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I feel like this question is probably wrong for MO, (too low level, perhaps unclear) but my curiosity has got the better of me:

I hear that the Riemann Zeta Function and its zeros have applications to quantum mechanics, as well as other fields. I do not understand these connections, and because of this the following question came up:

In theory, is it possible through physical experiments (particle experiments) to approximately calculate the first few zeros of the Riemann zeta function?

In other words, (using the explicit formula) could we write down the $n^{th}$ prime number (up to a given margin of error/probability of correctness) only from doing quantum mechanical experiments?

(If there are conjectures/facts that we cannot prove, but would answer the question, I would be happy to hear those too)

Thanks!

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Currently, no. The idea, which originated (maybe?) in a suggestion of Polya, and has been developed further by Keating, Berry, Deninger and others, is that perhaps there is a "natural" quantum mechanical system such that the eigenvalues of its Hamiltonian are the zeros of the zeta function. Certain symmetries of the system would imply RH. This is explained quite well in many googleable places. –  David Hansen Feb 6 '11 at 6:48
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I would suggest to look at Connes' articles related to the Riemann Zeta function, arithmetics and noncommutative geometry, etc. –  Thomas Riepe Feb 6 '11 at 10:16
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"In other words, could we write down the nth prime number (up to a given margin of error/probability of correctness) only from doing quantum mechanical experiments?". Yes, of course: Eratosthenes' Sieve can be implemented in our universe, and our universe is definitely quantum mechanical. But this isn't quite what you're after. There do exist physicists who think about thinks like gases of "primons", but the little that I've seen has left me unconvinced that it has much application to either math or physics. (I hope I'm wrong!) –  Theo Johnson-Freyd Feb 6 '11 at 18:50
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The idea behind this is that the statistical distribution of the zeros of the Riemann zeta function appears to be a distribution which also arises in the eigenvalues of Hamiltonians of quantum chaotic systems, in random matrix theory, and now in a number of other places. It's possible that there is a quantum chaotic Hamiltonian which gives rise to the zeros of the zeta function. However, one can also argue that expecting this is as unrealistic as expecting a direct correspondence between two stochastic processes which both give rise to normal distributions. –  Peter Shor Feb 6 '11 at 18:53
    
One clarification: by "appears to be" above, I mean "almost certainly is, although it has not yet been proven." –  Peter Shor Feb 6 '11 at 18:53
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6 Answers 6

I recently found the article about this topic which may give you some answer: "Physics of the Riemann Hypothesis" http://arxiv.org/abs/1101.3116

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For pleasurable background reading, permit me to suggest the oft-told tale of the encounter between Freeman Dyson and Hugh Montgomery. One source is The Riemann hypothesis: the greatest unsolved problem in mathematics by Karl Sabbagh, Chapter 9, available via Google Books here.

...when Hugh Montgomery was visiting Princeton in 1972 and was introduced to ... Freeman Dyson over tea, he answered perfectly truthfully when Dyson asked him conversationally what he was working on. His answer struck a chord with Dyson, who then supplied a piece of information that indirectly led to what today is seen as the most promising approach to proving the Riemann Hypothesis.

The same story is told in the book Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime, by Dan Rockmore; Google Books here. A bit more technical detail is in the Wikipedia article on Montgomery's conjecture. And here is a discussion by John Baez in his This Week's Finds column.

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M. Berry, Riemann's zeta function: a model of quantum chaos, Lecture Notes in Physics, Vol.263, Springer-Verlag, 1986.

I thought that this article was very useful.

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This may be buried in one of the references above, but for those don't wish to go through them all...

The zeta function can arise as the trace of Hamiltonians governing physical systems. For example in an experiment to measure the Casimir effect (two perfectly conducting plates placed very close to each other) the force they exert on each other has a formula that involved the the derivative of the Riemann zeta function evaluated at $-\frac{1}{2}$. This has been experimentally validated to a reasonable amount of precision.

This may not get you zeros, but it gets you certain values.

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There were some papers I heard of which sought to explore the connections between Riemann zeta functions and quantum theory (saw them on my brother's buzz and Anirbit's). (Never dare ask me about details!) (Some have been mentioned above.)

  1. NONCOMMUTATIVE GEOMETRY AND THE RIEMANN ZETA FUNCTION (by Alain Connes)

  2. RIEMANN HYPOTHESIS AND QUANTUM MECHANICS (by MICHEL PLANAT, PATRICK SOL´E, AND SAMI OMAR)

  3. A PHYSICS PATHWAY TO THE RIEMANN HYPOTHESIS (by Germán Sierra)

  4. The Physics of the Riemann Zeta function(David Lyon)

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Item 2 is not by van der Poorten; he was just quoted below the title. The authors are Michel Planat, Patrick Sole ́, and Sami Omar. –  Stopple Feb 7 '11 at 21:23
    
Oh, thanks. Corrected. –  Unknown Feb 7 '11 at 22:03
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This paper came up in a different context on MO a few days ago: http://crd.lbl.gov/~dhbailey/dhbpapers/ppslq.pdf

It shows the multiple zeta function (a generalization of the zeta function to multiple variables) showing up in QFT in what I saw as an amazing way. But I doubt there is any analogue physical apparatus that gets the values out in a way that tells us anything about RH.

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