Is a magnetic monopole really necessary for charge quantization?

Question

The usual remark that goes about in a first encounter of Dirac monopoles is that it solves the problem of electric charge quantization. I have also studied t'Hooft Polyakov monopoles which asymptotically quantize the charge. Basically they prove that $ e g \in k\mathbb Z$ where $k$ is a dimensionless number, $e$ is the electric charge and $g$ is the magnetic charge.

First of all, what do we mean by saying that charge should be quantized? Are we trying to say that all electrical charges we find in nature are integer multiples of the electronic charge ($e=1.6\times10^{-19}C$)? In that case, isn't the obvious explanation the fact that everything is made up of electrons and protons? Why take the trouble to invent monopoles to explain this matter of fact?

Perhaps we are trying to explain why the electronic charge is that particular number. Surely, the $U(1)$ charge can be any real number. But I do not see how quantization of the electrical charge helps in explaining this number anyway. Instead, it raises more questions. If $U(1)$ charge is quantized, which is to say that several integral multiples of a charge quantum are allowed to exist, where are all the other elementary particles with all the integral multiple charges that are allowed? How many times the charge quantum is the electronic charge? What particle has the minimum allowed charge? If that particle happens to be the electron (and why is that?), and you are going to explain charges of other composites of electrons in terms of the electronic charge, why bother invent the magnetic monopole machinery in the first place?

Answer 1

1. Yes, "quantization of charge" means that all charges are a multiple of some fundamental charge unit $e$.

2. Of course, everything being made up of electrons and protons with a fixed charge explains quantization. But it doesn't explain why there are only electrons and protons, or why the charge of the proton is a multiple of that of the electron. What Dirac quantization explains a priori, i.e. without any further experimental input about the number of charged fundamental particles, is that all charged particles must have a charge that is a multiple of $e$, regardless of whether they are fundamental or composite. This is different from the indeed rather trivial observation that composite particles made of two charged fundamental particles with charges $e^+ = -e^-$ are charged as multiples of $e^+$.

3. That other fundamental particles with other integral multiples of $e$ may exist doesn't mean they must exist. Dirac quantization only says that if other charged particles exist, they must be quantized in terms of the fundamental unit $e$. Note that, since we now know of the existence of quarks, the fundamental charge unit of our universe would be one third of the electron charge. Note also that the electron is not composite, so your explanation for the charges of atoms because they're composites of electron and proton utterly fails to explain why the electron charge is an integral multiple of the quark charge (as it does with the electron and the proton charge, but perhaps two equal fundamental charges could be seen as natural, while a 1:3 relationship certainly demands explanation for why it could not be irrational).

As an aside, Dirac quantization is not a useful argument if you already know the gauge group of electromagnetism is $\mathrm{U}(1)$ - the representations of $\mathrm{U}(1)$ are classified by integers, and their charges are integral multiples of the fundamental representation's charge. But, classically, it is impossible to decide whether the gauge group of electromagnetism is $\mathbb{R}$ or $\mathrm{U}(1)$. Dirac's argument in modern language essentially shows that if the Aharonov-Bohm effect exists and if magnetic monopoles exist, then the gauge group of quantum electromagnetism must be $\mathrm{U}(1)$, not $\mathbb{R}$, if it is to be consistent.