|Quantum field theory|
In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca.
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.
They are not invariant under the electromagnetic gauge transformations
where f is an arbitrary function, except for when m = 0.
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