[*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."
**J.-F. Burnol's homepage
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