# Number theory in Physics

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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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. –  Mark C Nov 9 '10 at 13:35
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 –  TreyK Dec 2 '12 at 7:19
In squirrel cage motors the bars are employed in prime numbers. –  Waqar Ahmad Dec 20 '13 at 16:40
@WaqarAhmad: That's not really physics - it's more of engineering. –  Sanath Devalapurkar Feb 2 at 3:24
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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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Welcome, Daniel, thank you for sharing. You should be able to post them now. I'm glad to have a researcher around! Oh, your blog is in Portuguese, too bad (for me). –  Mark C Nov 10 '10 at 6:08
Thanks Mark — it really helps (i thought my reputation score would be "transfered" from some of the other StackExchange sites i've already used, but...). ;-) Anyway, as for pt_BR, try Google Translate: not perfect but, gives you a flavor. 8-) –  Daniel Nov 10 '10 at 22:30
once you get 200 or more reputation on any one Stack Exchange site, you'll get a +100 rep bonus when you associate any account on another SE site with that one. Maybe that's what you were thinking of. –  David Z Nov 10 '10 at 22:35

Here's a journal link (full disclosure: I'm on the editorial board).

Communications in Number Theory and Physics

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Damn! I never thought number theory can be applied to anything other than cryptography!! enlightening! –  Pratik Deoghare Nov 9 '10 at 12:47

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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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. –  Ron Maimon Sep 18 '11 at 21:28
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. –  j.c. Oct 4 '11 at 22:59
@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. –  Ron Maimon Oct 5 '11 at 0:12

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 recall being stunned (they can do that?) by the mere mention of the technique called "Zeta Function Regularization", see http://en.wikipedia.org/wiki/Renormalization, http://en.wikipedia.org/wiki/Zeta_function_regularization , to sum divergent series like zeta(-n) to get finite results.

I thought that was the limit. Then I heard about p-adic strings, complex dimensions, and more. See e.g. http://inc.web.ihes.fr/prepub/PREPRINTS/2008/M/M-08-42.pdf .
The ideas are certainly very beautiful.

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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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You may be interested in this review article "Physics of the Riemann Hypothesis"

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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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If you are interested in analytic number theory, look at the paper Eisenstein series for higher-rank groups and string theory amplitudes by Michael Green (one of the founders of string theory), Stephen Miller (a number theorist), Jorge Russo (a physicist), and Pierre Vanhove (a physicist).

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Here is a nice article from a news channel:

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Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. –  Dimensio1n0 Jul 22 '13 at 16:45

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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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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The connections between number theory and physics are too powerful, too certain, to be ignored for much longer by the mainstream scientific community. At some point surely we will realise physics can be investigated through the analysis of number theory, and the implications of this are - well, interesting doesn't really cover it.

The universe is clearly and unambiguously mathematically consistent. There can only be, by definition, one sum total of existence. And at that scale, the only value we can ascribe to the sum total of existence is that it = 1. If then everything within the sum total of existence is composed of parts of that sum total, it's not illogical to suggest that whatever exists within the universe is 'made' of nothing more complex than fractions of the whole, self-organising at every scale from the astromnomical to the quantum - according to the laws of number theory.

Our human number system is only a system of classification - representative of the natural ordering of the universe into integer parts. We have names for the different parts of the electromagnetic spectrum, so we have names for the different parts of the numerical spectrum.

As the theory of gravity is a human description of a natural characteristic of the universe, so number theory is a human description of a natural characteristic of the universe. We don't pretend we invented gravity, simply because we worked out some of the 'rules' by which it operates, why then do we insist that we invented numericality and mathematicality? Particularly when it is so self-evident that for humans to have evolved, the universe HAD to have both those characteristics already.

But what really blows my mind isn't the amazing possibilities this has for science. No. What blows my mind is that these connections are so clear, and they so clearly make scientists uncomfortable - so uncomfortable they refuse to look any closer and when pressed begin to mumble excuses about anthropocentricity they never put forward when investigating gravity, or quantum mechanics - both of which we're only able to investigate because our existence is predicated on theirs, in exactly the same way.

Surely, surely if ANYONE should be refusing to look away from what makes them uncomfortable, it is the scientific community? Any true scientist, when they came across a natural phenomena that weirded them out so badly as this, would surely LOOK MORE CLOSELY?!

Come on people. Sort yourselves out! Solve the mystery! Bloody hell, whatever else the connections are - they're nothing if not seriously strange, and seriously cool.

Let's call it 'teh big bang theory' - like the big bang theory, but you need some humour to get it.

;)

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Hello user danny. Welcome to Physics.SE. We try not to discourage new users. But, a nice tip - "Please be precise of your answer. Don't be too chatty" as I'd say... –  Waffle's Crazy Peanut Dec 2 '12 at 6:18
Identical post by answerer: physics.stackexchange.com/q/45685/2451 –  Qmechanic Dec 2 '12 at 10:32