Examples of number theory showing up in physics

Question

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?

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Joe Fitzsimons

Comments

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.

This post has been migrated from (A51.SE)

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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.

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Artem Kaznatcheev

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@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.

This post has been migrated from (A51.SE)

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@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.

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Joe Fitzsimons

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Duplicate on Phys.SE: physics.stackexchange.com/q/414/2451

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Qmechanic

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MUBs is a really fascinating subjects. They are also linked with [Latin squares](http://en.wikipedia.org/wiki/Graeco-Latin_square#Mutually_orthogonal_Latin_squares), as e.g; shown [here](http://pra.aps.org/abstract/PRA/v79/i1/e012109). I find this link with number theory more surprising than the role of prime numbers.

This post has been migrated from (A51.SE)

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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.

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Frédéric Grosshans

Answer 1

there is also ZETA REGULARIZATION for divergent integrals of the form $\int_{a}^{\infty}x^{m}dx $ and $ \int_{0}^{b}x^{-m}dx $ for positive a and b integer and 'm' a real number this can be used in renormalization :)

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Jose Javier Garcia