A complex classical field Φ of particles is, by itself, invariant under global phase changes but not under local phase changes. It is made gauge invariant by coupling it with the EM potential, A, by substituting the covariant derivative for the normal partial derivative in the Lagrangian. But if the particles represented by Φ have zero electrical charge, the covariant derivative is the same as the normal derivative. Does that mean that a classical field of uncharged particles cannot be made gauge invariant? Thanks in advance.