Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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20. The Riemann Operator and Other Approaches
20. The Riemann Operator and Other Approaches
Pages 312-326
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From page 312... ...
The previous year, in fact, Michael Berry had published a paper titled "Riemann's Zeta Function: A Mo(lel for Quantum Chaos? " Using results that were being widely circulated and (liscusse(1 at the time, inclu(ling some of O(llyzko's, Berry tackled the following question.
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From page 313... ...
Still, the arithmetical component is there in quantum mechanics, whereas in classical mechanics it is almost entirely absent. The foundations of classical physics the physics of Newton an Einstein are quintessentially analytical, in the mathematical sense.
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From page 314... ...
Kepler's laws assure us that we will still get a regular ellipse, and the mathematics of the vis viva equation tell us that a 1 percent error in r typically turns up only a 0.5 percent error in v. The situation is manageable, predictable.
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From page 315... ...
It is true that there is a degree of uncertainty in quantum mechanics, but the controlling equations are nonetheless linear. Small perturbations lead to small consequences, as with the classical vis viva equation for two-body motion.
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From page 316... ...
The periodic orbits that underlie a cIassical-chaotic system correspond to the eigenvalues of the operator defining this "semicIassical" system. Michael Berry argued that if there is a Riemann operator, it models one of these semiclassical chaotic systems, and its eigenvalues, the imaginary parts of the zeta zeros, are the energy levels of that system.
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From page 317... ...
A rotation is a rotation, even if you forgot to (lraw in a pair of axes. The operators use(1 in mathematical physics operate on much more complicated spaces than that, of course.
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From page 318... ...
Each point of the space, in other words, represents a state of the system. The operators used in quantum mechanics encode observable features of the system most famously, the Hamiltonian operator, which encodes the system's energy.
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From page 319... ...
In 1897 the Prussian mathematician Kurt Henselii7 devised an entire new family of objects to deal with certain problems in the theory of algebraic fields, like that a + bow field that I discussed in Chapter 17.ii. These objects are called "p-a(lic numbers." There is one field of these exotic creatures, with infinitely many members in it, for any prime number p.
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From page 320... ...
In the 1990s mathematical physicists set about constructing adelic quantum mechanics, in which the actual rational-number measurements that show up in experiments were taken to be manifestations of these bizarre creatures hauled up from the lightless depths of the mathematical abyss. This is the kind of space Alain Connes built for his Riemann operator to play in, an a(lelic space.
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From page 321... ...
Having the prime numbers already built in to one side of the formula ought to make everything easy. In a way it (lees, an(1 Connes's construction is brilliant, and extremely elegant, with energy levels that are precisely zeta zeros on the critical line.
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From page 322... ...
That is, as I said, exactly equivalent to the Hypothesis. Analytic number theorist Dennis Hejhal of the University of Minnesota actually uses this as a way to present the RH to nonmathematical audiences, to avoid having to introduce complex numbers.
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From page 323... ...
If you can prove that it is true, you will have proved the RH.~9 VII. A less direct probabilistic approach concerns the so-called "Cramer model." Harald Cramer was, in spite of that accent on his name, Swedish, and yet another insurance company employee an actuary for Svenska Livforsakringsbolaget, but also a popular and inspiring lecturer on math and statistics.
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From page 324... ...
Any broad statistical property the primes have how many you expect to fin(1 in intervals of certain lengths, for instance, or the degree of clustering (what Hilbert, in stating his eighth problem, called"condensation") this random list will have, too.
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From page 325... ...
In courts of law, in medical consultations, in drawing up insurance policies, it is the balance of probabilities that we take into account, not ironclad certainties. Sometimes, of course, we use the actual mathematical theory of probability to quantify the matters un(ler (lispute that is why insurance companies employ actuaries.
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From page 326... ...
The rules of evidence can deliver very persuasive results, sometimes contrary to the strictly argued certainties of mathematics. Consider, for example, the very un-mathematical fact that a hypothesis might be seriously weakened by a confirming instance.
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