- Proca action
In

physics , in the area of field theory, the**Proca action**describes amass ive spin-1 field of mass m inMinkowski spacetime . The field involved is a realvector field **A**. TheLagrangian 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 thespeed of light and $hbar$ isDirac's constant . In thedimensionless units commonly employed in theoretical physics, these may both be taken to be one. TheEuler-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 theKlein-Gordon equation .The Proca action is the gauge-fixed version of the

Stückelberg action via theHiggs mechanism .Quantizing the Proca action requires the use of

second class constraints .

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