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:

:$mathcal\{L\}=-frac\{1\}\{16pi\}(partial^mu\; A^\; u-partial^\; u\; A^mu)(partial\_mu\; A\_\; u-partial\_\; u\; A\_mu)+frac\{m^2\; c^2\}\{8pi\; hbar^2\}A^\; u\; A\_\; u.$

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):

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

which is equivalent to the conjunction of

:$left(partial\_mu\; partial^mu+left(frac\{mc\}\{hbar\}\; ight)^2\; ight)A\_\; u=0$

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.