Dear Alan,

I enjoyed reading your essay, which I think is both well-motivated and well-explained. I have a couple of questions/remarks.

1. Like you, I tend to find certain aspects of quantum theory and quantum field theory less well-motivated than relativity, which is based on simple physical principles. In particular, even in ordinary quantum theory, I am not very fond of the Hilbert space/operator algebra view, which takes things for granted that are convenient mathematically but physically dubious. I prefer Feynman's sum-over-histories viewpoint, even though the math is harder, because the physical ideas are clear and refer only to entities with obvious physical meaning.

2. Have you read the essays by Torsten Asselmeyer-Maluga and Jerzy Krol in this contest? They together propose new methods of trying derive quantum gravity from smooth manifolds alone. I find their ideas interesting, but I will warn you that their approach is not something that can be fully understood the first time through, at least not for me. Anyway, since you focus on theories based on geometry, I thought you might find them interesting.

3. There are some interesting new ways in which Hopf algebras are showing up in quantum information theory these days, which is rather striking for something with essentially a geometric origin. There is a preprint about this by Sasakura, and I have written some about it myself, although unfortunately it isn't in any shape to publish yet.

Paradoxically, although my mathematical work involves mostly manifolds and algebraic varieties, my ideas about fundamental physics are quite different. I suspect that manifolds are "too good to be true" physically because of their special order-theoretic properties and the fact that they imply things like nonmeasurable sets. My view is that manifolds have dominated physics historically mostly because they are mathematically convenient, much like Hilbert spaces are mathematically convenient for quantum theory. However, a lot of the papers I have been reading lately are causing me to reconsider this, and I like to keep an open mind.

I prefer to try to build fundamental physics from more primitive notions like causal relations, which lend themselves to information-theoretic and even computational approaches. I note from your bio that you taught computer science; it's a bit ironic that a mathematician such as myself would try to think about physics in terms of information theory, while a computer scientist would think about it in terms of geometry!

Anyway, I enjoyed reading about your work! Take care,

Ben Dribus

    • [deleted]

    Dear Patrick,

    I read your essay with interest. I too have taken a Geometric approach. However, unlike yours, it concerns very simple Geometric relationships leading to trignometric expressions between related phenomena. It makes relativistic phenomena quite understandable visually. I am sure you would like my essay

    The gist of my essay: http://fqxi.org/community/forum/topic/1549

    1. It identifies the PRIMORDIAL Foundational Problems in Newtonian Mechanics (NM) that runs through ALL BRANCHES OF PHYSICS. (Please see the short attachment "Primordial Foundational Problems").

    2. It eliminates the problematic concept of POINT-MASS (common to NM, QM, SRT) to allow internal structure for a particle. This in turn enables to resolve the other interconnected primordial problems.

    3. The result: By taking these two steps, ALL THE EQUATIONS OF SRT are DYNAMICALLY derived by identifying the trignometric relations within the energy-momentum equation, and by restoring Galileo's principle of relativity. (I request you to have a glance at the attachment - "Geometrodynamics of Energy" to verify this claim). - See also comment by L.B Crowell below.

    4. This achievement will establish that I have not just treated these problems at the level a speculative discussion as in other essays, but that the problems discussed are real problems, by virtue of their solution leading to the unification of NM and SRT (by finding an equation of motion which is equally valid for slow and very fast motions).

    Here is the impartial comment made by Ben Dribus (essayist in no 2 position): "One thing I will say is that it appears as if you made an honest effort to answer the question posed by the essay contest rather than just writing down your favorite ideas about physics. You will notice that I made a similar effort..... I am not sure why it was rated so low, but my impression is that many authors automatically rate other essays low to boost their own standing".

    Here's the comment made by LB Crowell (essayist at no. 20 position): "The calculations I just looked at and they seem alright. ...... Your procedure appears to be some euclideanization of relativity. At the end you arrive at equations which are the same as special relativity".

    In order to enable follow up of your comments easier for me, I request you to reply to this under my essay : http://fqxi.org/community/forum/topic/1549

    Best regards,

    VirajAttachment #1: 2_Primordial_Foundational_Problems.docAttachment #2: 2_GEOMETRODYNAMICS_OF_ENERGY.doc

      Dear Peter,

      Thank you for your remarks on my post. I very much enjoyed reading your essay. The idea of an asymmetric component of the metric [possibly implemented through non-commutative geometry] leading from classical to quantum dynamics has always been fascinating for me. I have played around with it in some of my simplistic papers, but without much success yet. More promising, as of now, seems to be the the theory of Trace Dynamics developed by Stephen Adler and collaborators, which seems to have strong elements whereby noncommutative classical dynamics gets connected to quantum dynamics.

      Regards,

      Tejinder

      Hello Ben

      Thanks for your note. Some (maybe) short responses:

      1. Describing states with functions in a Hilbert space does not of itself worry me. The functions are functions on space-time, which seems to be what is wanted if particles are diffuse. The problems arise when one tries to combine this kind of account with GR.

