Riemann Rumbling On

I’ve just finished reading the delightful book by Sabbagh, Dr Riemann’s zeros. Although not a mathematician himself, Sabbagh competently launches into equations and diagrams, which he clearly explains for the lay reader. He spent a lot of time interviewing experts in the Riemann Hypothesis, to the point of attending their lectures, and he recorded the conversations.

The interaction of physics and the Riemann Hypothesis started with a memorable event, recounted in the book. Freeman Dyson was having afternoon tea as usual one day at Princeton in 1972. He was introduced to the visitor, Montgomery, who had been looking at the average gap in a long list of $\zeta$ zeroes. Montgomery mentioned his formula for the pair correlation, namely

$1 – ( \frac{\textrm{sin} \pi u}{\pi u} )^2$

at which point Dyson exclaimed that this was precisely the density of the pair correlation of eigenvalues of random matrices in the GUE.

Sabbagh was impressed by the awe that many mathematicians had for the Riemann Hypothesis. Conrey explained that the Riemann hypothesis is the most basic connection between addition and multiplication that there is, and Connes said: it is a basic primitive question about the adelic line which we don’t understand. It is a question about the way addition is fitting with multiplication.

Reminds one of categorical distributive laws, heh? Recall that for us addition and multiplication are monads (please don’t tell me you’ve forgotten about those). Anyway, a morphism $+ \times \rightarrow \times +$ which describes the commutativity of monads is a distributive law (wow – wikipedia is on to it). These are entities we need to think about in the context of quantum topos theory, because weak distributivity is the thing that separates quantum logic from the intuitionistic logic of an ordinary topos.

2 Responses so far »

  1. 1

    CarlBrannen said,

    Hmmmm. The Lebesgue measure on matrices is equivalent to the exponential of what I used to use as the energy for an element of a Clifford algebra, when represented as a matrix. That is, one sums over the squares of the absolute values of the entries, and this is equivalent to breaking the Clifford element into canonical basis elements and summing over the squares of the real multiples that describe the Clifford algebra as a vector space. The unexponentiated function is also a natural way of giving a probability measure to multiples of primitive idempotents of a Clifford algebra.

    But this didn’t work at getting the particle structure. Eventually I realized that I needed to give different weights to the different blades. The simplest guess was to assign the vectors a weight w > 1, and then the bivectors w^2, the trivectors w^3, etc.

    Then the particle structure appears if one assumes that nature cancels first the highest weight, then the next highest weight, etc., in that order.

    So the whole thing raises the question of what happens when you translate this modified weighting back into NxN matrices.

    The relation between unexponentiated and exponentiated measures in QM is very tight. The natural wave equation on a Clifford algebra is d \Psi = 0 where d is the natural differential (or massless Dirac) operator of the algebra. If \Psi solves this differential equation, then so does exp(\Psi). This is also how one relates the probabilities of QM, i.e. |(a|b)|^2, with the probabilities of statistical mechanics, exp(-\beta E).

  2. 2

    Anonymous said,

    Keep going!

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