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The Spectrum of Riemannium

Brian Hayes

Erbium and Eigenvalium

Among the spectra in Figure 1 is a series of 100 energy levels of an atomic nucleus, measured 30 years ago with great finesse by H. I. Liou and James Rainwater and their colleagues at Columbia University. The nucleus in question is that of the rare-earth element erbium-166. A glance at the spectrum reveals no obvious patterns; nevertheless, the texture is quite different from that of a purely random distribution. In particular, the erbium spectrum has fewer closely spaced levels than a random sequence would. It's as if the nuclear energy levels come equipped with springs to keep them apart. This phenomenon of "level repulsion" is characteristic of all heavy nuclei.

What kind of mathematical structure could account for such a spectrum? This is where those eigenvalues of random Hermitian matrices enter the picture. They were proposed for this purpose in the 1950s by the physicist Eugene P. Wigner. As it happens, Wigner was another Princetonian, who could therefore make an appearance in our movie. Let him be the kindly professor who explains things to a dull student, while the audience nods knowingly. The dialogue might go like this:

Wigner: Come, we'll make ourselves a random Hermitian matrix. We start with a square array, like a chessboard, and in each little square we put a random number....
Student: What kind of number? Real? Complex?
Wigner: It works with either, but real is easier.
Student: And what kind of random? Do we take them from a uniform distribution, a Gaussian...?
Wigner: Customarily Gaussian with mean 0 and variance 1, but this is not critical. What is critical is that the matrix be Hermitian. A Hermitian matrix—it's named for the French mathematician Charles Hermite—has a special symmetry. The main diagonal, running from the upper left to the lower right, acts as a kind of mirror, so that all the elements in the upper triangle are reflected in the lower triangle.
Student: Then the matrix isn't really random, is it?
Wigner: If you insist, we'll call it half-random. We fill the upper half with random numbers, and then we copy them into the lower half. So now we have our random Hermitian matrix M, and when we calculate its eigenvalues....
Student: But how do I do that?
Wigner: You start up Matlab and you type "eig(M)"!

Eigenvalues go by many names, all of them equally opaque: characteristic values, latent roots, the spectrum of a matrix. Definitions, too, are more numerous than helpful. For present purposes it seems best to say that every N-by-N matrix is associated with an Nth-degree polynomial equation, and the eigenvalues are the roots of this equation. There are N of them. In general, the eigenvalues can be complex numbers, even when the elements of the matrix are real, but the symmetry of a Hermitian matrix ensures that all the eigenvalues will be real. Hence they can be sorted from smallest to largest and arranged along a line, like energy levels. In this configuration they look a lot like the spectrum of a heavy nucleus. Of course the eigenvalues do not match any particular nuclear spectrum level-for-level, but statistically the resemblance is strong.

When I first heard of the random-matrix conjecture in nuclear physics, what surprised me most was not that it might be true but that anyone would ever have stumbled on it. But Wigner's idea was not just a wild guess. In Werner Heisenberg's formulation of quantum mechanics, the internal state of an atom or a nucleus is represented by a Hermitian matrix whose eigenvalues are the energy levels of the spectrum. If we knew the entries in all the columns and rows of this matrix, we could calculate the spectrum exactly. Of course we don't have that knowledge, but Wigner's conjecture suggests that the statistics of the spectrum are not terribly sensitive to the specific matrix elements. Thus if we just choose a typical matrix—a large one with elements selected according to a certain statistical rule—the predictions should be approximately correct. The predictions of the model were later worked out more precisely by Dyson and others.

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