Essay Abstract

Often in physical theories, mathematics plays a secondary role, in support of a primary physical picture. Here we entertain instead the idea that physical concepts can be drawn out of purely mathematical objects.

Author Bio

C. Furey is due to graduate with a PhD from the Perimeter Institute for Theoretical Physics / University of Waterloo. The ideas sketched in this essay have been the underlying motivation for three papers published during these graduate years.

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Hi Furey,

It has been few years now that I have been trying to get somebody to listen to my idea which is very much along the lines of your work. I have to admit that I have not tried hard but I could see that my idea is close to all the references that you mentioned. BTW, what happened to Fontini she seems to have been teleported to another planet, I liked her work which is related to mine.

Although my system might look very different from yours but I think fundamentally somehow they are linked. I think my main advantage is that time automatically does not appear and space is just an aspect of what makes a particle.

Thanks for you attention, and hopefully you can give me some feedback which is more important than rating.

I do hope you know basic programming, like JavaScript.

Essay 2015

    Cohl

    You've written a refreshingly thought provoking essay. I too, am very much drawn to the algebraic approach. From the matrix model perspective, particles indeed seem to have a purely algebraic description. For example, in Smolin's D=27 matrix model, fermions can be identified with rank one elements of the observable algebra, consisting of only self-adjoint operators. These rank one elements are eigenmatrix solutions of a characteristic equation in a Jordan formulation of quantum mechanics. When normalized, these rank one elements are primitive idempotents of the observable algebra.

    In Smolin's model, there is an F_4 symmetry, acting on sets of (at most) three orthogonal idempotents, which are easily seen when a given element of the observable algebra is expanded in a spectral decomposition as a sum of (scaled) rank one elements. Geometrically, the picture is in a projective space, where a given operator takes values on three charts. Imagine this as a 3-sphere generalization of the two charts on a 2-sphere. These charts, in Smolin's model, are 16-dimensional, reflecting the 16-dimensional structure of the fermions in his model. The F_4 symmetry arises from the octonion division algebra underlying the observable algebra. Using F_4 transformations, one can diagonalize a given element of the observable algebra, and ascertain the (eigen)values the operator takes on the three charts.

    In string theory, for coincident brane configurations, one has a U(N) symmetry. If we restrict to the case of three branes, we have a worldvolume with U(3) gauge symmetry. Think of this U(3) as the freedom the strings connecting the branes have, on the worldvolume. There are 3 scalars coming from the modes with coordinates transverse to the worldvolume, which are the positions of the 3 branes in the transverse space. The process of giving some of the scalars expectation values i.e., moving the branes away from each other in the transverse space, breaks the gauge symmetry from U(3) to a subgroup. This is the Higgs mechanism, arising from diagonalizing the hermitian operator for the D-brane system. The hermitian operator's eigenvalues give the classical positions for the D-branes, while the off-diagonal entries of the unitary operators (short strings) give fluctuations about classical spacetime.

    Applying the D-brane intuition to Smolin's model, he seems to have a type of octonion three brane model, with F_4 symmetry. As there are topological restrictions to any projective space higher than OP^2 with the octonions, Smolin's model is maximal, over the octonions.

      Dear C. Furey,

      I agree "that physical concepts can be drawn out of purely mathematical objects.", but I argue in my essay here that visual modelling of these mathematical objects makes them a great deal more "useful".

      Regards, enjoyed reading your essay and hope you get a chance to look at my visual modelling.

      Ed Unverricht

        24 days later

        Dear Adel Sadeq,

        Thank you for your comments. Yes, I have a lot of admiration for Fotini's work, and also wish that she would come back to physics. She was the source of some very brilliant ideas.

        Also, I will gladly take a look at your essay.

        All best wishes,

        CF

        Dear Ms Jagannathan,

        Great - thanks for reading!

        All the best,

        CF

        Dear Michael,

        Thanks - Great to hear from you again.

        Are you referring to Lee's exceptional Jordan algebra paper? If so, have you posted any work which extends it? Please feel free to forward your references to me if you have.

        All the best,

        Cohl

        Dear Ed Unverricht,

        Thanks for reading, I'll take a look.

        All the best,

        CF

        4 days later

        Dear C. Furey,

        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