On the Dirac charge quantisation, bare vs. renormalised$.$

Does the Dirac quantisation condition, $ge\in\mathbb Z$ (and its Schwinger-Zwanziger generalisation) refers to bare charges, or (on-shell) renormalised ones?

Both options seem natural to me, at least to some extent. From the point of view of path-integrals, one would expect the bare charges to be quantised, while from the point of view of observables (à la Aharonov-Bohm), one would expect the on-shell ones to be.

• – AccidentalFourierTransform May 15 '18 at 17:49
• Nice question! How do you define the renormalized magnetic charge? – pppqqq May 15 '18 at 18:54
• @pppqqq Thank you. To keep things as symmetric as possible (with respect to the electric charge), the renormalised magnetic charge is basically $g=Z_g g_0$, where $Z_g$ is determined by some physical condition (e.g., that the large $r$ limit of the effective electromagnetic field coincides with that of a monopole of charge $g$). This is similar to how we define $e$ as the coefficient of the Coulomb-like term that appears when you expand the effective potential of QED around $r\to\infty$. – AccidentalFourierTransform May 15 '18 at 18:59
• I see. My wild guess is that the relation should be preserved under renormalization, i.e. $Z_g = Z_e$ (much like what happens to the relation between particle and soliton mass in $1+1$ dimensional $\phi ^4$ theory, say). Perhaps we should give a look to what happens in the Georgi-Glashow model. Let's see what experts have to say :-) – pppqqq May 15 '18 at 19:24

An example of this type of nonrenormalization theorem is the Adler-Bardeen theorem (Please see the following review by Adler), which guarantees that no correction to the anomaly exists beyond one-loop. This theorem is related to the quantization of the coefficient of the Wess-Zumino-Witten term in $3+1$ dimensions.
The same is true for the Dirac quantization condition, The Dirac equation of a particle moving on a 2-sphere in the presence of a magnetic monopole has solutions only when the Dirac quantization condition is satisfied, and in this case $eg$ becomes half the number of zero modes of the Dirac equation, please see, for example, the following work by Deguchi and Kitsukawa for a derivation.