fundamental interaction. 't Hooft, for example, is trying to use anomaly cancellation in a 4d conformal invariant framework to generate all the masses and parameters of the standard model. An ambitious project, but he is maybe one of the few people in the world that could actually pull it off!

The \varphi clock is NOT a conformal clock. It's actually just the proper time as measured by a stationary observer. The conformal time, t, would be something like: tan(t/2) = tanh(\varphi/2). This keeps t finite. It's what people use to draw the Penrose diagram of dS.

Regarding the boundary condition: the picture I have now is a bit different (we should discuss why). It turns out that you can only have conformal invariance at the boundary. This means that "shape observers" in dS spacetime must be inherently holographic (I can tell you how in Munich). I think you'll like this picture a lot because the shape dynamics theory you get is like a correlation function between "in" and "out" shapes of the universe, as if the universe was a kind of scattering experiment. Anyway, I'm looking forward to seeing you in Munich and would be happy to discuss this more then!

Best of luck in the competition. I will read and comment on your essay but give me a couple of days: I'm very busy with the Loops conference.

Take care,

Sean.

Dear Sean

You wrote "...so that there can be no minimum length"

I agree but on other reason

http://vixra.org/abs/1301.0191

Good luck

Yuri

Nice essay, well-written and clearly explaining its topic. It does seem less deeply connected to the question of "It from bit" than some of the other essays I've read, though. I guess countability could be related to the notion of an exact description via a sequence of bits... but if one can imagine eliciting an infinite such sequence, the space of possible sequences could be uncountable...

I"ll be interested to see how this stereographic/de Sitter duality is or isn't useful in future physics... curious on how you view its relation to the conformal/anti-de Sitter duality that is talked about a lot by string theorists?

    Hello Sean

    I agree with Howard. A lovely essay. I wonder how your concept of there being no minimum length would play out in the context of a proof I created in a Master's thesis (see attached).

    Please read my essay on what I call the Armchair Universe which provides a foundation for spacetime, which arose initially after considering scaling bothers. I would be very interested to see how your work might be connected to mine.

    Best wishes

    Stephen AnastasiAttachment #1: 1_A_problem_for_geometry_1.pdf

      Very interesting. I also enjoyed your previous paper I found on the arXiV (Right about Time). I had not previously made the connection between countability (one to one relation with the integers) and the continuum problem. Of course, you are right, its all about manifesting discreteness.

      However, I would appreciate your opinion other aspects of countability. Consider, for example, the distinction between identity and indistinguishability. If we can identify particles as being discrete, but they are indistinguishable (for example in a Bose-Einstein condensate), then are they countable? More generally, do we need "identity" before the notion of countability is valid?

      Also, what about recurrence? Imagine a pendulum (or other simple harmonic oscillator). If there is no physical mechanism to count beyond a modulo (in this case 2), then there could be an infinite number of recurrences (cycles of oscillation) before the event we observe, and after the event we observe. Can we refer to them as countable?

      More importantly, if you are asleep, and someone wakes you, and you see the oscillator make a few cycles (you can even count them), and then you sleep for a while, but you don't know how long you have slept and someone wakes you again, can you still say that the number of cycles is countable -- if you have absolutely no external mechanism whatsoever to count the cycles that went by when you were asleep -- can you still say that the cycles are countable?

      These may be easy questions for you to answer, nonetheless, they are interesting questions to me regarding "Is Spacetime Countable". For example, we could just as easily say "are cycles countable in the quantum domain"?

      Kind regards, Paul

        Hi Sean,

        Thanks for your helpful response. I don't want to make too big an issue of this -- and given your busy schedule, please don't feel a need to respond -- but when you say (on p. 2 of your essay) that scale-invariance implies that a theory stops changing as we make our measurements more precise, this raises the question of whether there are any limits on how precise our measurements can be. On the face of it, the Heisenberg Uncertainty Principle suggests that there are such limits. However, your reference to the "infinitely small" on p. 8, and the surrounding discussion there, suggests that you're denying any such limits. But in that case, it appears that Planck's constant h must get "scaled down" to the scale of the infinitely small; otherwise, observers at the infinitely small scale would, in attempting to perform measurements there, be completely "engulfed" by quantum uncertainty and consequently unable to make definite measurements at all. But if h gets scaled down, then presumably related constants such as G and c would also get scaled down; in the case of c, this would mean abandoning relativity altogether, and in the case of G we would end up with some sort of "degravitation" -- and these major theoretical changes, as we go down in scale, clearly violate scale-invariance.

        On the other hand, if there are limits on the precision of measurements, then it would seem that we have to settle for some sort of minimal scale. Of course, there's still a kind of scale-invariance here, since increasing the energy scale that we use to probe this minimal scale doesn't lead to new results or change our theories. But this energy-scale-invariance goes hand in hand with a minimal (Planckian) length, as the Spallucci/Ansoldi paper shows. So, I'm questioning whether scale-invariance should be viewed as an actual alternative to a minimal length, and to discreteness. In saying this, of course, I'm not rejecting your idea of scale-invariance; I'm simply denying that scale-invariance and discreteness are inherently opposed.

        One additional, minor point: in using the word "fundamental" in my earlier comment, I was simply following your own usage on p. 2, where you say that there's no way to prove that a purportedly fundamental theory is truly fundamental.

        Best Wishes,

        Willard

        In re-reading the comment I just made, I see that, in the final sentence of my second paragraph, the clause following the semicolon should be deleted as irrelevant, since you're not claiming that there's an inherent opposition between scale-invariance and discreteness. I apologize for my carelessness.

        -Willard

        Sean,

        the "number of degrees of freedom" of a standard QFT is discrete. Not continuous. This is the entire point of the Fock space construction. The one-particle space describes, obviously, a finite number of degrees of freedom, and so the two-particle state space, and so the n-particle space state, so you can associate an integer number to each degree of freedom, which is your definition of a system with a finite number of degrees of freedom.

        ciao, carlo

        • [deleted]

        • Post #65

        Dear Sean,

        Very good essay that, in my opinion, that tackles the ambitious task of unifying QM and GR in the right way.

        My topic has to do with "dessins d'enfants" and contextuality. In the wikipedia paper you have a picture of the triangulated hyperbolic plane.

        http://en.wikipedia.org/wiki/Dessin_d'enfant

        In his book "The icosahedron and the solutions of equations of the fifth degree" (Dover, NY, 1956), Klein's antcipates the Belyi functions for platonic solids by using the stereographic projection.

        My essay is here

        http://www.fqxi.org/community/forum/topic/1789

        I don't look at scale invariance, I don't look at the large scale universe but strangely there are closely related mathematical objects at the starting point.

        Good luck,

        Michel

          Hello Sean

          Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html)

          said: "It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don't know why that is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature."

          I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

          The belief that "Nature is simple" is however being expressed differently in my essay "Analogical Engine" linked to http://fqxi.org/community/forum/topic/1865 .

          Specifically though, I said "Planck constant is the Mother of All Dualities" and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism ... and so on.

          Taken two at a time, it can be read as "what quantum is to classical" is similar to (~) "what wave is to particle." You can choose any two from among the multitudes that can be found in our discourses.

          I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

          Since "Nature is Analogical", we are free to probe nature in so many different ways. And you have touched some corners of it.

          Best

          Than Tin