Dirac's Argument and Charge Quantization in Composite Particles

[stevendaryl]

Dirac gave an argument once upon a time showing that quantization of angular momentum together with the presumed existence of a magnetic monopole implies quantization of charge. If there is, anywhere in the universe, a magnetic monopole of magnetic charge $Q_m$, then the only possible values for the electric charge of a particle are:

$Q_e = \dfrac{n \hbar}{Q_m}$

(or something like that). So there is some smallest charge, $Q_0$, and every other charge must be an integer multiple of that.

My question is how this applies to composite particles such as the proton. Let's assume (contrary to any evidence) that there is a magnetic monopole, so by Dirac's argument, charge is quantized.

If you have a composite particle made up of smaller particles, is it that the total charge must be a multiple of $Q_0$, or must every constituent particle be a multiple of that minimum charge? More specifically, is it possible that the minimum charge is the charge on the electron, rather than the charge on the quark (which is 1/3 or 2/3 the electron's charge).

I'm guessing the latter, but I don't know whether Dirac's argument depends on the field of the electric charges being long-range (the electric field due to quark charge isn't, since quarks always appear in combinations with integral charge).

 

[Meir Achuz]

Unless the quarks in a proton are all at the same point in space, result applies to quarks.

 

[king vitamin]

I agree with clem: all charges are integer multiples of some minimum charge, with the quarks currently being the smallest experimentally known. I don't think confinement arguments change this, since the argument proceeds identically for any U(1) charge coupled to electromagnetism.

 

[stevendaryl]

I found a paper that discusses how Dirac's argument applies to quarks, but the conclusion is more complicated than yes/no.

The correct conclusion, then, if quarks are confined, is not that the minimal magnetic charge is $g_D$, but rather that the monopole carrying magnetic charge $g_D$ must also carry a color-magnetic charge. The color­ magnetic field of the monopole becomes screened by nonperturbative strong-interaction effects at distances greater than $10^{-13}$ cm. We also conclude that there cannot exist both isolated fractional electric charges and monopoles with the Dirac magnetic charge, unless there is some other (as yet unknown) long-range field that couples to both the monopoles and the fractional electric charges.

I don't understand the paper, exactly, but the author does seem to be saying that the existence of the magnetic monopole only implies that there can't be isolated fractional charges. The emphasis on "isolated" is in the original. But his argument introduces the complication of magnetic color charge.

http://www.theory.caltech.edu/~preskill/pubs/preskill-1984-monopoles.pdf

 

[king vitamin]

Wow, interesting paper (as usual from Preskill). If I understand it correctly, the point seems to be that the QCD gauge field also contributes a phase when a quark is transported around a magnetic charge, so that quarks actually introduce a new quantization condition which mixes up fractional electric charges with color charges. So all deconfined particles are integer multiples of the electric charge, but particles with non-neutral color charges can exist in multiples of the quark charge. Similar arguments apply to fractional and integer charged magnetic monopoles.