The Euler-Lagrange equation of motion for this case, also called the Proca equation, is:
which is equivalent to the conjunction of
which is the Lorenz gauge condition. When m = 0, the equations reduce to Maxwell's equations without charge or current. The Proca equation is closely related to the Klein-Gordon equation, because it is second order in space and time.
In the more familiar vector calculus notation, the equations are:
and is the D'Alembert operator.
Read more about this topic: Proca Action
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