M Theory Lesson 18

I am very grateful to kneemo for spotting this paper by Carlos Castro on the connection between the Riemann hypothesis and dual string scattering amplitudes. Castro works at a place in Atlanta with a great name: the Center for Theoretical Studies of Physical Systems.

The paper begins by looking at scattering amplitudes of the form

$\frac{\zeta (ik) \zeta (1 – ik)}{\zeta (1 + ik) \zeta (- ik)}$

which appear in studies of particles moving in the hyperbolic plane. Observe that the poles of this expression occur when $k_n = i (0.5 + i \rho_n)$, corresponding to the zeroes of the Riemann zeta function. Thus one considers complex values of $k_n$ and the question arises as to what physical interpretation the zeroes should have.

Castro starts by looking at the Veneziano four point dual string amplitude, expressed in terms of the beta function. The zeros of $\zeta$ on the critical line appear to correspond to the real poles of the tachyonic amplitude.

On another note, check out the coolest blog post ever, on Never Ending Books!

10 Responses so far »

  1. 1

    Doug said,

    Hi Kea

    1 – WOW! I concur RE: Never Ending Books post.
    Thanks for this link.

    Terry Gannon, ‘Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics’ (Cambridge Monographs on Mathematical Physics) touches upon similar relationships.

    Gannon refers to Mathieu groups as ‘mini-monster’ and the bi-monster as ‘maxi-monster.

    Although Gannon discusses ribbon categories, he seems to have more interest in braid categories, particularly relating braid group B3 to the Dedekind eta function [p 164-167 in 2.4.3 Braided #2: from the trefoil to the Dedekind]..

    2 – Other zeros?: consider the reciprocals of the Georg Cantor theorem “… implies the existence of an ‘infinity of infinities’ …”. Does this mean there is an infinity of zeros?

  2. 2

    Kea said,

    Hi Doug. Unfortunately, I don’t have a copy of Gannon’s book, but it sounds like a great read!

  3. 3

    Doug said,

    Hi Kea

    While perusing the ‘Never Ending Book’ Site, I came across this link to the Gannon ArXiv paper ‘Monstrous Moonshine: the first 25 years’. Lieven le Bruyn refers to this as a survey paper.

    I am reading a copy of the Gannon book obtained through university interlibrary loan.

    I saw on the Carl Brannen blog that you are about to receive you PhD.

    Best wishes.

  4. 4

    Mahndisa S. Rigmaiden said,

    03 01 07

    Hey there Kea:
    I was wondering when someone would cite Mr. Castro! Yeah, he is at Clark Atlanta University, which isn’t known for physics but has developed their reputation over the past few years. I first came across a paper of his on a Cantorian rep of a string. He is really into padic physics and correlates to fractals cool dude!

  5. 5

    kneemo said,

    The Y.Z. Huang references in Gannon’s Monstrous Moonshine: The first twenty-five years seem to be the key to relating bubble operads to vertex operator algebras. Read over page 13 in Gannon’s paper to understand why.

  6. 6

    Mahndisa S. Rigmaiden said,

    03 02 07

    Hey Kea:
    Come by the blog at your leisure. I think you will like the practicality of this post. Have a great weekend!

  7. 7

    Kea said,

    The Huang reference seems to be a book. I’m sure my library doesn’t have it. Yes, spheres with tubes for the VOA. But it’s still not clear where the 2-operads come in. Most simply: a twist in the tube (or ribbon diameter) would be labelled by an ordinal, which could be interpreted as the 2nd level branches of a 2-tree.

    I must understand this better. Maybe I should try and get hold of Huang’s book … hmm …

  8. 8

    kneemo said,

    You can get a sneak peek courtesy of Google scholar. I went ahead and ordered the book through amazon marketplace at $39.95.

    As far as free material goes, his publications page has loads of pdf links.

  9. 9

    Kea said,

    Wow, kneemo! Great link. There’s a chapter called vertex partial operads – sounds promising. I suppose I could get hold of a copy on interloan.

  10. 10

    Doug said,

    Hi Kea

    Thanks for the comment on PF, RE: Gannon book.

    This paper is off-topic, but related to CarlB post on ‘The Fermion Cube’ at PF.

    M Bohner, A Peterson [U-MO-Rolla]
    ‘Dynamic Equations on Time Scales’

    Chapter 1: The Time Scales Calculus
    Figure 1.5, page 20 a Cantor Set in Time
    Only 50 of 353 pages with 253 references

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