Let us think for a moment about what MEs might be changed into if
magnetic monopoles were discovered. We would then expect all four
equations to be inhomogeneous:
& = & &rho#rho;_e (GLE)

×- t & = & _e
(AL)

& = & &rho#rho;_m (GLM)

×+ t & = & - _m (FL)
or, in a vacuum (with units of magnetic charge given as ampere-meters,
as opposed to webers, where 1 weber =
ampere-meter):
& = & 1 &rho#rho;_e (GLE)

×- t & = & _e
(AL)

& = & &mu#mu;_0 &rho#rho;_m (GLM)

×+ t & = & -&mu#mu;_0 _m (FL)
(where we note that if we discovered an elementary magnetic monopole of
magnitude
similar to the elementary electric monopolar charge of
we would almost certainly need to introduce additional constants - or
arrangements of the existing ones - to establish its quantized
magnitude relative to those of electric charge in suitable units as is
discussed shortly).

There are two observations we need to make. One is that nature could be
rife with magnetic monopoles already. In fact, every single charged
particle could have a *mix* of both electric and magnetic charge. As
long as the ratio
is a *constant*, we would be unable to tell.

This can be shown by looking at the following *duality
transformation* which ``rotates'' the magnetic field into the electric
field as it rotates the magnetic charge into the electric charge:
& = & ' (&Theta#Theta;) + Z_0 '(&Theta#Theta;)

Z_0& = & Z_0' (&Theta#Theta;) + '(&Theta#Theta;)

Z_0& = & - ' (&Theta#Theta;) + Z_0 '(&Theta#Theta;)

& = & -Z_0' (&Theta#Theta;) + '(&Theta#Theta;)
where
is the impedance of free space (and
has units of ohms), a quantity that (as we shall see) appears frequently
when manipulating MEs.

Note that when the angle
, we have the ordinary MEs we are
used to. However, all of our measurements of *force* would remain
unaltered if we rotated by
and
in the
old system.

However, if we perform such a rotation, we must *also* rotate the
charge distributions in exactly the same way:
Z_0 &rho#rho;_e & = & Z_0 &rho#rho;_e' (&Theta#Theta;) + &rho#rho;_m'(&Theta#Theta;)

&rho#rho;_m & = & - Z_0&rho#rho;_e' (&Theta#Theta;) + &rho#rho;_m'(&Theta#Theta;)

Z_0_e & = & - _e' (&Theta#Theta;) + _m'(&Theta#Theta;)

_m & = & -Z_0_e' (&Theta#Theta;) + _m'(&Theta#Theta;)

It is left as an exercise to show that the monopolar forms of MEs are
left invariant - things come in just the right combinations on both
sides of all equations to accomplish this. In a nutshell, what this
means is that it is merely a matter of convention to call all the charge
of a particle electric. By rotating through an arbitrary angle theta in
the equations above, we can recover an equivalent version of
electrodynamics where electrons and protons have only magnetic charge
and the electric charge is zero everywhere, but *where all forces
and electronic structure remains unchanged* as long as *all
particles have the same
ratio*.

When we search for magnetic monopoles, then, we are really searching for
particles where that ratio is *different* from the dominant one. We
are looking for particles that have zero electric charge and only a
magnetic charge in the current frame relative to
.
Monopolar particles might be expected to be a bit odd for a variety of
reasons - magnetic charge is a pseudoscalar quantity, odd under time
reversal, where electric charge is a scalar quantity, even under time
reversal, for example, field theorists would *really really like*
for there to be at least *one* monopole in the universe.
Nobel-hungry graduate students wouldn't mind if that monopole came
wandering through their monopole trap, either.

However, so far (despite a few false positive results that have proven dubious or at any rate unrepeatable) there is a lack of actual experimental evidence for monopoles. Let's examine just a bit of why the idea of monopoles is exciting to theorists.