Recently, Daniel made a comment that I accidentally deleted. He was reading Carl Brannen's paper on spin path integrals, and he noticed that the spin eigenvalues $\pm 1/2$ were obtained from the infinite series

$1 + 1 + 1 + 1 + \cdots = - \frac{1}{2} = \zeta (0)$

$1 - 1 + 1 - 1 + \cdots = \frac{1}{2} = \eta (0)$

which are explained on that old post at the Everything Seminar. Here $\zeta$ is of course the Riemann zeta function and $\eta$ the Dirichlet eta function. These closely related functions both have a functional relation responsible for extending the function to the complex plane.

In constructive number theory, this will be the typical fashion in which quantum numbers arise. The initial $\pm 1$ may be interpreted as the bosonic spin eigenvalues in the steps that create the emergence of fermionic spin. However, we should not forget the supersymmetry between fermions and bosons in the mathematical formulation of emergent geometry. For example, we know that

$1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = \sum \frac{1}{2^n}$

$-1 = 1 + 2 + 4 + 8 + \cdots$

is one means of obtaining a bosonic eigenvalue from a fermionic one. Two eigenvalues may be collected into a matrix:

6 years ago

Oops, I missed a zeroth power for the matrices. Well, as you can see, the zeta function has a great deal to do with duality and bootstrapping in quantum gravity.

ReplyDeleteThe generalised Riemann hypothesis says that no Dirichlet L-series has a zero with real part greater than $1/2$.

I am discussing this issue with Carl. He already answered me and today I indicated this blog post to him.

ReplyDelete