Is Elitzur's theorem valid only in lattice field theory? Elitzur's theorem, stating that spontaneous breakdown of a gauge symmetry is impossible, was originally proved for a lattice gauge theory.  Is it valid in continuum field theory? Any ref?
 A: Well, you have to specify what you mean by continuum gauge theory. The
only way I know how to regulate gauge theories in the continuum directly
is the perturbative way, which needs gauge fixing, which already breaks 
gauge symmetry. In this kind of contexts what can break is a global
symmetry, not a local one (this is the Higgs mechanism, which is
often sloppily called spontaneous gauge symmetry breaking).
I remember thinking about this question once. :) 
It particularly bothered me, because you would need symmetry breaking 
in an SU(2)-Higgs model in particle physics. And since you can 
define nonperturbatively such QFT by taking a continuum limit 
of the lattice gauge theory, you seem to have a problem, you 
seem to always have no symmetry breaking at all. It is easy
to show by formal manipulations of the path integral that
the Higgs VEV is always zero.
The only solution I could think of in this context was the same 
as for spontaneous symmetry breaking in general. I think the only 
way you can see symmetry breaking is to include an explicit 
breaking term, than do the infinite volume and continuum limit 
first, and than do the symmetry breaking term goes to zero limit. 
It is important that the order of the limits is not 
interchangeable. If you do it the other way you always get zero. 
Unfortunately I have not seen this kind of calculation carried out
anywhere, but this is my best guess.
If anyone has a better understanding of this, I am also very interested.
