The field can be expressed as tex2html_wrap186

tex2html_wrap187 where L is the lagrangian density. Action should be a lorentz scalar if this equation is to apply in all reference frames. Since tex2html_wrap188 is lorentz invariant, this means tex2html_wrap189 should be lorentz invariant.

  1. Easiest L: tex2html_wrap190 problem: changes with gauge transformations. L should be gauge invariant. tex2html_wrap191 and tex2html_wrap192 gives same electric and magnetic field. More simple expressed, this is: tex2html_wrap193 .
  2. Next L: use field strength tensor tex2html_wrap194 so tex2html_wrap195 so tex2html_wrap196 tex2html_wrap197 so tex2html_wrap198 tex2html_wrap199 . The lorentz gauge gives: tex2html_wrap200 so tex2html_wrap201 . This is just the wave equation.
  3. With a charge or current, tex2html_wrap202 , assume tex2html_wrap203 . Then tex2html_wrap204 so tex2html_wrap205 tex2html_wrap206 which gives: tex2html_wrap207 . Expanding into space and time, you get: tex2html_wrap208 . This equation splits into: tex2html_wrap209 and tex2html_wrap210 . The other 2 maxwell's equations are automatically satisfied using the dual tensor.

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