# Examples of number theory showing up in physics

My question is very simple: Are there any interesting examples of number theory showing up unexpectedly in physics?

This probably sounds like rather strange question, or rather like one of the trivial to ask but often unhelpful questions like "give some examples of topic A occurring in relation to topic B", so let me try to motivate it.

In quantum computing one well known question is to quantify the number of mutually unbiased (orthonormal) bases (MUBs) in a $d$-dimensional Hilbert space. A set of bases is said to be mutually unbiased if $|\langle a_i | b_j \rangle|^2 = d^{-1}$ for every pair of vectors from chosen from different bases within the set. As each basis is orthonormal we also have $\langle a_i | a_j \rangle =\delta_{ij}$ for vectors within the same basis. We know the answer when $d$ is prime (it's $d+1$) or when $d$ is an exact power of a prime (still $d+1$), but have been unable to determine the number for other composite $d$ (even the case of $d=6$ is open). Further, there is a reasonable amount of evidence that for $d=6$ there are significantly less than $7$ MUBs. If correct, this strikes me as very weird. It feels (to me at least) like number theoretic properties like primality have no business showing up in physics like this. Are there other examples of this kind of thing showing up in physics in a fundamental way?

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MUBs is a really fascinating subjects. They are also linked with Latin squares, as e.g; shown here. I find this link with number theory more surprising than the role of prime numbers. – Frédéric Grosshans Dec 1 '11 at 19:30

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The MUB connection is really to finite fields (and latin squares as @FrédéricGrosshans mentions) and only through that to prime numbers. I guess we could say this is a connection to number theory, but really seems like a connection to abstract algebra, which is not nearly as surprising. – Artem Kaznatcheev Dec 2 '11 at 4:22
@Artem: It's true that you can arrive at above result via finite fields, but the structure of the partial results is governed by number theoretic properties. I don't really see the way of arriving at a given result as particularly fundamental, as there are often multiple paths to the result. – Joe Fitzsimons Dec 2 '11 at 12:43
– Qmechanic♦ Aug 25 '12 at 10:26

## 9 Answers

There are many attempts for a physical proof of the Riemann hypothesis. The major work in this direction was summarized in a recent review by: Schumayer and Hutchinson.

One of these attempts was proposed by: Berry and Keating. Their suggestion is within the framework of the Hilbert–Pólya conjecture, according to which, the Hilbert–Pólya Hamiltonian, whose spectrum is the imaginary part of the zeta zeros, can be obtained by quantizing a classical Hamiltonian of a chaotic system having periodic orbits with log prime periods. They argue that the classical Hamiltonian can be $xp$ (with appropriate yet unknown boundary conditions).

Another suggestion is due to Freeman Dyson in his Birds and Frogs lecture who suggests that the Riemann hypothesis might be proved through the classification of one dimensional quasicrystals.

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This seems more like engineering a physical system to embody certain number theoretic properties, rather than them occurring unexpectedly. – Joe Fitzsimons Dec 1 '11 at 15:15
I wouldn't count it as "engineering". It's more like using physical intuition in order to make a breakthrough in maths. We know the properties the Hilbert-Polya hamiltonian should have, so we whether it might be implemented in a physical system. – Javier Rodriguez Laguna Dec 3 '11 at 19:12

Here is a toy example; I don't know how interesting this will be to physicists. The eigenvalues of the Laplacian acting on, say, smooth functions $\mathbb{R}^k/(2\pi \mathbb{Z})^k \to \mathbb{C}$ are given by $$\{ m_1^2 + ... + m_k^2 : m_i \in \mathbb{Z} \}.$$

as a multiset (that is, with multiplicities). These are the energy eigenvalues of $n$ free non-interacting quantum particles on a circle. The multiplicity of a given eigenvalue is therefore the number of ways to write it as a sum of $k$ (integer) squares.

This is a classical number-theoretic problem. For example, it is a classical result that the number of ways to write a non-negative integer $n$ as the sum of two squares is $$r_2(n) = 4 \sum_{d | n} \chi_4(d)$$

where $\chi_4(d)$ is equal to $0$ if $d \equiv 0, 2 \bmod 4$, equal to $1$ if $d \equiv 1 \bmod 4$, and equal to $-1$ if $d \equiv 3 \bmod 4$. In general, the number of ways $r_k(n)$ to write a non-negative integer $n$ as the sum of $k$ squares has generating function $$\sum r_k(n) q^n = \left( \sum_{m \in \mathbb{Z}} q^{m^2} \right)^k = \theta(q)^k.$$

The function $\theta(q)$ is a theta function. Theta functions are closely related to modular forms, an important topic in number theory, and in fact the classical proof of the closed form $$r_4(n) = 8 \sum_{d | n} [4 \nmid d]$$

(where we have used the Iverson bracket above) proceeds by showing that $\theta(q)^4$ is a modular form; see Wikipedia.

