# Number theory in Physics

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As a Graduate Mathematics student, my interests 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!

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Chandrasekhar

asked Nov 9, 2010
retagged May 4, 2014
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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Mark C
Related question on TP.SE: theoreticalphysics.stackexchange.com/q/609/189 now physics.stackexchange.com/q/26856/2451

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Qmechanic
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

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user TreyK
In squirrel cage motors the bars are employed in prime numbers.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Waqar Ahmad
@WaqarAhmad: That's not really physics - it's more of engineering.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Sanath Devalapurkar

@Dilaton Was the "mp.mathematical-physics" tag added by you? I am removing it, as Mathematical Physics, Theoretical Physics, Experimental Physics, etc., have their own categories.

Huh, why is mathematical physics a category, is it not part of theoretical physics? They used this tag on TP too, and I regularly use it on MO

@Dilaton I thought that it could deserve it's own category, since it is done from a "mathematican's perspective", not a physicist's, but then a tag could be enough to distinguish it, so OK.

@Dilaton Ok, just fixed all the mathematical-physics tagged questions, and have merged the Mathematical Physics category into the Theoretical Physics category.

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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... ;-)

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Daniel
answered Nov 10, 2010 by (685 points)
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).

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Mark C
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-)

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Daniel
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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user David Z
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Here's a journal link (full disclosure: I'm on the editorial board).

Communications in Number Theory and Physics

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Eric Zaslow
answered Nov 9, 2010 by (385 points)
Damn! I never thought number theory can be applied to anything other than cryptography!! enlightening!

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Pratik Deoghare
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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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user j.c.
answered Nov 9, 2010 by (260 points)
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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Ron Maimon
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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user j.c.
@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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Ron Maimon
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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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user grautur
answered Jun 20, 2011 by (50 points)
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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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user sigoldberg1
answered Nov 9, 2010 by (180 points)
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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.

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Marcel
answered Feb 13, 2011 by (300 points)
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One of the more recent connections between number theory and physics appeared in the context of the (refined) counting of degeneracies of half and quarter-BPS states in four-dimensional string theories with $\mathcal{N}=4$ supersymmetry. The canonical example here is given by the compactification of type II string theory on $K3\times T^2$ which is related to the heterotic string compactified on a six-torus. A larger family of examples is obtained by considering asymmetric orbifolds leading to CHL models. It also has lead to a connection with the sporadic simple group Mathieu group $M_{24}$ (and $Co_0$) and is called Mathieu moonshine. I will list three objects that appear in this context, all of number theoretic interest. Let $\rho=1^{a_1}2^{a_2}\ldots N^{a_N}$ denote a conjugacy class of $M_{23}\in M_{24}$ (for concreteness) and we will stick to the simplest situation i.e., type IIA compactified on $K3\times T^2$, again for concreteness.

1. The generating function of half-BPS states twisted by a symplectic automorphism of order $N$ ($N\leq 8$) is given by the multiplicative eta product $\eta_\rho(\tau)$ defined by the map: $$\rho=1^{a_1}2^{a_2}\ldots N^{a_N} \longrightarrow \eta_\rho(\tau):=\prod_{j=1}^N \eta(j\tau)^{a_j}\ ,$$ where $\eta(\tau)$ is the Dedekind eta function. This connection first appeared in  http://arxiv.org/abs/0907.1410 and connects to the work of Geoff Mason. In fact, there exist multiplicative eta products for all conjugacy classes of $M_{24}$ -- the connection with the physics application for all cases is not known.
2. The elliptic genera of $K3$ twisted by an arbitrary element of $M_{24}$ (also called twining genera) is given by a Jacobi form of weight $0$ and index $1$. All Jacobi forms can be obtained as a twisted trace over a graded $M_{24}$-module analogous to how the various J-functions were obtained as traces over a Monster module. This story is the most complete and the first paper here is due to Eguchi, Ooguri and Tachikawa. The picture was completed by Cheng, Gaberdiel, Hohenegger, Volpato, Eguchi, Hikami and others.
3. The generating function of degeneracies of twisted quarter-BPS states is given by genus-two Siegel modular forms that can be constructed in two ways: (i) as an additive lift that is determined by the multiplicative eta product (appearing in 1 above) and (ii) as a Borcherds product determined by the Jacobi form (appearing in 2 above). Some of these Siegel modular forms appear in the work of Gritsenko and Nikulin and are also related to the ones constructed by Clery and Gritsenko.

The story listed above is a glimpse into what is an active area of research and I am sure to have missed out a lot (sorry). There is a beautiful interplay here between, string theory, number theory, sporadic groups and Lie algebras (not mentioned here).

Modular forms appear in string theory and conformal field theory when one computes partition functions on a torus or higher genus surface.

answered May 5, 2014 by (1,545 points)
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You may be interested in this review article "Physics of the Riemann Hypothesis"

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user xavimol
answered Mar 1, 2011 by (60 points)
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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.

;)

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user danny
answered Dec 2, 2012 by (10 points)
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...

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Waffle's Crazy Peanut
Identical post by answerer: physics.stackexchange.com/q/45685/2451

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user Qmechanic
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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).

This post imported from StackExchange Physics at 2014-03-24 09:18 (UCT), posted by SE-user cubetwo1729
answered Dec 17, 2012 by (10 points)

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