excerpt from "On Fourier and Zeta(s)" by J.-F. Burnol

[This is taken from the end of the paper - pages 34-35. Emphases have been added.]

"We turn now to some speculative ideas concerning the zeta function, the GUE hypothesis and the Riemann hypothesis. When we wrote "The Explicit Formula and a propagator" we had already spent some time trying to think about the nature of the zeta function. Our conclusion, which had found some kind of support with the conductor operator log |x| + log |y|, stands today. The spaces H^ and especially Theorem 7.2 have given us for the first time a quite specific signal that it may hold some value. What is more, Theorem 7.2 has encouraged us into trying to encompass in our speculations the GUE hypothesis, and more daring and distant yet, the Riemann Hypothesis Herself.

We are mainly inspired by the large body of ideas associated with the Renormalization Group, the Wilson idea of the statistical continuum limit, and the unification it has allowed of the physics of second-order phase transitions with the concepts of quantum field theory. Our general philosophical outlook had been originally deeply framed through the Niels Bohr idea of complementarity, but this is a topic more distant yet from our immediate goals, so we will leave this aside here.

We believe that the zeta function is analogous to a multiplicative wave-field renormalization. We expect that there exists some kind of a system, in some manner rather alike the Ising models of statistical physics, but much richer in its phase diagram, as each of the L-functions will be associated to a certain universality domain. That is, we do not at all attempt at realizing the zeta function as a partition function. No, the zeta function rather corresponds to some kind of symmetry pattern* appearing at low temperature. But the other L-functions too may themselves be the symmetry where the system gets frozen at low temperature.

Renormalization group trajectories flow through the entire space encompassing all universality domains, and perhaps because there are literally fixed points, or another more subtle mechanism, this gives rise to sets of critical exponents associated with each domain: the (non-trivial) zeros of the L-functions. So there could be some underlying quantum dynamics, but the zeros arise at a more classical level, at the level of the renormalization group flow.

The Fourier transform as has been used constantly in this manuscript will correspond to a simple symmetry, like exchanging all spins up with all spins down. The functional equations reflect this simple-minded symmetry and do not have a decisive significance in the phase picture.

But we do believe that some sort of a much more hidden thing exists, a Kramers-Wannier like duality exchanging the low temperature phase with a single hot temperature phase, not number-theoretical. If this were really the case, some universal properties would hold across all phases, reflecting the universality exemplified by the GUE hypothesis. Of course the hot phase is then expected to be somehow related with quantities arising in the study of random matrices. In the picture from Theorem 7.2, \Lambda seems to play the role of an inverse temperature (coupling constant).

We expect that if such a duality did reign on our space it would interact in such a manner with the renormalization group flow that this would give birth to scattering processes. Indeed the duality could be used to compare incoming to outgoing (classical) states. Perhaps the constraints related with this interaction would result in a property of causality equivalent to the Riemann Hypothesis.

Concerning the duality at this time we can only picture it to be somehow connected with the Artin reciprocity law, the ideas of class field theory and generalizations thereof. So here our attempt at being revolutionary ends in utmost conservatism."

* Burnol later explained:

"I am using "symmetry pattern" with a meaning more akin to common-sense language than to precise physicist's technical vocabulary. Of course in the physics of phase transitions, symmetry is restored at high temperature: lowering the temperature breaks symmetries. From this point of view a continuum is more symmetric than a lattice, as the continuum has a larger group of invariances. But from the common sense point-of-view there is more symmetry in a diamond than in a continuum homogeneous material. It is in this second, common sense language, that I am using the word "symmetry pattern": at high temperature all potential "symmetry patterns" are mixed, part of one very big symmetry; at low temperatures phases with broken symmetries arise, one for the zeta function, others for the other L-functions."


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