The field involved is a complex 4-potential , where is a kind of generalized electric potential and is a generalized magnetic potential. The field transforms like a complex four-vector.
The Lagrangian density is given by:
where is the speed of light in vacuum, is the reduced Planck constant, and is the 4-gradient.
The Euler–Lagrange equation of motion for this case, also called the Proca equation, is:
which is equivalent to the conjunction of
with (in the massive case)
which may be called a generalized Lorenz gauge condition.
When , 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 vector calculus notation, the equations are:
and is the D'Alembert operator.
The Proca action is the gauge-fixed version of the Stueckelberg action via the Higgs mechanism. Quantizing the Proca action requires the use of second class constraints.
If , they are not invariant under the gauge transformations of electromagnetism
where is an arbitrary function.
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