This is a great question, but I'm afraid my answer will be a little
feeble. If a colleague can come up with something better, we will
First, let's just make sure other readers understand what's so
mysterious. Various different particles have all sorts of different
ratios of masses. Those ratios aren't even whole numbers, and there's
no detailed pattern in them. Yet those same particles have very
specific charges: zero, or either plus or minus one electron charge.
What's special about electrical charge that makes it come in these
fixed-size lumps, or charge quanta?
The argument I've heard runs something like this. It starts with an
analogy. In a superconducting loop the quantum wave function of the
superconducting fluid in the loop changes as it goes around the loop in
a way that depends on the amount of magnetic flux in the loop. The wave
must have the same value after once around that it has at the start,
because that's the same actual spot. Only the special quantized values
of the flux let that happen: zero or some integer multiple of some flux
Now what would make electrical charge (equivalent to the electrical
flux leaving a particle) be quantized? The claim is that if there are
magnetically charged particles (magnetic monopoles) they too must have
well-defined quantum states, and that this requirement places a
constraint on electrical fluxes. That constraint leads to the
requirement that electrical charge be quantized.
Since no one has seen a magnetic monopole, that may seem like a pretty indirect argument.
Another incomplete argument comes from the standard electroweak
gauge theory, which requires that quarks and leptons come in
generations in order for what are called "anomalies" to cancel.
Higher-order quantum corrections to the interaction of three gauge
bosons by exchanging a fermion in a triangle-shaped loop cause big, big
problems when computing anything at all in high-energy theories. They
can predict infinite masses for the force carriers (W, Z, and photon).
The only way these can cancel (in the case that the force carrier is
the photon, which interacts with charged particles with a strength
proportional to their charges), is for the sum of all of the charges of
the quarks and leptons within a generation to add up to zero.
For example, up-type quark has charge 2/3, the down-type quark has
charge -1/3, and the electron has charge -1. There are three colors
associated with each of the quarks, and so the anomaly-cancelling
charge relationship is
3(2/3 - 1/3) + (0 - 1) = 0
which is satsified by what we know about quark and lepton charges.
Now the charge on a proton is that of two up-type quarks and one
down-type quark, (2*2/3 - 1/3) = 1, and so the above restriction does
not at all predict that the proton and the electron have equal and
opposite charge, but it does give reasons why the charges on the quarks
and leptons are not completely unrelated.
One observation in nature is that a neutron will decay
spontaneously into an electron and an electron antineutrino and a
proton. What happens at the quark level is that a down-type quark emits
a W- boson, turning into an up-type quark in the process. The W- boson
then couples to the electron and electron antineutrino. This decay
indicates that the difference between an up-quark's charge and a down
quark's charge is the charge on an electron. That, and the anomaly
relation above are enough to show that protons and electrons have
But this still doesn't answer "why", except to say that there's
another piece of experimental evidence to indicate this equality of
charge. More speculative models unifying the strong and electroweak
forces (called "GUT"'s, for "Grand Unified Theories"), propose that the
separate symmetries of the strong and electroweak interactions are
really just pieces of a much larger class of symmetries which have not
yet been observed. Some of these GUT's naturally predict the charge
equality of the electron and proton (with the minus sign), but also
predict other weird stuff that we don't see, like leptoquarks, leaving
many physicists skeptical.
(Sources: unfortunatley, they are rather dense, and are more
suitable to graduate physics students: Perkins: "An Introduction to
High-Energy Physics", and Peskin and Schroeder, "An Introduction to
Quantum Field Theory").
(published on 10/22/2007)