Re: EPSILON_ZERO, OK ?

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Concerning:
> I think that is the same thing, isn't it David? Your interpretation
> (and perhaps the way that Dirac stated it originally) is essentially
> the Bohr-Stoner quantization criterion, another guise of the law of
> quantization of angular momentum.

Like Jack, I too had Dirac's 1931 paper in mind.  Although I went from
memory/memorism I didn't recall the angular momentum of the EM field
playing a crucial role in the quantization condition.  Rather, the
single-valuedness of the wave function of an electrically charged
particle's wave function in the presence of a magnetic monopole was
required where the monopole was formally considered as at the end of an
infinitely long and narrow solenoid which carried the magnetic flux back
to the monopole from 'infinity'.  Since the solenoid's location in space
was to be completely unobservable to the charged particle (where
changing the path of the solenoid was accomplished by a particular
version of a gauge transformation), this would only be the case if the
product of the magnetic and electrical charges obeyed the Dirac
quantization condition.  Otherwise, gauge invariance & single-valuedness
of the wave function could not be preserved.

Leigh, since you brought up the issue of EM angular momentum I thought
I might have forgotten something so I looked up the discussion of
monopoles in Jackson's E&M book (1975 version).  In there I discovered
that, indeed, there are arguments involving the angular momentum of the
EM field (but they are due to others than Dirac).  jackson gives one due
to Goldhaber involving the net change in orbital angular momentum of an
electrically charged particle moving past a monopole.  He also points out
a calculation by Thomson of the total angular momentum of the EM field
due to two particles at rest where one is electrically charged and the
other one is magnetically charged.  Jackson goes on to say that Saha &
Wilson used the result of this calculation to obtain the Dirac
quantization condition via a semiclassical analysis a la Bohr-Sommerfeld
by equating this EM field angular momentum to an appropriate multiple of
h-bar.  So apparently it seems, Leigh, that you had remembered this other
method of derivation of the quantization condition, and I had remembered
Dirac's method.

> As I recall one integrates ExB dV
> for a monopole and a (quantized) electric charge over all space and
> one obtains a total angular momentum in the field.  ...

Actually, ExB is proportional to the EM momentum density--not the
angular momentum density.  The angular momentum would be found from an
integration of the triple vector product: rx(ExB) over space.

> I haven't performed the calculation in all my years of teaching, so
> I'm not at all clear on the details, but I seem to remember reading
> a Dirac paper in the sixties in Science in which he coined the term
> "dyon" for the magnetic monopole.

As I recall (again, my memory isn't all that clear here either), a
'dyon' is supposed to be a particle that has *both* an electric charge
*and* a magnetic charge in the same particle, rather than just an
electrically uncharged magnetic monopole.

David Bowman
David_Bowman@georgetowncollege.edu