At tea in 1972 a graduate student started writing a formula about prime numbers and Dyson finished it, because he had derived the same function for the energy levels of heavy nuclei. Two rooms, one function, nobody in both.
Spring 1972, afternoon tea at the Institute for Advanced Study. Hugh Montgomery was a graduate student visiting from Cambridge; he had been computing the pair correlation of the zeros of the Riemann zeta function — how the zeros sit relative to one another once you have removed the fact that they get denser as you go up. Chowla insisted on introducing him to Dyson. Dyson asked what he worked on. Montgomery said the pair correlation of the zeta zeros, and started to write
$$R_2(r) = 1 - \left(\frac{\sin \pi r}{\pi r}\right)^2$$
and Dyson said, before he had finished, that it was the pair correlation of eigenvalues of a random Hermitian matrix. He knew because he had derived it, for a completely different reason, in the early sixties. CITED
Number theory and the energy levels of heavy nuclei had produced the same function, and nobody had noticed because nobody was in both rooms. That is the whole content of this chapter and it is worth more than the anecdote it is usually told as.
The claim is checkable, and it is checkable from nothing — no table of zeros, no
library. ch-dyson-verify.py computes the zeros itself with the
Riemann–Siegel formula, finds 3 681 of them below $t = 4200$, agreeing with the
published values of the first ten to a mean error of $2.1\times10^{-3}$. It unfolds them
by $w_n = \vartheta(t_n)/\pi$, so the mean spacing is one, keeps the upper half where the
asymptotics are better, and histograms every pair.
r observed GUE Poisson
0.05 0.0000 0.0082 1.0000
0.15 0.0380 0.0719 1.0000
0.25 0.0760 0.1894 1.0000
0.45 0.5269 0.5119 1.0000
0.95 1.1298 0.9973 1.0000
1.95 0.9125 0.9993 1.0000
RMS deviation over $r < 1.5$: 0.078 from GUE, 0.502 from Poisson — a factor of 6.4. COMPUTED
The first bin is the one to look at. Uncorrelated points would give $1$. The zeros give $0$. They repel each other, and they repel with exactly the strength that eigenvalues of a random Hermitian matrix do.
Dyson recognised it because of what he had done a decade earlier, and that is the part of his work this corpus should be reading.
In 1962 he asked what kinds of random matrix ensemble are possible at all. Not which ones are convenient — which ones exist. A quantum system with a symmetry has a Hilbert space carrying a representation, and the matrix elements live in whatever field the commutant allows. Frobenius proved in 1878 that there are exactly three finite-dimensional associative division algebras over the reals: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$.
β = 1 orthogonal GOE ℜ time-reversal, integer spin
β = 2 unitary GUE ℂ no time-reversal
β = 4 symplectic GSE ℍ time-reversal, half-integer spin
Three, and only three, and the number is not a modelling choice. Frobenius’ theorem removes every other option.
This is ortogênese in the sense the corpus uses it: the classification is generated by a constraint, in the directions the constraint leaves open. Dyson did not choose three ensembles. He counted what was left.
And the zeta zeros landed on $\beta = 2$ — the one with no time-reversal symmetry. Whatever operator has the zeros as its spectrum, if one exists, is not time-reversal invariant. That is a statement about arithmetic, derived from a theorem about division algebras, and it is still the strongest structural hint anyone has about what Hilbert–Pólya would have to be. OPEN
He also wrote the essay this gallery could have been named after. Birds fly high and see far and unify; frogs live in the mud and see the flowers close up and solve problems. Dyson said mathematics needs both and that he was a frog, which from the man who unified Feynman's, Schwinger's and Tomonaga's electrodynamics was either modesty or a very precise self-assessment.
The corpus should notice one more thing. In 1952 Dyson proved that the perturbation series of quantum electrodynamics — the series whose agreement with experiment is the most accurate in physics — diverges. Not converges slowly: has zero radius of convergence. The argument is a page long and turns on what happens if the coupling is negative: the vacuum becomes unstable, so the function cannot be analytic at the origin.
A divergent series that predicts to twelve digits is the cleanest existing example of the thing this corpus keeps insisting on: a result can be correct, useful, experimentally confirmed, and resting on something that is not established. Physics kept using the series, correctly, and wrote down what it had not proved. That is the epistemic standard, demonstrated by someone who had every incentive not to.
| Operator | In this chapter | In dm³ |
|---|---|---|
| C | unfolding — the density removed, only the correlation left | compression: strip what is not structure |
| K | Frobenius: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and nothing else | the constraint that removes options |
| F | $\beta \in \{1,2,4\}$ — three ensembles, not a continuum | the fold — threshold, not scale SHOWN |
| U | the zeta zeros landing on $\beta = 2$ | the branch arithmetic chose COMPUTED |
Every number on this page is produced by book7/ch-dyson-verify.py. It records in its own closing block what it establishes and what it does not.
F. J. Dyson, “The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics”, J. Math. Phys. 3, 1962, 1199–1215.
F. J. Dyson, “Divergence of perturbation theory in quantum electrodynamics”, Phys. Rev. 85, 1952, 631–632.
H. L. Montgomery, “The pair correlation of zeros of the zeta function”, Proc. Symp. Pure Math. 24, 1973, 181–193.
A. M. Odlyzko, “On the distribution of spacings between zeros of the zeta function”, Math. Comp. 48, 1987 — the large-scale numerical confirmation.
F. J. Dyson, “Birds and Frogs”, Notices of the AMS 56, 2009, 212–223.
F. G. Frobenius, “Über lineare Substitutionen und bilineare Formen”, Crelle 84, 1878 — the three real division algebras.