UPDATE Jan 14, 2019. Michael Atiyah passed away on January 11, 2019. He was born on April 22, 1929. Rest in peace.
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Short answer: no. The derivation is almost certainly wrong in many ways.
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According to Lubos Motl, the Riemann hypothesis proof that Michael Atiyah gave out today:
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Short answer: no. The derivation is almost certainly wrong in many ways.
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According to Lubos Motl, the Riemann hypothesis proof that Michael Atiyah gave out today:
is flawed in many ways:
Also, several anonymous commentators on the Internet have claimed that the definition of a weakly analytic function in the paper is wrong.
In another paper, Michael Atiyah claims that he can derive the exact value of the fine structure constant α from renormalization principles and geometric mathematical principles:
The α paper is of interest to our blog. The fine structure constant can be defined, for example, as
α = k_e e^2 / h-bar c,
in SI units, where k_e is the Coulomb constant. In natural units,
α = e^2 / 4π.
In natural (Planck) units, α is the force between two electrons when they are placed at a 1 natural unit distance from each other.
Our goal in this blog is to show that the correct way to calculate scattering amplitudes is to use a cutoff for the energy or the momentum of virtual pairs. No renormalization should be used. In our approach, the force between two electrons at one unit distance of each other can be set anything we like - it is truly a constant of nature and not a geometric mathematical constant like π is. That is, the physics is sensible in all models regardless of the value of α.
What about π? Can we construct sensible physical models with different values of π? The mathematical value of π is based on an idealized plane geometry which happens to be approximately the geometry which we observe in physical space. Can we construct a physical model where there are no idealized planes at all, at any level of abstraction?
If we have a physical model where the space is smooth at small distances, then we can define π as the limit of the circumference / diameter when we study smaller and smaller circles.
If our physical model would be a discrete model, then the mathematical π might have no bearing on our physical model.
If our physical model would be a fractal, then the mathematical value of π might not be relevant.
Thus, in smooth and continuous models of physics, the mathematical value of π will be very relevant. In other kinds of models, it might not be.
If we return back to the case of α, Michael Atiyah would have hard time proving that the physics will not work with any other value of α than what we have in this universe. A priori, the result of Michael Atiyah is most probably wrong, but we have not yet studied his paper in detail. Michael Atiyah does not seem to use much physics in his paper. It would be strange if one could derive the value of the fine structure constant without a physical analysis.
This is a test comment posted by Heikki Tuuri.
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