# Dirac magnetic monopoles and quark fractional electric charge quantization

When applying the Dirac quantization rule for electric and magnetic charge, I assume one is considering unit electric charges such as electrons. How does the Dirac quantization rule apply for the fractional electric charges of quarks?

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– Qmechanic Jun 12 '13 at 19:15
I answer this question at physics.stackexchange.com/questions/268709/…. – tparker Jul 26 at 0:23

The Dirac quantization rule comes from integrating the angular momentum of the superposed electromagnetic field of a charge and a monopole. This angular momentum turns out to be finite and independent of the distance $h$ between the charge and the monopole. The argument then goes that if it's possible to isolate a single fundamental charge $e$ and a single fundamental monopole $g$ in some region of space, then the total angular momentum in that region has to be a multiple of $\hbar$. Here "isolation" means that the distance to any other particle is $\gg h$.

But note that the isolation of the particles is critical. If you put a monopole near a hydrogen atom, the total angular momentum of the electromagnetic field vanishes, because the angular momentum density is of the form $\textbf{E}\times\textbf{B}$, which is bilinear in the fields.

Since quarks are confined, the argument can never be applied to a quark.

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This is a good point. Nonetheless, the argument does apply to QFTs where the number of flavors is high enough that confinement doesn't happen. – user1504 Jun 12 '13 at 14:55
What if you raised the temperature above the Hagedorn temperature? Then quarks would deconfine, and you could transport a single quark around the Dirac string. – tparker Jul 14 at 18:59
Your argument doesn't apply in the case where you move a quark around a monopole in a circle smaller than the confinement radius, or above the Hagedorn radius. See physics.stackexchange.com/questions/268709/… for the complete answer. – tparker Jul 26 at 0:16

The logic is the same: If $q$ is an electric charge and $g$ is a magnetic charge, one must have $gq \in 2\pi\mathbb{Z}$ (in lazy theorist units). So if there is a largest magnetic charge, then there is necessarily a smallest electric charge. You don't have to assume that you are dealing with unit charges to make this argument.

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