Deligne at Cambridge

Although a recent Cambridge lecture by Deligne introduces the cohomology of algebraic varieties as gently as possible, by the end of the hour he is amusingly talking about, as he puts it, motivic reasons for things.

The lecture carefully describes the differences between three types of cohomology:
1. Betti: topological, eg. the windings of a path around a circle
2. de Rham: about differential forms (associated to the variables in the polynomial that define the space)
3. $p$-adic cohomology: this is really nice when considering base number fields that are not necessarily $\mathbb{C}$.

For the de Rham cohomology functor, one ends up with vector spaces over some number field $k$. For $p$-adic cohomology, one instead has modules over the ring $\mathbb{Z}_{p}$ of $p$-adic integers.

This is just what happens in quantum mechanics when we stop worrying about Hilbert spaces. The mutually unbiased bases in dimension $d$, for $d$ a prime power, are given by the structure of the finite field on $d$ elements. The qubit component of quantum mechanics, for example, uses all dimensions $d = 2^{n}$, where $n$ is the number of qubits. A qubit set of observables therefore only needs these finite fields. If we throw in an appropriate categorical limit, we end up with the $2$-adic integers. Let us say that qubits are not about vector spaces then, because they are more naturally about modules over the $2$-adic integers.