# Gauge fixing and gauge transformations

#### "Don't panic!"

If a theory is gauge invariant and one chooses to fix a particular gauge, having done this is it then possible to make a gauge transformation from this chosen gauge to another gauge, or have we already "spent" the gauge symmetry?
Apologies if this is a really basic question, but I've got myself a bit confused in a particular example in reading a paper on the brane origins of Galilean invariance of galileons http://arxiv.org/pdf/1008.1305v2.pdf (starting on page 9). They carry out this procedure of fixing a particular gauge and then subsequently making a gauge transformation, but I'm slightly confused as to why they can do this?

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#### haushofer

Gaugefixing often doesn't completely fix the gauge. You should check if the next gauge transformation they perform leaves the gauge choice invariant.

#### "Don't panic!"

Gaugefixing often doesn't completely fix the gauge. You should check if the next gauge transformation they perform leaves the gauge choice invariant.
They do a Poincare transformation which breaks the gauge choice and so then use a gauge transformation to "fix" it such that the gauge choice is invariant.

Is it the case that if one simply chooses a gauge (thus fixes a gauge), then one can still transform to other gauges? Is that what you mean by "Gauge fixing often doesn't completely fix the gauge."?

(here is the paper I've been reading by the way: http://arxiv.org/pdf/1008.1305v2.pdf the discussion I'm referring to starts on page 9).

#### haushofer

Ah, ok, i see. They use a compensating gauge transformation!

One has two kinds of gauge transfo's for branes: Poincare in the targetspacetime, and gct's on the worldvolume. This means that if you make a gauge transfo which takes you out of your gauge choice for the embedding coordinate X, you can make a compensating worldsheet transfo to bring you back into the gauge choice. The X is a scalar under these worldvolume gct's! See e.g. green, schwarz, witten chapter 2.3, eqn. 2.3.19.

The same kind of thing is apparent in string theory, where you can compensate some gct's on the worldsheet, which changes the Minkowski-choice of gauge, by a Weyl rescaling on the worldsheet. this is the reason why string theory is a 2-dim. CFT.

#### haushofer

So forget what I said before, it is not relevant here.

#### "Don't panic!"

One has two kinds of gauge transfo's for branes: Poincare in the targetspacetime, and gct's on the worldvolume. This means that if you make a gauge transfo which takes you out of your gauge choice for the embedding coordinate X, you can make a compensating worldsheet transfo to bring you back into the gauge choice. The X is a scalar under these worldvolume gct's! See e.g. green, schwarz, witten chapter 2.3, eqn. 2.3.19.
So is the idea that one picks a gauge that specifies the embedding in the bulk, and then performs a Poincare transformation on the bulk which breaks this gauge choice, however one can simply re-parameterize the brane which amounts to performing a gauge transformation of the worldvolume coordinates on the brane?

#### haushofer

Yes. You have two different kinds of gauge transfo's, which gives you the freedom to still to a particular combination of them without leaving your gauge choice. This is not trivial, of course; you have to check explicitly that this is possible! The different parameter will then be related, like in your paper.

In the gsw section i recommended you one choses the lightcone gauge, but then you have to check the Lorentz algebra again, this time with th compensating transfo's included. The algebra should still close.

#### "Don't panic!"

You have two different kinds of gauge transfo's
Are they choice of how you embed the brane in the bulk and choice of how you parametrize the coordinates on the brane?

Also, in general (for gauge theories), once one has fixed a gauge can one then subsequently perform a gauge transformation to another gauge as long as the new gauge satisfies the gauge fixing conditions that one specified in fixing the original gauge?

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#### haushofer

Yes, and yes. Once you chose a gaugecondition, you're still free to gauge whatever you want as long as those gauge conditions are still satisfied. As i said, choosing a gauge often doesn't completely eliminate all the gauge freedom you have.

#### "Don't panic!"

Ok, I think I'm starting to understand it a little better now. Thanks for your help.

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