Proca action

The field involved is a complex 4-potential ${\displaystyle B^{\mu }=\left({\frac {\phi }{c}},\mathbf {A} \right)}$, where ${\displaystyle \phi }$ is a kind of generalized electric potential and ${\displaystyle \mathbf {A} }$ is a generalized magnetic potential. The field ${\displaystyle B^{\mu }}$ transforms like a complex four-vector.

The Lagrangian density is given by:[2]

${\displaystyle {\mathcal {L}}=-{\frac {1}{2}}(\partial _{\mu }B_{\nu }^{*}-\partial _{\nu }B_{\mu }^{*})(\partial ^{\mu }B^{\nu }-\partial ^{\nu }B^{\mu })+{\frac {m^{2}c^{2}}{\hbar ^{2}}}B_{\nu }^{*}B^{\nu }.}$

where ${\displaystyle c}$ is the speed of light in vacuum, ${\displaystyle \hbar }$ is the reduced Planck constant, and ${\displaystyle \partial _{\mu }}$ is the 4-gradient.

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

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

which is equivalent to the conjunction of[3]

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

with (in the massive case)

${\displaystyle \partial _{\mu }B^{\mu }=0\!}$

which may be called a generalized Lorenz gauge condition. For non-zero sources, with all fundamental constants included, the field equation is:

${\displaystyle c{{\mu }_{0}}{{j}^{\nu }}=\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{\partial }^{\nu }}{{\partial }^{\mu }}\right){{B}_{\mu }}}$

When ${\displaystyle m=0}$, the source free equations reduce to Maxwell's equations without charge or current, and the above reduces to Maxwell's charge equation. This Proca field equation is closely related to the Klein–Gordon equation, because it is second order in space and time.

In the vector calculus notation, the source free equations are:

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

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

and ${\displaystyle \Box }$ is the D'Alembert operator.

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.

If ${\displaystyle m\neq 0}$, they are not invariant under the gauge transformations of electromagnetism

${\displaystyle B^{\mu }\rightarrow B^{\mu }-\partial ^{\mu }f}$

where ${\displaystyle f}$ is an arbitrary function.

• Electromagnetic field
• Photon
• Quantum electrodynamics
• Quantum gravity
• Vector boson
• Relativistic wave equations
• Klein-Gordon equation (spin 0)
• Dirac equation (spin 1/2)

• Supersymmetry Demystified, P. Labelle, 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 0-07-145546 9