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I understand gauge theory as the theory of continuous transformation group which keeps Lagrangian (or dynamics) invariant. So some integral invariants could be found. In terms of classical electromagnetism, we choose gauge so that obtain different vector potentials $A$. My question is, since now different potentials doesn't affect dynamics, is that meaningless to choose a gauge? I just have no idea how gauge theory play a role.

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Basically, when one speaks about a gauge theory, what is meant is a theory invariant under the action of a local continuous group, i.e. the group might act differently on distinct points in spacetime. This is unlike a translation, where the whole system is taken to a different position.

The presence of a gauge symmetry implies the presence of a redundancy in the description of the degrees of freedom of the theory. Take for example the Lagrangian of the gauge part of electrodynamics, which is given in terms of the gauge field $A_\mu$ by

$$\mathcal{L}=-\frac12\partial^\mu A^\nu\partial_\mu A_\nu + \frac12\partial^\mu A^\nu\partial_\nu A_\mu + J^\mu A_\mu,$$

contains no time derivative of $A_0$, as can be seen by explicitly writing out the index contractions. As a consequence, there is no canonically conjugate momentum and therefore the field has no dynamics. Therefore, it has to be removed from the theory. One can do this by imposing a gauge condition on the gauge field. The interesting thing is that choosing such a gauge is not unique, there are many ways in which one can restrict the field. The choice now depends on what one wants to do with the theory, which kind of calculations one wants to perform. Common gauges appearing all over the place would be the Coulomb gauge (also called transverse or radiation gauge) $$\nabla\cdot A=0,$$ the Lorenz gauge $$\partial^\mu A_\mu=0,$$ or a family of conditions given by $$v^\mu A_\mu=0,$$

where $v^\mu$ is a constant vector. One refers to temporal gauge if it is timelike, lightcone gauge if it is lightlike and axial gauge if it is spacelike.

To adress your question directly: choosing a gauge is not meaningless, it is important in order to write down a consistent theory free of redundancies.

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  • $\begingroup$ Thank you! Can I say gauge doesn't help understand dynamics? $\endgroup$ – Shuchang Oct 7 '13 at 14:33
  • $\begingroup$ I would not say that. I would rather say that imposing a gauge condition helps reducing the problem to the relevant dynamics. $\endgroup$ – Frederic Brünner Oct 7 '13 at 15:35

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