Warning! This set of interlinked pages is very much under construction. If you are already familiar with the subject matter, I would appreciate any help you might provide. If you are seeking information on the subject matter, I would be interested to hear what you would ultimately like this page to contain. Thank you.



Gutzwiller Trace Formula
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Selberg Trace Formula
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Riemann's zeta function
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Riemann-Weil Explicit Formula
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Connes'(hypothetical)
trace formula
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(Deninger's) Lefschetz
Trace Formulae



From p.1 of Connes' "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function"

"We shall give in this paper a spectral interpretation of the zeros of the Riemann zeta function...The spectral interpretation of the zeros of zeta will be as an absorption spectrum, i.e. as missing spectral lines. All zeros will play a role in the spectral side of the trace formula, but while the critical zeros will appear per se, the noncritical ones will appear as resonances and enter in the trace formula through their harmonic potential with respect to the critical line."

Connes also relates his his ideas to the Riemann-Weil explicit formula and to the Selberg Trace Formula fairly directly. He doesn't mention the Gutzwiller Trace Formula by name, but goes into quantum chaos and refers to Berry's work heavily.

Connes is trying to prove the Riemann Hypothesis (which concerns the zeros of the Riemann zeta function) by proving "a certain classical trace formula".



In her New Scientist article, E. Klarreich says [of the Clay Foundation prize for a proof of the Riemann Hypothesis]:

"The winner could well be Alain Connes, a mathematician based at the Institute of Advanced Scientific Study in Bures-sur-Yvette, France. Connes has a startlingly direct approach to the problem: create a system that already includes the prime numbers. To understand how, you have to imagine a quantum system not as a particle bouncing around an atom, say, but as a geometrical space. It sounds odd, but it represents one of the weird things about quantum systems: they can be two or more things at once.

Like Schrödinger's cat, which is a peculiar mixture of dead and alive, any quantum object can find itself in a "superposition" of different states. To characterise this messy existence, physicists use what they call a state space. For each kind of possibility (say "alive" and "dead"), you draw a new axis and add a dimension to the space. If there are just two possible states, as is the case for Schrödinger's cat, the space is two dimensional, with three states it is three dimensional, and so on.

Then in the Schrödinger's cat space, you would mark a cross one unit along the x-axis to represent a fully alive cat. Similarly, a stone dead cat would be one unit up the y-axis, and a part-alive, part-dead cat would appear somewhere along an arc between these points.

The "shape" of the space affects how the state moves around in it, and therefore how the system works, including the way its energy levels are arrayed. This depends not just on the number of dimensions, but also on the geometry of how they are stuck together.

Connes decided to build a quantum state space out of the prime numbers. Of course, the primes are a bunch of isolated numbers, nothing like the smooth expanses of space in which we can measure things like angles and lengths. But mathematicians have invented some bizarrely twisted geometries that are based on the primes. In "5-adic" geometry, for example, numbers far apart (in the ordinary way) are pulled close together if they differ by 5, or 15, or 250--any multiple of 5. In the same way, 2-adic geometry pulls together all the even numbers.

To put all the primes in the mix, Connes constructed an infinite-dimensional space called the Adeles. In the first dimension, measurements are made with 2-adic geometry, in the second dimension with 3-adic geometry, in the third dimension with 5-adic geometry, and prime numbers.

Last year Connes proved that his prime-based quantum system has energy levels corresponding to all the Riemann zeros that lie on the critical line. He will win the fame and the million-dollar prize if he can prove that there aren't any extra zeros hanging around, unaccounted for by his energy levels.

That last step is a formidable one. Has Connes simply replaced the Riemann hypothesis with an equally difficult question? Some experts advise caution. "I still think that some major new idea is needed here," says Bombieri."
 


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