|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.
Lagrangian density 
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.
Gauge fixing 
They are not invariant under the electromagnetic gauge transformations
where f is an arbitrary function, except for when m = 0.
- W. Greiner, "Relativistic quantum mechanics", Springer, p. 359, ISBN 3-540-67457-8
- Supersymmetry P. Labelle, Demystified, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
- Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
- Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
See also 
- Maxwell's equations
- Vector boson
- Electromagnetic field
- Quantum electrodynamics
- Quantum gravity
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