# Proca action

Proca action

In physics, in the area of field theory, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The field involved is a real vector field A. The Lagrangian density is given by:

:$mathcal\left\{L\right\}=-frac\left\{1\right\}\left\{16pi\right\}\left(partial^mu A^ u-partial^ u A^mu\right)\left(partial_mu A_ u-partial_ u A_mu\right)+frac\left\{m^2 c^2\right\}\left\{8pi hbar^2\right\}A^ u A_ u.$

The above presumes the metric signature (+---). Here, "c" is the speed of light and $hbar$ is Dirac's constant. In the dimensionless units commonly employed in theoretical physics, these may both be taken to be one. The Euler-Lagrange equation of motion is (this is also called the Proca equation):

:$partial_mu\left(partial^mu A^ u - partial^ u A^mu\right)+left\left(frac\left\{mc\right\}\left\{hbar\right\} ight\right)^2 A^ u=0$

which is equivalent to the conjunction of

:$left\left(partial_mu partial^mu+left\left(frac\left\{mc\right\}\left\{hbar\right\} ight\right)^2 ight\right)A_ u=0$

with

:$partial_mu A^mu=0 !$

which is the Lorenz gauge condition. The Proca equation is closely related to the Klein-Gordon equation.

The Proca action is the gauge-fixed version of the Stückelberg action via the Higgs mechanism.

Quantizing the Proca action requires the use of second class constraints.

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