Re: Actions, symmetries, and gauge theories

On 2007-02-06, coalquay404 <coalquay404.2lkya2@xxxxxxxxxxxxxxxxx> wrote:

Suppose that we have some theory which is invariant under the action
of
a gauge group, G. Since this theory is a gauge theory, it can be
derived from a singular Lagrangian or Lagrangian density, L. Now
suppose that the action for this theory,

S = \int L(q,\dot{q})

is invariant under the action of some group H. What is the
relationship
between the groups G and H?

H may be larger than G. For instance, translation invariance is usually
not considered a gauge symmetry. But an action functional may well be
translation invariant and invariant under local gauge transformations.
This is in fact the case for electrodynamics.

I guess the reason I'm interested in asking this is to find out
whether
or not the symmetries of an action functional must necessarily be the
same as the symmetries of the resultant equations of motion for the
theory. As an extension, I wonder if the symmetries of an action could
generate a different symmetry group in the equations of motion. It
strikes me that this is an obvious question, but I can't seem to find
any (rigorous) information on it.

A symmetry of the action functional implies a symmetry of the equations
of motion (Noether's theorem). But, a priori, the reverse need not be
true. However, I'm not sure I can provide an example off the top of my
head. Would such an example answer your question, though?

Hope this helps.

Igor

.

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