# Introduction to Gauge Symmetries: Good, Bad or Ugly?

I'm trying to come up with a good (as in intuitive and not 'too wrong') definition of a gauge symmetry.

This is what I have right now:

A dynamical symmetry is a (differentiable) group of transformations that respects system dynamics, ie maps solutions to solutions.

A rigid symmetry is a dynamical symmetry that maps solutions to different solutions. A rigid symmetry has a Noether charge that is only conserved on-shell, ie dependent on the equations of motion.

A gauge symmetry is a dynamical symmetry that maps solutions to identical solutions up to 'parametrization' or 'gauge'; in particular, the solutions correspond to the same initial conditions and physics and only differ in their mathematical description. A gauge symmetry has a Noether charge that is conserved off-shell, ie independent of the equations of motion.

As an example, we take classical mechanics: In general, time dependence of the solutions matter as reparametrization changes velocities. However, in the relativistic case 4-velocities are constrained to 'length' $c$ and dynamics need to be independent of the particular choice of the 'unphysical' 3-velocities.

First of all, is this correct? If so, is there a better choice of wording? Should anything be added?

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Related: physics.stackexchange.com/q/13870/2451 and links therein. –  Qmechanic Dec 10 '12 at 16:17

## 1 Answer

Actually, the second Noether theorem doesn't tell us that the Noether current associated to a local (i.e. gauge) symmetry is conserved off shell, but that it rather vanishes on shell. See the paper of M. Forger and H. Römer, "Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem", Ann. Phys. 309 (2004) 306-389, arXiv:hep-th/0307199 for a thorough discussion on this matter. Nonetheless, it has an equivalent formulation in terms of an off-shell conservation law (then called a "Noether identity") through the map taking infinitesimal local gauge parameters to infinitesimal field variations, but the conserved current in this case is not the canonical Noether current associated to the symmetry. It is only so if the equations os motion hold, in which case it vanishes. In some cases, however, this off-shell conservation law is indeed a covariant conservation law (i.e. with respect to some connection field which is natural to the theory) for an "improved" partial canonical Noether current corresponding to a "pure gauge" sector of the model. This is the case, for instance, of Yang-Mills theories minimally coupled to matter. The corresponding "covariant" conservation law in the matter sector then only holds on shell. Such a structure, however, may not be present in all models which admit local symmetries.

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