Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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18. Number Theory Meets Quantum Mechanics
18. Number Theory Meets Quantum Mechanics
Pages 280-295
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From page 280... ...
In 1917, just around the time of the Conjecture, Ernest Rutherford observed the splitting of the atom; 15 years later, Cockroft and Walton split the atom by artificial means. This led in turn to Enrico Fermi's work, to the first controlled chain reaction in 1942, and to the first nuclear explosion on July 16, 1945.
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From page 281... ...
In the case of seriously heavy elements like uranium, the whole blob is teetering on the edge of instability. It might, in fact, (lepen(ling on the precise mix of protons and neutrons, be actually unstable, liable to fly apart of its own volition.
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From page 282... ...
The necessary work was done, the necessary statistical tools for complex quantum dynamical systems like heavy-element nuclei were developed in the late 1950s and early 1960s, key players being the nuclear physicists Eugene Wigner and Freeman Dyson. One central concept was that of a random matrix.
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From page 283... ...
They are plucked at random from a Gaussian-normal distribution the famous "bell curve" that crops up all over the place in statistics. Imagine the standard bell curve drawn on a sheet of fine-ruled graph paper, so that there are hun(lre(ls of graph-paper squares under the curve (Figure 18-1~.
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From page 284... ...
In particular, their eigenvalues turned out to provide an excellent fit for the energy levels observed in experiments. Therefore these eigenvalues, the eigenvalues of random Hermitian matrices, became the subject of intensive study through the 1960s.
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From page 285... ...
. 2 4 6 8 10 FIGURE 18-2 The eigenvalues of a 269-by-269 random Hermitian matrix.
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From page 286... ...
repulsion elect in Figure 18-2 the random scattering of Figure 18-3 has more adjacent pairs of values very close together than has the eigenvalue distribution (and, inevitably, more far apart, too)
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From page 287... ...
The number theorist was Hugh Montgomery, a young American (loin" graduate work at Trinity College, Cambridge G.H. Hardy's old college.
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From page 288... ...
The expression that Hugh Montgomery mentioned, the expression that had emerged from his inquiries into the Riemann zeta function's nontrivial zeros, was precisely the form factor associated with a random
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From page 289... ...
To illustrate the point, I am going to take all the non-trivial zeros of the Riemann zeta function up to the height 500i that is, on the critical line (they all are on the critical line; the Riemann Hypothesis is certainly true down at these low levels) from 2 to 2 + 500i.
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From page 290... ...
By 1973 a vast amount of mathematical literature consisted of theorems that assumed the truth of the Hypothesis.~09 Today the quantity is correspondingly vaster, and if the RH (as I shall henceforth term it, fohowing Montgomery and all other modern researchers) proves false, this entire superstructure will become unstable; though if the counterexamples are few, much coul(1 be rescued.
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From page 291... ...
Hugh Montgomery lectured on what was, by that time, the "Montgomery pair correlation conjecture" at Princeton in 1978. Among those present was Andrew Odlyzko, a young researcher from the AT&T facility.
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From page 292... ...
Further work cleared up the discrepancies noted in the 1987 paper, and the Montgomery Pair Correlation Conjecture became the Montgomery-Odlyzko Law. The Montgomery-Odllyzko Law The (listribution of the spacings between successive nontrivial zeros of the Riemann zeta function (suitably normalized)
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From page 293... ...
Again, this doesn't affect the relative spacing; I have merely switched rulers. This final form of my sequence starts like this: 0, 1.2473,2.5840, ..., and ends like this: 9997.3850,9999.1528, 10,000.
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From page 294... ...
This shows the spacings for my 10,000 zeros against the curve predicted by GUE theory. It's not a sensationally good fit, but then my sample isn't very big, or very high up the critical line.
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From page 295... ...
The eigenvalues of a random Hermitian matrix arise from inquiries into the behavior of systems of subatomic particles under the laws of quantum mechanics. What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?
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