Proof that the E.M Field is invariant under guage transformation.

[hob]

To prove:

F \overline{} \mu\nu = \nabla\overline{} \muA \overline{} \nu - \nabla\overline{} \nuA \overline{} \mu

is invariant under the gauge transformation:

A \overline{} \mu \rightarrow A \overline{} \mu + \nabla\overline{} \mu\LambdaI end up with:

F \overline{} \mu\nu = F \overline{} \mu\nu + [\nabla\overline{} \mu,\nabla\overline{} \nu]\Lambda

Which I guess is invariant provided \nabla\overline{} \mu & \nabla\overline{} \nu commute?

Do they commute? and if so why?

Many thanks.

 

[xepma]

Yes, they commute. In the case of normal minkowski space time and the Abelian gauge group U(1) the differential operators reduce to ordinary derivatives.