Electrons as math whizzes

Physicists hint that quantum systems could verify Riemann’s famous conjecture about prime numbers

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If two physicists are right, a single electron might know more about numbers than all of the world’s mathematicians. In an upcoming Physical Review Letters, the researchers hint that the dynamics of an electron can embody the solution to the nearly 150-year-old Riemann hypothesis, a crucial unsolved problem that has wide and deep consequences for number theory.

Germán Sierra of the Spanish National Research Council in Madrid and Paul Townsend of the University of Cambridge in England have proposed that when an electron is confined to moving in two dimensions, its possible energy level values might encode the key to the Riemann hypothesis.

Another new study suggests that electrons in certain quantum systems could embody all the information necessary to solve the conjecture. The graph above shows points (where all colors meet, along one vertical and one horizontal axis) at which calculation of Riemann’s zeta function would give zero values.
The possible energy levels of an electron confined on a plane may embody a solution to the Riemann hypothesis, a nearly 150-year-old problem in mathematics that’s at the heart of the study of prime numbers. The hypothesis says that a certain formula can give only the value zero when calculated at certain points that lie along one of two possible lines on the plane. In this visual representation, some of those points are indicated by the full color wheel. Jeffrey Stopple/University of California, Santa Barbara

“They have gone a step forward toward giving a physical description of the Riemann hypothesis,” comments Jonathan Keating of the University of Bristol in England. He warns, though, that the problem may not have gotten any easier as a result.

The hypothesis, or conjecture, was formulated by the great German mathematician Bernhard Riemann in 1859. It is mostly regarded as important because it would help reign in the apparent chaos in the world of prime numbers — whole numbers such as 2, 3, 5, 7, 11, and so on, that can’t be wholly divided by other numbers except 1 and themselves.  

Every whole number is either a prime number or the product of prime numbers, which makes primes one of the most pivotal concepts in mathematics. The Riemann hypothesis also has a $1 million “wanted” sign: The Clay Mathematics Institute in Cambridge, Mass. offers a cash prize in exchange for proof.

Mathematicians at least since Euclid have known that the list of prime numbers is infinite. No obvious pattern has been found in how they show up in the list of all numbers, with one possible exception: The prime number theorem, proven toward the end of the 19th century, describes how rare prime numbers are, on average.

The theorem predicts, for example, that from 1 to 1 million (or 106), about one in every six numbers is a prime; between 1 and 1 billion (or 109), it’s about one in every nine, and so on. In general, the theorem says that among the whole numbers between 1 and 10n, about one number for every n is a prime. (This is not literally true, due to a correction factor, but it describes approximately how primes become less and less frequent.)

At first sight, the Riemann hypothesis has nothing to do with prime numbers. It is a conjecture about a formula called Riemann’s zeta function, which calculates a number for every point on a plane. Riemann’s intuition was that the “zeros” of the zeta function — the points where zeta calculates the value zero — can lie only along one of two straight lines on the plane.

Mathematicians have shown that if the hypothesis is true, it would give further strength to the prime number theorem. It would imply that there are no wild statistical fluctuations in the distribution of primes. While still unpredictable, the primes would at least not be ruled by complete chaos.

Mathematicians have long suspected that there might be a way to convert the Riemann hypothesis into an equation similar to those used in quantum physics. The zeros of the zeta function could then be calculated the same way physicists calculate the possible energy levels for an electron in an atom, for example.

Following ideas by Keating and Michael Berry of the University of Bristol and also by Alain Connes of the Collège de France in Paris, Sierra and Townsend have now made that connection a bit more concrete. They have suggested that an electron constrained to move in two dimensions, and subjected to electric and magnetic fields, might have energy levels that precisely match the zeros of the zeta function. Demonstrating the existence of such a system, even on paper, would confirm the Riemann hypothesis.

The physicists didn’t quite do that yet, though. “We were able to get a piece of the Riemann formula” for the zeta function, says Sierra. Their explicit model gives only an approximation of the energy levels they needed. “Maybe you can prove the Riemann hypothesis thanks to nature,” says Sierra. “But this has not been done.”

In fact, don’t expect the Riemann hypothesis to be proved anytime soon. The paper constitutes “rather modest progress,” in the opinion of Enrico Bombieri, a mathematician at the Institute for Advanced Study in Princeton, N.J. Bombieri says that physicists still haven’t demonstrated a true connection between the zeta function and physics. Until then, he says, “attempts of this type belong to the works based on ‘wishful thinking,’ or even ‘pie in the sky.’

Keating, however, is more optimistic about Sierra and Townsend’s idea. “Maybe it will suggest further developments in the subject,” he says. If nothing else, it shows a possible connection between abstract arithmetic and physics, he says. “I think it’s a lot of fun.”

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