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.^[1] 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^[2]

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

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.

References

↑ Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
↑ McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3

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