Hi Sean,

Fantastic Essay! You really have a knack for this level of explanatory writing; you should do more of it. (George Musser, if you're reading this, go commission an article by Sean post-haste!)

What really came through for me was the first argument that I've really bought/understood as to why scale invariance is so attractive. Sure, I've always disliked the idea of some fundamental Planck length and prefer the continuum, but mainly because I have too much respect for Lorentz and Poincare invariance, not because of this type of argument. Beautiful stuff.

I vaguely remember that the only non-scale-invariant piece of the standard model is the Higgs...? Any thoughts on how that might play out in this story?

Am I right that your clock variable \varphi is what would be measured by a conformal clock? (Meaning, say, Einstein's light clock where the two mirrors are changing their relative distance along with the expansion of the universe.) That almost fits with my limited understanding of these things, except that I had thought the universe was of finite duration as measured by such a clock, while your clock parameter still (logarithmically) diverges as t->\infty. Does the small cosmological constant limit this to a finite \varphi even as t->\infty?

And if I'm right that the universe *is* of finite duration (as measured by such a clock), does this imply anything in particular for the way you see cosmological boundary conditions as coming into the story? (Would one have a future boundary condition at the final conformal-time boundary?)

Again, excellent job! (Arguably 2 contests too late, but I, for one, am very glad you fit it into this year's topic.) Keep it up and you might even strong-arm me into seriously thinking about cosmology again... :-)

Best,

Ken

PS -- Looking forward to catching up in Munich!

    Hello,

    I liked your essay and found it interesting, but I'm wondering if the scale-invariance that you describe is really opposed to fundamental discreteness. I believe that the paper "Regular black holes in UV self-complete quantum gravity," by E. Spallucci and S. Ansoldi (arXiv:1101.2760), is illuminating in this regard. (As an aside, I've already discussed this paper with Douglas Singleton over at his essay; you might find his paper (co-authored with Elias Vagenas & Tao Zhu), with its use of self-similarity, relevant to your own perspective.) Spallucci & Ansoldi argue (drawing on earlier ideas of G. Dvali)that the Planck scale is a scale-invariant limit, or "anchor," of the very kind that you describe on p. 2 (section 2) of your essay. Yet they also take the Planck length to be a minimal length - in the sense that it is impossible to probe shorter distances - so that it sets a fundamental discreteness scale.

    So, I'm not sure that discreteness needs to be rejected, even if one accepts scale-invariance. Admittedly, there is the objection you mention about it being impossible to prove that a given discrete theory T is truly fundamental; but to me, the mere possbility of a more fundamental theory is hardly a devastating objection to such a T.

    Anyway, best wishes and good luck in the contest,

    Willard Mittelman

      Dear Dr.Sean Gryb:

      You didn't read my post of july 14. Maybe you would like to read Einstein unic short verbal "space-time" description in Einstein "ideas and Opinions" page 365.

      With my very best whishes

      Héctor

        Dear Sean,

        You said:

        "a. That space is closed and has the shape of a 3 dimensional sphere. This means that an observer can head in the same direction and (eventually) come back to their original location.

        b. That space is expanding and that this expansion if fueled by a very small, positive parameter called the cosmological constant, which we will discuss briefly."

        This is exactly what I describe in my essay . I have also come up with a scale invariant theory in which I show that the proton's diameter is a scaled up version of the Planck length and that the proton's mass is a scaled down version of the Planck mass.

        If you take the Planck length and multiply it by 1020 (the scale factor) and divide it by 1-1/8Pi, you get the exact value of the proton's diameter measured with a muon. If you divide that SAME value again by the SAME 1-1/8Pi, you get the exact value of the proton's diameter measured with an electron (solving the proton radius measurement problem ). This 1-1/8Pi is explained in my theory (it is (8Pi-1)/8Pi).

        If you take the Planck mass and multiply it by 10-20(the scale down factor) and multiply it by 8-1/Pi, you get the exact value of the proton's mass. Again, 8-1/Pi is explained in my theory (it is (8Pi-1)/Pi).

        The 8Pi-1 also appears in the proton/electron mass ratio formula that I present in my essay but also in a lot more formulae that I won't describe here but that you can find here.

        I would love to have your expert opinion on my findings, I hope that you will take the time to read my essay.

        Patrick

          Thanks! And good luck to you in this competition!

          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