From: jbaez@zermelo.mit.edu (John C. Baez)
Subject: Re: Symmetries, groups, and categories
Organization: MIT Department of Mathematics, Cambridge, MA
Distribution: na
Date: Fri, 14 Aug 92 20:28:26 GMT

In article <1992Aug14.024852.23897@nuscc.nus.sg> matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:
>I agree, keep it up, JB!
>BUT I was very disppointed to see you talking about "general covariance"
>as the symmetry of GR. Come on, you know perfectly well that [for
>example] Maxwellian electrodynamics in Minkowski space is "generally
>covariant".

Indeed, there is an important sense in which this is true, which is too
little known (I had to discover it myself; I've never seen a reference
to it though anyone who thinks about will realize it).  Namely, if one
takes special relativity, together with the vacuum Maxwell's equations
for example, one can think of this as a theory about a 1-form A (the
vector potential) and a Lorentzian metric g.  The laws of nature in
this simple system, in explicitly generally covariant form, are that g is
flat and d*dA = 0.  Here * is the Hodge dual which depends on g.   Given
any diffeomorphism we can transform (mathematicians would say "pull
back") a solution (g,A) to get another solution (g',A').  

So, yes, Maxwellian electrodynamics has the diffeomorphism group as a
symmetry in one sense.   Note however that g plays a funny role in
Maxwellian electrodynamics.  It doesn't wiggle around, it just sits there.
That is, the equation that g is flat has only ONE solution on R^4 up to
coordinate transformations.  It's not what physicists would call a
"dynamical field" in Maxwellian electrodynamics, while it is in GR.

In other words, *if we wish* we can reduce the symmetries of Maxwellian
electrodyamics from the whole diffeomorphism group down to the Poincare
group in a natural manner: *pick* coordinates in which g is given by the
usual metric dt^2 - dx^2 - dy^2 - dz^2 (which we always can do) and
define the Poincare group to be the group of diffeomorphisms preserving
this form.  Because we can do this we just say that Maxwellian
electrodynamics has the group of diffeomorphisms as a "gauge" group of
symmetries, while it has the Poincare group as its "physical" group of
symmetries.   Here I am just following some people in using the words
"gauge" and "physical" in this technical sense.  I don't want to argue
about why the Poincare group is more "physical" than the diffeomorphism
group in any touchy-feely sense.  And I am not using the word "gauge" in
the narrow sense of Yang-Mills theory.

 It has absolutely nothing to do with GR. A realistic GR
>spacetime has no symmetries. I suggest that you banish "general
>covariance" from your mind. 

In GR, as you point out, your garden-variety solution has no special
"physical" symmetries, but the whole theory of GR still has gauge
symmetries.  (You'll note that in my post I took care to mention that
they were only gauge symmetries.)  If one wants an analog of "physical"
symmetries for GR or other topological quantum field theories one needs
to do something like generalize the notion of symmetry to
representations of CATEGORIES, as I described.  

It is hard to banish diffeomorphism-invariance from ones mind in working
with GR.  Agreed, it's not a physical symmetry.  But its presence as a
gauge theory is crucial in, for example, the Arnowitt-Deser-Misner
formulation of the Cauchy problem for GR.  One sees that in such
theories the Hamiltonian serves as a constraint equation H = 0 -- and
many other curious things!


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