Sunday, February 13, 2011

All the Weil

The proven Weil conjectures for varieties over finite fields outline the properties of zeta functions for such spaces. For example, consider the projective line over the finite field with $p^n$ elements. It's zeta function is

$\zeta (s) = \frac{1}{(1 - p^{-s})(1 - p^{1-s})}$

and higher dimensional projective spaces just have more factors on the bottom. Grothendieck and others eventually understood the Weil conjectures using an $l$-adic cohomology theory, where $l$ is a prime different from $p$. The zeta function was expressed in terms of determinants for $l$-adic cohomology groups, and a Poincare duality explains the famous duality symmetry of zeta functions.

In quantum gravity we find fields with $p^n$ elements in the mutually unbiased bases for dimension $p^n$. As for a projective line, there are always $p^n + 1$ elements in a set of MUBs. Considering $n$ qupits for all $n$, we are naturally led to a generalisation of $p \times p$ matrix entries to the $p$-adic numbers. But constructing the complex numbers correctly in this axiomatic setting does not mean taking the complex numbers as they are usually given to us, complete with an unmeasurable continuum and a set theoretic continuum hypothesis. As any topos theorist knows, the real and complex numbers are not even uniquely defined in a topos. Quantum gravity will use the numbers it requires, and no more. A useful complicated space is, after all, described by a finite amount of data.


  1. That is not to say that we cannot start thinking about concrete infinite mass matrices of countable index, or rather, square root mass matrices that can be paired.

  2. Despite feeling rather hopeless about getting into grad school I'm nevertheless honing my physics. And I'm getting drawn deeper into understanding what you and Rios and others are doing.

    I just figured out that I can rewrite my weak quantum number paper so as to make it a very obvious and natural calculation in a Hopf algebra. I'd overlooked this because my Basic Algebra I book (Jacobson) calls it by a different name, i.e. K[G] a field over a group.

    If I don't have it written up Tuesday it's cause I'm too lazy to be in this business.

  3. What is the difference between l and p primes?

  4. That's good, Carl. Ulla, when studying specific spaces over general number fields, it turns out that different primes behave differently, and it is useful to play them off against each other. Here, $l$ and $p$ are just two distinct prime numbers. See, for instance, the definition of ramification and Galois module.

  5. Kea, would you be so kind as to write a post on Kopf algebra, especially as it connects with what you're doing? I know it's related, and I'd like to reference it, but it's difficult theory.

  6. Amazing! A couple introductions to Hopf algebra are finally moving me forward on understanding category theory. I.e. and a paper by Selig, "A Very Basic Introduction to Hopf Algebras".


Note: Only a member of this blog may post a comment.