# Why does zeta(3) appear in the g-factor of the electron - its anomalous magnetic moment?

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The anomalous magnetic moment, to second order in the fine structure constant, contains $\zeta(3)$, the Riemann function at the real value 3. The value appears inside the pre-factor, as calculated by Petermann in 1957.

a: Is there a simple way to explain its appearance?

b: Some people have speculated on the internet that this appearance is related to knots. How can one see this?

asked Oct 19, 2019 in quant-ph
edited Oct 19, 2019

$\zeta(3)$ appeared in Petermann's calculations due to his method of calculating.

It is nuts to say that its appearance is related to knots. Tha anomalous magnetic moment takes into account the "radiative corrections", which is nothing but taking into account permanent coupling to omnipresent EMF, electron-positron pair, etc. oscillators. This reveals a "complicated" rather than "elementary" caracter of the electron. In other words, it reveals the electron being a quasi-particle of a huge system.

$\zeta(3)$ appears independently of the method of calculation. Please look at the various other people who did the calculation. In all expressions, $\zeta(3)$ arises.

And surely it is not "nuts" to say that this value of the Riemann function is related to knots. I just found various papers by Kreimer on arxiv on this topic. But they are not easy to digest.

The rest of what you write is correct. But no part of your comment helps to answer either question (a) or question (b).

I disagree. Arising or not arising $\zeta(3)$ must depend on the calculation method. The exact value of the anomalous magnetic moment my not contain $\zeta(3)$ at all.

$\zeta(3)$ itself may be connected to "knots", but who cares?

@Vladimir If you can show an error in Petermann's 1957 calculation (the g-factor to order $\alpha^2$), you will be world famous overnight. Even Petermann himself became famous because he found an error in the first calculation from 1949 (which also contained $\zeta(3)$).

Your chances to become famous are small however, because in the meantime,  the calculation has been checked by computer algebra many times. Even in the term of order $\alpha^3$, known since 1996, the number $\zeta(3)$ arises; also a term with $\zeta(5)$ arises.

If you have a proof why $\zeta(3)$ cannot appear in the g-factor, you will also be famous and be cited in all textbooks. Please publish the proof - or tell us all about it. Facts tell otherwise: so far, the calculation agrees with measurements completely, to all measured digits.

It might well be that you do not care about knots, but I do. That is why I asked. Your statements get more and more wrong. And they still do not help answering question (a) or (b).

I am not looking for fame, as a matter of fact, so this was not my motivation. The calculations you refer to are made within the usual QED or in its electroweak extension ; both including renormalizations and soft mode summations. I believe that in a correct QED formulation there will not be those inconsistencies and the "calculation method" will be different.

As far as I know, the anomalous magnetic moment "belongs" to the electron only in weak magnetic fields. Otherwise the interaction term contains non-linear magnetic field contributions, so the matter here is more complicated than just being proportional to $\zeta(3)$.

in fact, zeta(3) appears inside a sum ( of positive and negative values ) with powers of pi , of ln(2) and one another integer based infinite sum. It is common to see zeta in solutions of pde with powers of e. Perhaps it is counter intuitive but it is not astonishing to find such expressions after tricks and even high quality approximations. This doesn't discard a not obvious deep mathematical reason. Notice that in the Peskin-Schroeder, the part of the integral involving zeta is not affected by UV or IR divergences.

$\zeta(3)$ is just a number, not a function. And it is not sole in the (approximate) expression, so let us not exagerate its meaning.

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