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As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications of Number theory to physics. I have heard Applications of linear algebra and analysis to many branches of physics, but not number theory.

Waiting forward in receiving interesting answers!

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  • $\begingroup$ Here's a journal link (full disclosure: I'm on the editorial board). Communications in Number Theory and Physics $\endgroup$
    – Eric Zaslow
    Commented Nov 9, 2010 at 12:26
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    $\begingroup$ Good question, I was wondering the same when I was writing a question or answer recently. I had to take number theory out because I realized I did not know of any obvious connections to physics. $\endgroup$
    – Mark C
    Commented Nov 9, 2010 at 13:35
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    $\begingroup$ Related question on TP.SE: theoreticalphysics.stackexchange.com/q/609/189 now physics.stackexchange.com/q/26856/2451 $\endgroup$
    – Qmechanic ♦
    Commented Dec 2, 2011 at 1:23
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    $\begingroup$ As if the Riemann Zeta function wasn't already cool enough, it has plenty of applications to physics. The temperature at which matter changes phase to be a Bose-Einstein condensate uses $\zeta(3/2)$ in its calculation. Also, this may be of interest: en.wikipedia.org/wiki/Riemann_zeta_function#Specific_values $\endgroup$
    – TreyK
    Commented Dec 2, 2012 at 7:19
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    $\begingroup$ In squirrel cage motors the bars are employed in prime numbers. $\endgroup$
    – user28737
    Commented Dec 20, 2013 at 16:40

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I'm not sure i'll be able to post all the links i'd like to (not enough 'reputation points' yet), but i'll try to point to the major refs i know.

Matilde Marcolli has a nice paper entitled "Number Theory in Physics" explaining the several places in Physics where Number Theory shows up.

[Tangentially, there's a paper by Christopher Deninger entitled "Some analogies between number theory and dynamical systems on foliated spaces" that may open some windows in this theme: after all, Local Systems are in the basis of much of modern Physics (bundle formulations, etc).]

There's a website called "Number Theory and Physics Archive" that contains a vast collection of links to works in this interface.

Sir Michael Atiyah just gave a talk (last week) at the Simons Center Inaugural Conference, talking about the recent interplay between Physics and Math. And he capped his talk speculating about the connection between Quantum Gravity and the Riemann Hypothesis. He was supposed to give a talk at the IAS on this last topic, but it was canceled.

To finish it off, let me bring the Langlands Duality to the table: it's related to Modular Forms and, a such, Number Theory. (Cavalier version: Think of the QFT Path Integral as having a Möbius symmetry with respect to the coupling constants in the Lagrangian.)

With that out of the way, I think the better angle to see the connection between Number Theory and Physics is to think about the physics problem in a different way: think of the critical points in the Potential and what they mean in Phase Space (Hamiltonian and/or Geodesic flow: Jacobi converted one into another; think of Jacobi fields in Differential Geometry), think about how this plays out in QFT, think about Moduli Spaces and its connection to the above. This is sort of how I view this framework... ;-)

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A semi-silly idea that I've read about is the Primon gas, a model where the Riemann zeta function arises as the partition function of a quantum statistical mechanical system.

More seriously, take a look at the papers of Yuri Manin and Matilde Marcolli on the hep-th arxiv, which attempt to connect the holographic principle to arithmetic geometry. I think there's a lot of hope that the techniques in physics inspired by quantum field theory and string theory might have applications to various branches of mathematics including number theory (for this sort of thing, I can't do better than point you to the writings of John Baez) -- I am not as aware of applications of number theory to the kind of physics that can be tested experimentally (though I'd love to be corrected).

One unrelated example -- Freeman Dyson has made vague speculations on quasicrystals and the Riemann hypothesis, you can read about it along with some entertaining history in this article.

