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Q & A: why quantized charge?

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Most recent answer: 10/22/2007
Q:
A proton is about 1,842 times the size of an electron. It is composed of completely different sub-atomic particles than the electron. Yet, it has exactly the opposite charge of an electron. Why?
- Anonymous (age 16)
University of Washington
A:
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 update it.

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 quantum.

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.

Mike W.

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 opposite charges.

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").

Tom

(published on 10/22/2007)

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