### Prime Evil Cats

Category theorists are fond of replacing the counting numbers by sets, in the category Set. The trouble is that prime numbers don't make a lot of sense for ordinary sets. If I buy 3 oranges and 5 apples, I can add them by putting them all in the same fruit bowl. On the other hand, what is special about three oranges, as opposed to four or six? The category Set does have Cartesian product, but there is nothing special about sets with a prime number of elements.

The simplest object for which the primes define a space of some kind are the integers, as a special object in the category of rings. But as Kapranov and Smirnov point out in their paper, this space has dimension greater than zero, with respect to any reasonable topology. Where are the points?

This question leads to a study of the field with one element. Recall that ordinary sets are like vector spaces for this field $F$, but sets can also come with extra structure. In particular (for an extension $F(n)$ of the magical field $F$) one has pointed sets and actions by roots of unity. Now sets are usually acted on by permutations, so we want mixtures of permutations and roots of unity, which is what MUBs were about!

The correct actions for $F(n)$ are given by the wreath product of permutations $S_{d}$ and roots of unity. That is, $d \times d$ matrices with only one non zero element in each row and column, with every element an $n$-th root of unity. When $n$ is prime, and when $n$ corresponds with $d$, we get nice MUB type operators.

Unfortunately, although quantum mechanics is suddenly a whole lot closer to arithmetic, it is still not clear why quantum $3$-oranges are so much better than quantum $4$-oranges, except that $n$-oranges had better be built from prime ones.

The simplest object for which the primes define a space of some kind are the integers, as a special object in the category of rings. But as Kapranov and Smirnov point out in their paper, this space has dimension greater than zero, with respect to any reasonable topology. Where are the points?

This question leads to a study of the field with one element. Recall that ordinary sets are like vector spaces for this field $F$, but sets can also come with extra structure. In particular (for an extension $F(n)$ of the magical field $F$) one has pointed sets and actions by roots of unity. Now sets are usually acted on by permutations, so we want mixtures of permutations and roots of unity, which is what MUBs were about!

The correct actions for $F(n)$ are given by the wreath product of permutations $S_{d}$ and roots of unity. That is, $d \times d$ matrices with only one non zero element in each row and column, with every element an $n$-th root of unity. When $n$ is prime, and when $n$ corresponds with $d$, we get nice MUB type operators.

Unfortunately, although quantum mechanics is suddenly a whole lot closer to arithmetic, it is still not clear why quantum $3$-oranges are so much better than quantum $4$-oranges, except that $n$-oranges had better be built from prime ones.

## 1 Comments:

A commenter, with spam like features, asked what was an n-orange. This is just a way of talking about quantum sets which are acted on by nxn matrices.

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