Gauge Potential Transformations

[michael879]

Can someone please show me what transformation the lagrangian is invariant under in a theory with just an SU(N) gauge field (with or w/o a source term I don't think it matters). I tried to find this on my own and I got some confusing answers.

Except for U(1), every source I found said that the transformation is of the form [itex]A_\mu \rightarrow UA_\mu U^\dagger - iU\partial_\mu U^\dagger[/itex]. However, this clearly doesn't apply to a U(1) gauge field where the transformation is [itex]A_\mu \rightarrow A + \partial_\mu f[/itex]

*edit* I just realized that they are actually the same, but I can't figure how to delete this thread.

 

[AndyW]

Thank you for your question. In a theory with just an SU(N) gauge field, the Lagrangian is invariant under the following transformation:

A_\mu \rightarrow UA_\mu U^\dagger - iU\partial_\mu U^\dagger

where U is an SU(N) matrix. This transformation is known as a gauge transformation and it is a symmetry of the theory. It can be shown that this transformation leaves the Lagrangian invariant, meaning that the equations of motion and physical observables remain unchanged.

For the case of U(1) gauge field, the transformation is slightly different:

A_\mu \rightarrow A + \partial_\mu f

where f is an arbitrary function. This is still a gauge transformation and it also leaves the Lagrangian invariant.

I hope this helps clarify any confusion you may have had. If you have any further questions, please don't hesitate to ask. Keep up the good work in your research!