From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Did electric/magnetic symmetry "break"?
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Sun, 6 Sep 92 19:22:39 GMT

In article <5SEP199219330991@zeus.tamu.edu> dwr2560@zeus.tamu.edu (RING, DAVID WAYNE) writes:
>carlip@landau.ucdavis.edu (Steve Carlip) writes...
>>The electromagenetic symmetry you are asking about is called duality.  
>>In the absence of charged matter, you can freely rotate the electric and
>>magnetic fields into each other.  Charged sources provide an _explicit_
>>symmetry-breaking, by uniquely picking out one "direction" to couple
>>to.  This isn't the kind of broken symmetry you're asking about, I
>>think; for instance, there's no phase transition between a symmetric
>>and a nonsymmetric ground state. 
>
>'rotate' and 'direction' suggest a continuous symmetry. But taking the dual
>is discrete. Is that right?
>
>Dave Ring
>dwr2560@zeus.tamu.edu
>

I just gave a bit of the spiel about duality last time.  The 
discrete symmetry

F -> *F

can be embedded in a continuous symmetry group - just the circle
group, U(1) - as follows:

F -> (cos t)F + (sin t)*F

Here t is any old real number and has nothing to do with time.  (The
problem with talking to physicists is that to them letters all have
pre-assigned meanings!)   In fact this stuff is old hat; it is what
you're doing when you make up a complex electromagnetic field E + iB
and then multiply it by a phase exp(it)!  (Remember your old electromagnetism
course?)  The reason for dressing it up in fancy new language by using
the electromagnetic field strength two-form F and the Hodge star
operator * is just to prove that I'm a mathematician.  :-)  No, seriously,
the point is to show that duality symmetry doesn't depend on a choice
of coordinate system, and in fact works on any curved four-dimensional spacetime. 
I.e., it's not just a hack.

So one can make up a *complex* two-form, let's call it G, by setting

G = F + i*F

so that the vacuum Maxwell's equations just become

dG = 0

and duality symmetry as described above is encoded in the fact that
this equation is invariant under 

G -> exp(it)G.

Now one can imagine introducing charges and currents by setting

dG = *J

where 

K = J + iM

Here J is the electic charge/current one-form and M is the magnetic
charge/current one-form (which seems to be zero in real life).  (Note
that I said d*F = J in a previous post but I meant d*F = *J, up to a
sign that I am too lazy to remember!)

What Steve was referring to was that the absence of magnetic monopoles
just amounts to saying that K has no imaginary part.  However, due to
duality symmetry this is sort of a matter of convention: if there were
no electic monopoles and ONLY magnetic monopoles, the physics would be
the same.  As long as the current K is of the form exp(it)J for some
real one-form J, we can use duality symmetry to rotate things so that
K is real!  I.e., we rotate the complex plane until a given line through
the origin gets sent to the real axis.

Duality symmetry can be souped up.  The person who has played it most
definitively seems to be Kostant, who taught a course on Maxwell's
equations at MIT in which he used his considerable group-theoretic
prowess to uncover a grand SO(3,3) symmetry lurking around somewhere,
if I recall correctly.  More recently Guillemin and Sternberg wrote a
fun book that covers some of this stuff, if I recall correctly.  I
forget the title.

By the way, we just had an earthquake here at Riverside, but
everything's still standing.  I am posting from MIT because the