From: jbaez@zermelo.mit.edu (John C. Baez)
Subject: Length scales in physics 3 - the classical electron radius
Organization: MIT Department of Mathematics, Cambridge, MA
Distribution: na
Date: Tue, 18 Aug 92 17:54:28 GMT

Another characteristic length scale is the length scale at which 
renormalization becomes really important.  Renormalization is an
aspect of field theory which deals with such issues as the
fact that the electromagnetic field produced by an electron has energy
and thus should be counted as part of the mass of the electron!
The length scale at which these effects become really important is
called the classical electron radius.  It's important to note that it
is really CLASSICAL, not quantum mechanical, because it only
depends on classical electromagneticsm, which doesn't involve hbar, and 
the formula for the rest energy of an electron, which involves c but not hbar. 
Indeed, renormalization was an issue in classical field theory before quantum 
field theory came along.

So the classical electron wavelength should just depend on the mass of
the electron, its charge, and the speed of light.  Recall these have units

m = M
e = L^{3/2}M^{1/2}/T
c = L/T

so to get a length out of these we should form e^2/mc^2.  So, without
doing any real work, we can guess

r_e = e^2/mc^2.

We can derive the classical electron radius by working out the
electric field outside of a ball having charge equal to that
of the electron, e, and radius L, then working out the energy of this
electric field, and then setting that energy equal to the electron mass m.
Solving for L we get a formula for the electron radius r_e.   In other
words, the classical electron radius is the radius the electron would
have to have for all of its mass to be due to the electric field it
produced, assuming it was a charged shell.  Up to miscellaneous factors we get

r_e = e^2/mc^2,

of course; since the actual calculation is not very exciting I'll skip
it.

It's worth noting that the classical electron radius is 1/137 as big
as the Compton wavelength of the electron - the all-important fine
structure constant again!  So we have 3 length scales:

Bohr radius r 
Compton wavelength L_{compton} 
Classical electron radius r_e

each of which is 1/137 as big as the previous one.  The Bohr radius
depends only on hbar, e, and m.   The Compton wavelength depends only
on hbar, c, and m.  The classical electron radius depends only on e,
c, and m.  Nice set-up, huh?  I suppose I should relent and tell you
that this mysterious number 1/137, the fine structure constant, is just 

e^2/hbar c.

It's a dimensionless constant depending only on hbar, e, and c.  In
this respect it's more fundamental than any of the length scales
mentioned, because all the length scales mentioned involve the
electron mass, and one could work them out for particles other than
the electron, whereas

e^2/hbar c

is truly universal, once you remember that the "electron charge" is
nothing specific to the electron but is a basic aspect of
electromagnetism that applies to all charged particles.  (Yes, quarks
apparently have charge 1/3, but that doesn't really affect my point.)  In
other words, the fine structure constant is a dimensionless measure of
how strong the electromagnetic force is, and we have seen that it sets
the ratio of 3 important length scales.
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