Maxwell's Equation's From Klein Gordon

[bhobba]

This is a follow on from the following thread where I put a little 'challenge' to the OP.
https://www.physicsforums.com/threads/when-can-klein-gordon-equation-be-used-for-photon.906767/

It probably didn't really interest him so he didn't do it, but I thought I would post it anyway - its quite interesting - you wouldn't think it would be this easy.

First you need to read the following to see how you get the source free Maxwell's equations from the Klein Gordon:
http://cds.cern.ch/record/944002/files/0604169.pdf

I will work in units c=1

First you see what happens if the E field is no longer source free by defining p =∇.E. Call p the charge density and reasonably assume whatever it is, is conserved so the continuity equation δp/δt + ∇.J = 0 applies. Reasonably we call J the current density. Let U = ∂E/∂t + J. Then ∇.U = 0 ie you can find a B' ∇XB' = ∂E/∂t + J. Of course this holds even when J=0 so ∇XB' = ∇XB and ∇XB = ∂E/∂t + J.

You then have the 4 Maxwell equations.

Does anyone want to see how you then get the Lorentz Force Law? Its from writing out the Lagrangian but will be happy to post the outline.

Thanks
Bill

 

[hilbert2]

This can't be used to show that there's no magnetic monopoles, right? Isn't that just an empirical observation?

 

[bhobba]

This can't be used to show that there's no magnetic monopoles, right? Isn't that just an empirical observation?

Correct. That's why I only gave a source to the E field.

Thanks
Bill