Proca action

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In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca.

This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.

Lagrangian density

The field involved is the 4-potential A^μ = (φ/c, A), where φ is the electric potential and A is the magnetic potential. The Lagrangian density is given by:

$\mathcal{L}=-\frac{1}{16\pi}(\partial^\mu A^\nu-\partial^\nu A^\mu)(\partial_\mu A_\nu-\partial_\nu A_\mu)+\frac{m^2 c^2}{8\pi \hbar^2}A^\nu A_\nu.$

where c is the speed of light, ħ is the reduced Planck constant, and ∂^μ is the 4-gradient.

Equation

The Euler-Lagrange equation of motion for this case, also called the Proca equation, is:

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

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. When m = 0, the equations reduce to Maxwell's equations without charge or current. The Proca equation is closely related to the Klein-Gordon equation, because it is second order in space and time.

In the more familiar vector calculus notation, the equations are:

$\Box \phi - \frac{\partial }{\partial t} \left(\frac{1}{c^2}\frac{\partial \phi}{\partial t} + \nabla\cdot\mathbf{A}\right) =-\left(\frac{mc}{\hbar}\right)^2\phi \!$

$\Box \mathbf{A} + \nabla \left(\frac{1}{c^2}\frac{\partial \phi}{\partial t} + \nabla\cdot\mathbf{A}\right) =-\left(\frac{mc}{\hbar}\right)^2\mathbf{A}\!$

and $\Box$ is the D'Alembert operator.

Gauge fixing

The Proca action is the gauge-fixed version of the Stueckelberg action via the Higgs mechanism. Quantizing the Proca action requires the use of second class constraints.

They are not invariant under the electromagnetic gauge transformations

$A^\mu \rightarrow A^\mu - \partial^\mu f$

where f is an arbitrary function, except for when m = 0.

Textbooks

• W. Greiner, "Relativistic quantum mechanics", Springer, p. 359, ISBN 3-540-67457-8
• Supersymmetry P. Labelle, Demystified, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
• Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
• Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9

See also

• Maxwell's equations
• Photon
• Vector boson
• Electromagnetic field
• Quantum electrodynamics
• Quantum gravity