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Consider a electric charge at the origin and an monopolar charge
at an arbitrary point on the axis. From the generalized form of
MEs, we expect the electric field to be given by the wellknown:

(8.125) 
at an arbitrary point in space. Similarly, we expect the magnetic field
of the monopolar charge to be:

(8.126) 
where
.
The momentum density of this pair of fields is given as noted above
by:

(8.127) 
and if one draws pictures and uses one's right hand to determine
directions, it is clear that the field momentum is directed around
the axis in the right handed sense. In fact the momentum follows
circular tracks around this axis in such a way that the field has a
nonzero static angular momentum.
The system obviously has zero total momentum from symmetry. This means
one can use any origin to compute the angular momentum. To do so, we
compute the angular momentum density as:

(8.128) 
and integrate it:
over all space. Using the vector identity:

(8.130) 
this can be transformed into:

(8.131) 
Integrating by parts:

(8.132) 
The surface term vanishes from symmetry because
is radially away
from the origin and averages to zero on a large sphere.
Thus we finally obtain:

(8.133) 
There are a variety of arguments that one can invent that leads to an
important conclusion. The arguments differ in details and in small ways
quantitatively, and some are more elegant than this one. But this one
is adequate to make the point. If we require that this field angular
momentum be quantized in units of :

(8.134) 
we can conclude that the product of must be quantized. This
is an important conclusion! It is one of the few approaches in physics
that can give us insight as to why charge is quantized.
This conclusion was originally arrived at by (who else?) Dirac.
However, Dirac's argument was more subtle. He created a monopole as a
defect by constructing a vector potential that led to a monopolar
field everywhere in space but which was singular on a single line.
The model for this vector potential was that of an infinitely long
solenoid stretching in from infinity along the axis. This solenoid
was in fact a string  this was in a sense the first quantum
string theory.
The differential vector potential of a differential magnetic
dipole
is:

(8.135) 
so

(8.136) 
This can actually be evaluated in coordinates for specific lines ,
e.g. a line from to the origin along the axis (to put a
``monopole'') at the origin. If one takes the curl of this vector
potential one does indeed get a field of:

(8.137) 
everywhere but on the line , where the field is singular. If
we subtract away this singular (but highly confined  the field is
``inside'' the solenoid where it carries flux in from ) we are
left with the true field of a monopole everywhere but on this line.
Dirac insisted that an electron near this monopole would have to not
``see'' the singular string, which imposed a condition on its
wavefunction. This condition (which leads to the same general
conclusion as the much simpler argument given above) is beyond the scope
of this course, but it is an interesting one and is much closer to the
real arguments used by field theorists wishing to accomplish the same
thing with a gauge transoformation and I encourage you to read it in
e.g. Jackson or elsewhere.
Next: Plane Waves
Up: Magnetic Monopoles
Previous: Magnetic Monopoles
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Robert G. Brown
20150823