The coupling of scalar particles to an electromagnetic field is done using the gauge invariant approach explained in the previous section. Using the minimal coupling, , and writing the Klein-Gordon equation so that all terms involving the electromagnetic potential appear on one side, we have
where the potential operator is
As the Klein-Gordon equation is second order, the coupling term has a quite complicated structure. It contains gradients and moreover is nonlinear in because of the quadratic last term.
Using minimal coupling, the conserved current becomes
Multiplying by to get the usual normalization gives the time and space components of the conserved current
Returning to our initial normalization and using natural units, we have
For a stationary state, , equation 4.72 gives
If , and the charge density has the same sign as of the particle. But if , and the charge density has the opposite sign as of the particle. In this case the field is strong and we would need to invoke field theory to show that particles are created in this case.