Riemann Zeta Function connection to Quantum Mechanics. - MathOverflow [closed]

Riemann Zeta Function connection to Quantum Mechanics.

2011-02-06T06:36:36Z

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!

Answer by Ferenc Szalai for Riemann Zeta Function connection to Quantum Mechanics.

2011-02-06T12:13:32Z

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

Answer by Joseph O'Rourke for Riemann Zeta Function connection to Quantum Mechanics.

2011-02-06T18:38:19Z

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.

Answer by To be cont'd for Riemann Zeta Function connection to Quantum Mechanics.

2011-02-06T20:11:21Z

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.)

Answer by none for Riemann Zeta Function connection to Quantum Mechanics.

2011-02-08T04:31:28Z

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.

Answer by shb for Riemann Zeta Function connection to Quantum Mechanics.

2011-02-08T08:50:35Z

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.

Answer by BSteinhurst for Riemann Zeta Function connection to Quantum Mechanics.

2011-05-17T14:33:51Z

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.

Answer by Mathman for Riemann Zeta Function connection to Quantum Mechanics.

2012-03-06T18:48:44Z

the Xi function appears as the functional determinant $ \frac{ \xi(s)}{\xi(0)}= \frac{det(H+s(s-1)+1/4)}{det(H+1/4)}$ of a certain Hamiltonian with the potential $ V^{-1}(x)= A \sqrt D n(x) $

with $ n(x) \pi = Arg\xi (1/2+ i \sqrt x) $

here 'n' plays the role of Eigenvalue staircase $ n(x)= \sum_{n=0}^{\infty}H(x-E_{n})$ and 'H(x)' is the Heaviside function.