Proca action

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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}{16\pi}(\partial^\mu A^\nu-\partial^\nu A^\mu)(\partial_\mu A_\nu-\partial_\nu A_\mu)+\frac{m^2 c^2}{8\pi \hbar^2}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(\partial^\mu A^\nu - \partial^\nu A^\mu)+ \left(\frac{mc}{\hbar}\right)^2 A^\nu=0

which is equivalent to the conjunction of

\left(\partial_\mu \partial^\mu+ \left(\frac{mc}{\hbar}\right)^2\right)A_\nu=0

with

\partial_\mu A^\mu=0 \!

which is the Lorenz gauge condition.

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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