# Theoretical reasons for charge quantization

I'm aware of Millikan's oil drop experiment and I've read that quarks have fractional eletric charge, but I was wondering if there's any theoretical argument that makes us believe charge is quantized.

I've also read about Dirac's famous work in which he demonstrated that if a magnetic monopole exists then electric and magnetic charge have to be quantized.

But if it doesn't exist? Is there any other theoretical reason for the charge to be quantized or is that the only way we know?

I know of two related arguments, one of which you allude to.

If a magnetic monopole exists, Dirac showed that if it is to be compatible with quantum mechanics, charge must be quantised.

The second related argument is technical. If all forces - strong, weak and electromagnetic - are unified at a high energy, charge must be quantised, because of the algebra of the symmetry group describing the unified force.

The relation between the arguments by Polyakov and 't Hooft is also technical. I can say simply that monopoles must result anyhow when the unified force is spontaneously broken to our three known forces.

• I have downvoted this answer, because not all guts predict charge quantization. To quote 't Hooft: "Only if the underlying gauge group is compact, and has a compact covering group, must electric charges in the $U(1)$ gauge groups be quantized [...], and whenever the covering group of the underlying gauge group is compact, magnetic monopole solutions can be constructed." So, if charge is quantized according to a GUT, then this goes hand in hand with magnetic monopoles (solitons). Commented Feb 5, 2014 at 18:32
• @Hunter, that is all true, of course, but I was typing out a quick answer on my tablet :) Commented Feb 6, 2014 at 10:34
• Fair enough, if you edit your answer, then I'll change my vote. Commented Feb 6, 2014 at 12:54

Charge quantization means that, in contrast to standard electromagnetism, Gauss law cannot return any real charge inside a closed surface, but only integer multiplicity of some fundamental charge (e or e/3).

So the question is how to repair electromagnetism to restrict Gauss law to integer values. The only way I have heard of is using topological analogue of Gauss law: Gauss-Bonnet theorem which says that integration of curvature over a closed surface, returns topological charge inside it - charge which has to be integer.

So defining EM field as curvature of some deeper field, and using standard Lagrangian for it, we can recreate electromagnetism with included charge quantization: Gauss law restricted to integer values. Additionally, it also repairs the problem of infinite energy of such charge, e.g. as electron - some article.