# Local phase invariance of our lagrangian

So in lectures we have just started looking at classical field theory. We were introduced to symmetries and we were told in general our lagrangian wont be invariance wrt to local phase changes but if we require it to be we can introduce the covariant derivative. This introduces a gauge field that acts to restore local phase invariance.

My question is what is the physical motivation for wanting our lagrangian to be invariant wrt to local phase changes? What are some cases where we want local $$U(1)$$ invariance (& why) and some cases where we dont (again & why)?

• In quantum mechanics, the overall phase change does not affect the probabilty amplitude. So it is sensible to perform the phase transform and at the same time, keep the Lagrangian invariant. At the same time, if a phase transformation is induced by some physical process, it is meaningful to require the local invariance under the local phase transformation. In some sense, the phase transformation at one spacetime point cannot be immediately transported to somewhere else. Some quantum numbers, such as the baryon number, correspond to the global U(1) since they are constant no matter where. Nov 15 '19 at 2:17

If you use Noether's theorem, there is a locally conserved current associated with local $$U(1)$$ symmetry. We can then identify this current as electric current. Without the symmetry, we wouldn't have a locally conserved current to use for electric current, so it's pretty important. As an additional note, we can find Lagrangians that have $$SU(2)$$ or $$SU(3)$$ local gauge invariance. When we do this, we find that the Lagrangian for $$SU(3)$$ describes the behavior of the strong interaction between quarks.