Level Repulsion Is Evidence About Symmetry, Not About Mechanism — Epoche C2
The distribution of gaps between neighbouring levels has been measured in three settings that share no subject matter — neutron resonances in heavy nuclei, the quantum spectra of systems whose classical counterparts are chaotic, and the heights of the non-trivial zeros of the Riemann zeta function — and in all three it is the same curve. The usual reaction is that this is a striking coincidence, and that somewhere there must be a common physical mechanism waiting to be found. This review sets out the case that the reaction is misdirected. The mathematics that explains the recurrence says that the shared distribution reports a shared symmetry class : a statement about which transformations leave the governing operator invariant, and about nothing else. No shared mechanism is required or implied. The argument is best made by accumulating the cases, deriving what they have in common, and then asking how much of that common structure a finite data set can actually detect. Case 1: nuclear resonances The first case is the one that started the subject, and it began from an admission of ignorance rather than from a model. In the 1950s Wigner faced the spectra of compound nuclei — hundreds of resonances per nucleus, produced by a many-body Hamiltonian nobody could write down, let alone diagonalise. His move was to stop trying. Replace the Hamiltonian by a matrix drawn at random from an ensemble that respects the same symmetries as the true one, and ask what the eigenvalues of a typical member look like. For a nucleus invariant under time reversal, a basis exists in which the Hamiltonian matrix is real and symmetric, so the ensemble is the set of real symmetric matrices with independent Gaussian entries weighted by $\exp(-\mathrm{Tr}\,H^{2}/4\sigma^{2})$ — the Gaussian orthogonal ensemble. His 1955 paper on such matrices gives the global answer: the eigenvalue density of a large one converges to a semicircle. That is not what the resonance data test. The data test the local statistics, the gaps between adjacent levels, and for those Wigner produced a surmise from the smallest non-trivial case. The derivation is short enough to give in full, and it is worth giving because it explains where every factor comes from. Write a $2 \times 2$ real symmetric matrix $H$ with diagonal entries $a$ and $b$ and off-diagonal entry $c$. Its two eigenvalues differ by $s = \sqrt{(a-b)^{2} + 4c^{2}}$. Put $x = a-b$ and $y = 2c$; then $$\mathrm{Tr}\,H^{2} = a^{2}+b^{2}+2c^{2} = \tfrac{1}{2}(a+b)^{2} + \tfrac{1}{2}\left(x^{2}+y^{2}\right),$$ so under the ensemble weight the trace of the centre of mass separates off and $x$ and $y$ are independent Gaussians of equal variance. The gap is therefore the radius of a two-dimensional isotropic Gaussian vector, and the radial measure in two dimensions supplies a factor $s$: $p(s) \propto s\,e^{-\kappa s^{2}}$. The two constants are then fixed, not free. Writing $p(s) = A s e^{-Bs^{2}}$, normalisation gives $\int_0^\infty A s e^{-Bs^2}\,ds = A/2B = 1$, and measuring $s$ in units of the local mean spacing means demanding $\int_0^\infty s\,p(s)\,ds = \tfrac{1}{4}A\sqrt{\pi}\,B^{-3/2} = 1$. The first equation gives $A = 2B$; substituting into the second gives $\tfrac{1}{2}\sqrt{\pi}\,B^{-1/2} = 1$, hence $B = \pi/4$ and $A = \pi/2$: $$p_{\mathrm{GOE}}(s) = \frac{\pi s}{2}\,\exp\!\left(-\frac{\pi s^{2}}{4}\right).$$ Two qualifications keep this honest. This is the exact spacing law for a $2\times 2$ matrix, not for the limit of large matrices; the exact limiting laws have no elementary closed form, and the surmise is used because it is a simple expression that the data cannot distinguish from them. And the derivation says nothing about nuclei — it is a statement about a measure on matrices. The empirical claim needs a source and a number, since the strength of the case rests on it. The standard test is the Nuclear Data Ensemble analysed by Haq, Pandey and Bohigas in 1982: 1726 measured resonance energies, drawn from 36 separate level sequences across 32 nuclei, rescaled so that each sequence has unit mean spacing and then pooled. Both the spacing distribution and the long-range rigidity of that ensemble agree with the orthogonal ensemble and are incompatible with an uncorrelated spectrum. The word sequences is doing essential work there and will return below: levels of different total spin and parity do not repel one another, so pooling them without sorting destroys the effect one is trying to measure. Case 2: quantum systems with chaotic classical limits The nuclear case involved an ensemble, so a sceptic can attribute the result to the averaging. The second case removes that escape. Bohigas, Giannoni and Schmit computed, in 1984, the spectrum of a Sinai billiard — a single particle bouncing freely inside a square containing a circular obstacle, a system whose classical trajectories separate exponentially and are chaotic in the technical sense. There is no ensemble here and nothing random anywhere: the Hamiltonian is one fixed differential operator with one fixed spectrum, computed to several hundred consecutive levels. The spacings nevertheless follow the same curve. Their conjecture, still a conjecture, was that this holds generically for quantum systems whose classical dynamics is chaotic; semiclassical arguments based on sums over periodic orbits make it plausible without establishing it. What converts this from a suggestive observation into evidence is the control case, and the control is decisive. For a classically integrable system the motion is confined to invariant tori and the levels are labelled by independent quantum numbers, one per degree of freedom. Berry and Tabor argued in 1977 that the resulting spectrum is, for a generic integrable system, a superposition of unrelated arithmetic progressions whose pooled spacings are Poissonian, $p(s) = e^{-s}$, with no repulsion at all. Integrability and chaos therefore give different, and maximally different, answers to the sa