I am working with a lagrangian on a homework problem. I expect it to have some gauge invariance. I can show that the Lagranian is invariant under those (gauge) tansformations but I have to use equations of motions to prove so. In typical textbook examples, I have seen gauge invariance established without using equations of motion. This lead to me this question:
Is gauge invariance supposed to hold off-shell as well ? Or is it okay to have gauge invariance that holds only on-shell?
By definition, a symmetry of an action has to hold without using equations of motion, so naviely I should not be allowed to use EOMs to establish gauge invariance. But on a second thought, gauge invariance refers to those transformations in phase space which map physical configurations to physical configurations. So I should be allowed to use EOMs. These two viewpoints are making me confused.
