Definitions

# 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:

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

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):

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

which is equivalent to the conjunction of

$left\left(partial_mu partial^mu+$
left(frac{mc}{hbar}right)^2right)A_nu=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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