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    $\begingroup$ The primon gas is not silly, just under-developed. It's the reason people believe that the Riemann hypothesis has something to do with eigenvalues of random matrices, and the Lee Yang circle theorem. $\endgroup$
    – Ron Maimon
    Commented Sep 18, 2011 at 21:28
  • $\begingroup$ As far as I know, the primon gas so far has not been rigorously related to the Hilbert-Polya conjecture that you are referring to (in particular, the conjectured operators in the latter look nothing like the "hamiltonian" of the primon gas). Please correct me if I'm wrong though. $\endgroup$
    – j.c.
    Commented Oct 4, 2011 at 22:59
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    $\begingroup$ @j.c.--- you aren't wrong, there is not much rigor in these things. But the main reason that the operators don't look alike is that the "primon" gas is in the infinite occupation number regime in the critical strip. There are no solid conjectures for the Hilber-Polya Hamiltonian in the infinite strip, as far as I know. The primon-gas business is mostly useful for recasting standard zeta-function identities so that they become obvious to someone who knows statistical mechanics. $\endgroup$
    – Ron Maimon
    Commented Oct 5, 2011 at 0:12
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There's a fantastic article on the relationship between the Riemann Hypothesis and "quantum chaos" at www.msri.org/ext/Emissary/EmissarySpring02.pdf (starts on page 1, continues on page 12).

Here's an excerpt (recall that Montgomery's Conjecture is a conjecture about the expected number of zeros of the Riemann zeta function that follow a zero in an interval of a certain length):

Montgomery was taken aback to discover that Dyson knew very well the rather complicated function appearing in Montgomery’s conjecture, and even knew it in the context of comparing gaps between points with the average gap. However — here’s the amazing thing: It wasn’t from number theory that Dyson knew this function but from quantum mechanics. It is precisely the function that Dyson himself had found a decade earlier when modelling energy levels in complex dynamical systems when taking a quantum physics view- point. It is now believed that the same statistics describe the energy levels of chaotic systems; in other words, quantum chaos!

The article describes some other surprising connections as well, between different zeta functions and the energy levels of other kinds of chaotic systems. Instead of copying those out here (I can't summarize, since I don't understand it well myself), I'll just end with a quote from the article:

In summary, the more intuitive development of quantum chaos allows more fruitful predictions about the distribution of primes (and beyond). On the other hand the more cautious development of prime number theory leads to more accurate predictions in quantum chaos.

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I wasn't aware of this until very recently, when i casually read this article about Ramanujan expressions for modular forms (which are a form of holomorphic functions that leave invariant certain lattices, and are extensively studied for their number-theoretic applications). Apparently there is something called "modular black holes" which i don't have the faintest idea what is about, but it mentions that they are thermodynamically close to normal black holes, so they can be used to compute certain scrambling functions of the event horizon degrees of freedom

I would rather have someone provide an authoritative answer mentioning more details about this, as my ramblings are more or less extracted unmodified from the article. I hope someone that really understand about this gets annoyed enough by my answer and provides a real one.

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I am on thin ice here, but I know that people in number theory study modular forms, and this is connected to partition functions for example of conformal field theory.

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The 2-volume "Frontiers in Number Theory, Physics, and Geometry", edited by Cartier et al is a great collection of articles.

My other suggestion would be to have a look at this page ("Number Theory and Physics at the Crossroads" workshop held at Banff) - the bottom half of the page lists a significant number of those areas where physics and number theory flourish together.

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there is a conjecture so called RIEMANN HYPOTHESIS , which have a deep relationship between the roots of the Riemann Zeta function and the eigenvalues of a Hamiltonian

http://arxiv.org/abs/1101.3116

http://findarticles.com/p/articles/mi_m1200/is_7_174/ai_n30887057/

and my humble paper about the subject http://vixra.org/pdf/1007.0005v8.pdf in fact RIEMANN HYPOTHESIS is juts to use WKB to find a Hamiltonian whose eigenvalues are the square of the RIemann zeros (imaginary part) and its FUNCTIONAL DETERMINANT is just the RIemann Xi function

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