# 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

Any term in the action, subject to a quantization condition should possess an appropriate nonrenormalization theorem protecting its coefficient. Please, see the following Wikipedia article . (The article refers also to other nonrenormalization theorems due to the holomorphy of the superpotential which are not relevant to our case.)

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 deep reason is related to the index theorem which states in the case of chiral anomaly that the integrated axial charge deficit is equal to the index of a Dirac operator axially coupled to the gauge field.

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.

Thus, the quantization condition should place a constraint on the renormalization forcing the product of the renormalized electric and magnetic charges to be equal to an integer.