# Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site.

Several years ago, I tried to define a notion of "hyperholomorphy" for functions of the quaternionic variable and, as I wrote down the equivalent of the Cauchy Riemann equations for such "hyperholomorphic" functions, I obtained a matrix that, when multiplied on the left by the metric tensor of special relativity (that is, passing from an euclidian metric to a lorentzian one), provided a wave equation and a continuity equation, suggesting a strong connexion between those "hyperholomorphic" functions and physics. I would like to know if, defining the Riemann Zeta function on quaternions and then using this "metrick" (multiplying by the metric tensor of special relativity), one could obtain some kind of an explanation of the famous observation of Dyson that the formula in Montgomery's pair correlation conjecture for the Riemann Zeta function was the same as the pair correlation function of random Hermitian matrices. For example, can the vertical distribution of the Riemann zeros under RH be shown to be invariant under the application of the considered "metrick"? Once again, I apologize for the vagueness of the question, but it would be great if someone could use it to get interesting results about the connexion between number theory and physics.

Edit: writing the analogue of the Cauchy-Riemann conditions for a quaternionic function $P(q)+iQ(q)+jR(q)+kS(q)$ defined for a quaternion $q=(t, x, y, z)$ gives $(\partial_{t}+i\partial_{x}+j\partial_{y}+k\partial_{z})(P+iQ+jR+kS)=0$ hence the "hyper-Cauchy-Riemann" matrix is:

$M=\begin{pmatrix} \partial_{t} & -\partial_{x} & -\partial_{y} & -\partial_{z} \\ \partial_{x} & \partial_{t} & -\partial_{z} & \partial_{y} \\ \partial_{y} & \partial_{z} & \partial_{t} & -\partial_{x} \\ \partial_{z} & -\partial_{y} & \partial_{x} & \partial_{t} \\ \end{pmatrix}$

so that $M.\begin{pmatrix} P \\ Q \\ R \\ S \\ \end{pmatrix}=0$.

Multiplying on the left by the transpose of $^{4}\nabla=\begin{pmatrix} \partial_{t} \\ \partial_{x} \\ \partial_{y} \\ \partial_{z} \\ \end{pmatrix}$ and then the transpose of the resulting matrix on the left by $\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$ gives $\partial_{t}^{2}=\nabla^{2}$ and $\partial_{x}+\partial_{y}+\partial_{z}=0$, hence the wave equation and the continuity equation.

The fact that multiplying the matrix $M$ on the left by $\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$ (that is, the application of the "metrick") yields the same matrix with the opposite signs of the coefficients of the first row shows that any hyperholomorphic function defined on an euclidian 4-manifold remains hyperholomorphic when defined on the correspondent lorentzian 4-manifold. This is thus an invariant property when one passes from the purely mathematical realm to the physical one. The key idea is to show that this holds true for the vertical distribution of the Riemann zeros, those ones having to be the mathematical equivalent (i.e on the euclidian side) of the energy levels of an atom (on the lorentzian side).

-
Why didn't you show us in math symbols the multiplication? – Enjoys Math Jun 5 '14 at 16:01
Just because I performed the considered camculations several years ago and lost the sheets of paper where they were written. I can perform them once again but it will take some time. The "metrick" is to multiply the "hyper-Cauchy-Riemann" matrix on the left by the $4\times 4$ matrix $(a_{ij})$ such that $\vert a_{ij}\vert=\delta_{ij}$, $a_{11}=-1$ and $a_{ii}=1$ for $i>1$. This multiplication is involutive. – Sylvain Julien Jun 5 '14 at 16:31