Definitions

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{L}=-frac{1}{16pi}(partial^mu A^nu-partial^nu A^mu)(partial_mu A_nu-partial_nu A_mu)+frac{m^2 c^2}{8pi 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)^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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