Proca action

Equation

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

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

which is conjugate equivalent to

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

and for m ≠ 0 implies

${\displaystyle {\mathrm{\partial }}_{\nu }{B}^{\nu }=0,}$

equivalent to a generalized Lorenz gauge condition. For the massive case however, this is a physical constraint rather than an optional gauge condition. For non-zero sources, with all fundamental constants included, the field equation is:

${\displaystyle c{\mu }_0{j}^{\nu }=\left[g^{\mu \nu }\left({\mathrm{\partial }}_{\sigma }{\mathrm{\partial }}^{\sigma }+\frac{m^2{c}^2}{{\hslash }^2}\right)-{\mathrm{\partial }}^{\nu }{\mathrm{\partial }}^{\mu }\right]B_{\mu }}$

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:

${\displaystyle ◻\varphi -\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\frac{1}{c^2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{A}\right)=-{\left(\frac{mc}{\hslash }\right)}^2\varphi }$

${\displaystyle ◻\mathbf{A}+\mathrm{\nabla }\left(\frac{1}{c^2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{A}\right)=-{\left(\frac{mc}{\hslash }\right)}^2\mathbf{A}}$

and ${\displaystyle ◻}$ 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.

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

${\displaystyle B^{\mu }\mapsto B^{\mu }-{\mathrm{\partial }}^{\mu }f}$

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