The Spectrum of Riemannium
The Operator of the Universe
Is it all just a fluke, this apparent link between matrix eigenvalues, nuclear physics and zeta zeros? It could be, although a universe with such chance coincidences in its fabric might be considered even stranger than one with mysterious causal connections.
Another possible explanation is that the statistical distribution seen in these three cases (and in several others I have not discussed) is simply a very common way for things to organize themselves. There is an analogy here with the Gaussian distribution, which turns up everywhere in nature because many different processes all lead to it. Whenever multiple independent contributions are summed up, the outcome is the familiar bell-shaped Gaussian curve: This observation is the essence of the Central Limit Theorem. Maybe some similar principle makes the eigenvalue distribution ubiquitous. Thus for Montgomery and Dyson to come up with the same correlation function would not be such a long shot after all.
Still another view is that the zeros of the zeta function really do represent a spectrum—a series of energy levels just like those of the erbium nucleus, but generated by the mathematical element Riemannium. This idea traces back to David Hilbert and George Pólya, who both suggested (independently) that the zeros of the zeta function might be the eigenvalues of some unknown Hermitian "operator." An operator is a mathematical concept that seems on first acquaintance rather different from a matrix—it is a function that applies to functions—but operators too have eigenvalues, and a Hermitian operator has symmetries that make all the eigenvalues real numbers, just as in the case of a Hermitian matrix.
If the Hilbert-Pólya thesis is correct, then random-matrix methods succeed in number theory for essentially the same reason they work in nuclear physics—because the detailed structure of a large matrix (or operator) is less important than its global symmetries, so that any typical matrix with the right symmetries will produce statistically similar results. Behind these approximations stands some unique Hermitian operator, which determines the exact position of all the Riemann zeros and hence the distribution of the primes.
Is that universal operator really out there, waiting to be discovered? Will it ever be identified? For the answers to those questions you'll have to see the movie. I don't want to give away the ending.
© Brian Hayes