      2. Asselmeyer-Maluga's essay is not easy reading, but it is an eye opener. I have looked at a few of his papers on arXiv, and at a summary of a book cited in one of them. The idea that matter arises through exotic smooth structures appears at first outlandish, but the effect of the sort of surgery he considers on connections seems to fit rather neatly with Einstein's notion of matter in GR. I am as yet far from understanding it all properly.

      3. Hopf algebras seem to pop up in lots of places.

      It will take a lot to disconnect physics from manifolds, after the success of GR. I like manifolds. Have you seen John Milnor's little book: "Topology from the differentiable viewpoint"? I read that long ago, and was hooked. The only caveat is that he says nothing about differential forms. I ought to have understood them too.

      bw

      Alan H.

      Hello Viraj

      Thanks for your comments. As there are so many submissions, it is hard to know where to start reading. I have been trying to understand Asselmeyer-Maluga's ideas. I don't know if his essay will be the winner, but certainly there are enough new ideas in it to keep me occupied for a few months. A lot of the fancy maths which gets talked about seems to be just that. He and the other writers he cites make it seem that the notion of exotic smooth structures could fit neatly into GR. Have you looked at his work?

      As to assessment: it does seem to give each of us reason to not vote as perhaps we should. I don't know what to do about it. Maybe just live with it, and vote with one's heart. It is a bit like the prisoner's dilemma.

      bw

      Alan H.

      Dear Alan,

      I never read that one, which is funny seeing that I've read a fair bit of his other work (K-theory, etc.) But I would consider myself "hooked" as well, at a mathematical level: I like Griffiths and Harris's Algebraic Geometry (complex manifold viewpoint), and Spivak's 5-volume set on differential geometry is good. Claire Voisin has written two excellent books on Hodge theory and complex algebraic geometry. I also like Hatcher's algebraic topology and Adams' old book about spectra. I think Carroll, Wald, and Hawking and Ellis are all good for GR, though at different levels. These days it seems more and more profitable to look for material online, however.

      I agree that "it will take a lot to disconnect physics from manifolds," but "disconnect" is a rather strong word. To believe that manifolds are an immensely useful and astonishingly accurate tool (given their highly idealized properties) to use for physics is not incompatible with a recognition that they also raise some serious issues, which always seem to arise precisely in regard to those properties which seem least physically relevant. In criticizing manifolds as fundamental, I probably come across as much more "anti-manifold" than I actually am.

      By the way, there are some other contributions from the "pro-manifold forces" here which you might enjoy if you haven't already read them: the shape dynamics essays (Barbour, Gryb/Mercati, Alves), the "desingularization" paper by Abhijnan Rej and the "singular" one by Cristinel Stoica, and of course the one by Krol that I mentioned before.

      Anyway, I am sure manifolds have a lot more to offer physics, regardless of my opinions! Take care,

      Ben

      Dear Ben

      Thanks for your note of this morning. In particular, thanks for all the references. Someone once offered to get a copy of the 5-volume Spivak set for me, and I didn't take him up. I should have done.

      In particular, thanks for the suggested reading list of other essays. The range of contributions is enormous, from no maths to the wildest ideas, right up to topos theory. I am just looking at the one by Abhijnan Rej, which seems to touch on the theory of infinite Galois groups and, perhaps, something called the Krull topology which was the subject of the very last lecture of a course which went beyond the syllabus of the final year of my first degree. Out of a large class, only 5 or 6 of us attended it.

      Much of it is wonderful stuff, but what all this might have to do with physics mystifies me. It seems that one can get a very long way with just the ideas underlying the theories of Einstein and Dirac: a 4-dimensional Riemannian manifold, and the algebra derived from its Riemannian structure, and maybe a variational principle. Anything beyond that counts as good clean fun, but not physics until there is evidence otherwise.

      bw, Alan H.

      • [deleted]

      Dear Alan,

      I agree with you about the role of geometry in physics. Indeed, bundles over manifolds, having as fiber various structures, are responsible for much of the progress attained so far, and even more is expected to come. Quantum mechanics too has much to gain from geometry, which indeed may be the connection between QM and GR. I sketched somethign about this in these slides. Congratulations for the essay and good luck with the contest.

      Best regards,

      Cristi Stoica

        Dear Cristi

        Thanks for reading my essay, and for your kind remarks on it, and for the link to your slides which I have read. Quite apart from your discussion, they taught me some physics I didn't understand before.

        My immediate reaction is that your conclusion "indeterminism exists only in the initial conditions" sounds almost right. Maybe one might add that this indeterminism will never be resolved by us poor mortals because "we, our memories, and any possible measuring apparatus are part of the solution. No solution is capable of discovering and internally representing what it itself is". One aspect of this is that, in this universe of which we are part, no computer can have more gates and memory cells than there are elementary particles. We cannot construct a computer which will extrapolate numerically from any given possible initial condition with unlimited precision.

        Best wishes, and trusting your slide presentation will be well received. Alan H.

        4 days later

        If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

        Sergey Fedosin

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