Local gauge symmetries Lagrangians and equations of motion

by FunkyDwarf
Tags: equations, gauge, lagrangians, local, symmetries
 P: 1,058 What do you mean about the equation of motion being invariant? for example a non-interacting spin-1/2 field EOM is just the simple Dirac Equation. If you allow for the existence of the spin-1 "photonic" field, coming from the local U(1) gauge symmetry, then the Dirac equation changes (you apply the minimal coupling $p^{\mu} \rightarrow p^{\mu} - q A^{\mu}$). In that sense, since your Lagrangian is invariant, then the EOM are also going to remain invariant (however they won't be the same for the 2 cases I mentioned). In the last case, both a transformed and not transformed Lagrangian are the same. In addition invariances (in general) are mainly to keep the action invariant and not the Lagrangian (for example the last can change up to a total derivative, and yet yield the same EoM). Now if the Lagrangian happens to remain invariant, so does the action.