Two complex fields + interaction, conserved current?

[Spinnor]

Say we have two complex fields, f_1 and f_2 which each are solutions of the D'Alembert operator or the wave operator in three space dimensions. Say we subtract the square of the magnitude of the difference of f_1 and f_2 from the Lagrangian for the fields f_1 and f_2. In that case the Lagrangian is unchanged if both f_1 and f_2 each under go a global phase change of exp(i*theta)? If so does that imply a conserved current?

Assume solutions of the form f_1 = f_2 and f_1 = - f_2 , such waves are mass-less and massive respectively?

Can we demand local phase invariance and get some type of interaction?

Thanks for any help!

 

[Greg Bernhardt]

@Spinnor did you find any more insight on this topic?