Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Saturday, May 26, 2007

Riemann Revisited III

OK, even if the Hypothesis turns out to be false, that was hilarious, but seriously now ... one promising route to the Riemann Hypothesis is in fact to show that it is undecidable with the standard axioms. It is hard to imagine how this could be done. Even if the zeta zeroes were completely re-characterised in terms of higher categorical invariants, in a way that seemed utterly natural and compelling, that does not imply that we must look at the zeta function that way. Well, mathematically that is. Physically speaking, we only care about the zeta function in so much as it can describe measurable quantities.

But the zeta function soon gets swamped by its generalisations, inflating the difficulty of the problem. These are certainly physically relevant. Recall, for example, Brown's paper for MZVs, which contains a 1-operad computation of Veneziano amplitudes. M Theory cannot avoid considering this construction outside set theory.

Update: Thanks to K. L. Lange for further interesting remarks about the possibility that Pati has inadvertently made progress on showing that RH is unprovable within standard analysis.

2 Comments:

Blogger Matti Pitkanen said...

There is also a physicist's approach to RH. For about 7 years ago I proposed what I called "Strategy for Proving Riemann Hypothesis", actually meant to be a proof but of course with the standards of physicist. See math@arXiv.org/0111262 or this.

I had two big ideas.

a) Conformal invariance is the symmetry of complex analysis. Why not to try to find a quantum physical system with conformal invariance for which the zeros of Zeta have clear physical interpretation?

b) Generalize the Hilbert Polya conjecture which identifies zeros of Zeta as eigenvalues of some to be identified Hamiltonian. Now these eigenvalues would not correspond "energies" but to complex numbers characterizing mutually non-orthogonal coherent states of some quantum system and vanishing of Riemann Zeta says that physical states are orthogonal to tachyonic state corresponding to conformal weight s=0. Inner products reduce to values of Zeta at line Re(s)=1.

I managed to build a physicist's proof of RH and have not been able to detect any flaw. I even saw the trouble of getting it published (Acta Math. Univ. Comeniae, vol. 72) since the promise was that I would get a permanent job from the Physics Department of Helsinki University in this case! I was of course fooled!

May 26, 2007 5:18 PM  
Blogger Kea said...

Hi Matti. Yes, I have seen your ideas referenced by mathematicians who work on RH. A pity your University did not appreciate your work. I have not read this link, so thanks.

Of course, I think the 'quantum system' that people are trying to find for the zeroes is in fact Quantum Gravity itself. There will be further posts!

May 27, 2007 12:48 PM  

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