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The $\theta$ page is crowded with greeks like the acropolis. Can you specify which $\theta$ you mean? – draks ... May 14 '12 at 18:23
@draks: it's defined immediately before I use the symbol. – Qiaochu Yuan May 14 '12 at 18:26
Ah, that was too close for me, thanks. (i) How are $k$, the number of squares and $n$, the number of particles on the circle related? And (ii) are there comparable formulas for $r_k(n)$ when one sums powers other than 2? – draks ... May 14 '12 at 18:32
@draks: $k$ is the number of particles. $n$ is (up to some normalization) an energy eigenvalue. The situation when summing powers other than $2$ is considerably more complicated because the close relationship to modular forms disappears and I don't know what's known about it. – Qiaochu Yuan May 14 '12 at 18:34
Thanks a lot.  – draks ... May 14 '12 at 18:35

I have an example of my own :). It appeared trying to calculate the dimension of a Hilbert space associated with rotationally invariant systems of n spins. The dimension was given in terms of the Moebius function. for details, check the appendix of Phys. Rev. E 76, 061127 (2007) or arXiv:quant-ph/0702164.

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That's weird, and certainly interesting. I'll give the paper a look. – Joe Fitzsimons Dec 2 '11 at 12:59

I've encoutered Diophantine equations (a variant of Pell's equation) in an (unpublished) attempt to turn a molecular system into a classical logical gate. The goal was to (approximately) synchronize incommensurable oscillations, and successives solution to the Diophantine equation gave me better fidelities.

I don't know if it qualifies for number theory, or even for physics, but I was surprised to find this equation as a good tool for my physics problem.

If anyone is interested, I can probably unearth my old notes and write something more detailed on the problem and the solution I found. Just ask in the comment.

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This is a general property--- diophantine approximation is related to resonance in classical systems like in KAM. – Ron Maimon May 8 '12 at 15:45

There are many theorems in quantum information which only apply to qudits of prime dimension. In particular, this seems to happen with graph states. In that case many theorems rely on the fact that multiplication modulo a prime is an invertible operation.

The Chinese Remainder Theorem can be used to show that graph states made of qudits of square-free dimension are equivalent to collections of graph states of qudits of prime dimension (the primes being the prime factorization of the original dimension).

Related to number theory is algebra. Group theory in particular tends to play an important role in quantum computing (e.g. the hidden subgroup problem).

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Thanks for taking the time to compose an answer. I don't really consider quantum algorithms as fundamental physics in the sense of this question, particularly given that the hidden subgroup stuff is driven by a generalization of problems from number theory (factoring/discrete logs). The graph state observation seems more related to the fact that you are looking at factoring a Hilbert space, which directly relates to primality of the dimesnionality, etc. – Joe Fitzsimons Dec 2 '11 at 12:50

For quantized cat maps, the inverse of Planck's constant is an integer N . There are various results for the special cases, where N is a power of a prime. So, the arithmetic properties of N play an important role here.

For references, see http://www.math.kth.se/~rikardo/cat2.pdf .

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Could you elaborate on this a bit? – András Bátkai Dec 2 '11 at 7:11
Added a reference. – jjcale Dec 3 '11 at 16:52

This answer is closely related to jjcale's answer. In this article, Gurevich and Hadani prove Rudnick's quantum ergodicity conjecture about the Berry-Hannay model. To do it they construct a number-theoretical description of the quantization of a torus phase space at rational values of hbar, involving l-adic sheaves on an algebraic variety of positive characteristic.

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This sounds more like physics showing up in number theory. – MBN Dec 5 '11 at 11:47
@MBN not quite. The result they prove belongs to the realm of quantum dynamical systems, not number theory. Number theory, or, more precisely, arithmetic geometry is a tool to solve the problem. – Squark Dec 5 '11 at 21:52

in general the Riemann xi function can be proved to be a functonal determinant

$\frac{\xi(s)}{\xi(0)}= \frac{det(H+1/4+s(1-s)}{det(H+1/4)}$ with $H=p^{2}+ V(x)$ and $V^{-1} (x)= 2 \sqrt \pi \frac{d^{1/2}}{dx^{1/2}}\frac{1}{\pi}arg\xi(1/2+i\sqrt x)$

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There's the Langlands program in supersymmetric quantum gauge theories, and string theory.

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