In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime.
The Proca equation is involved in the Standard Model and describes there the three massive vector bosons, i.e. the Z and W bosons.
This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.
is a kind of generalized electric potential and
(the massless case) reduces to which may be called a generalized Lorenz gauge condition.
For non-zero sources, with all fundamental constants included, the field equation is: When
the source free equations reduce to Maxwell's equations without charge or current, and the above reduces to Maxwell's charge equation.
This Proca field equation is closely related to the Klein–Gordon equation, because it is second order in space and time.
In the vector calculus notation, the source free equations are: and
Quantizing the Proca action requires the use of second class constraints.
they are not invariant under the gauge transformations of electromagnetism where