Editing Proca action (section)

==Equation==

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

:<math>\partial_\mu \Bigl(\ \partial^\mu B^\nu - \partial^\nu B^\mu\ \Bigr) + \left( \frac{\ m\ c\ }{\hbar} \right)^2 B^\nu = 0</math>

which is conjugate equivalent to<ref>{{cite encyclopedia |editor-first=C.B. |editor-last=Parker |year=1994 |title=conjugate equivalence |encyclopedia=McGraw Hill Encyclopaedia of Physics |edition=2nd |publisher=McGraw Hill |place=New York, NY |ISBN=0-07-051400-3 }}</ref>

:<math>\left[\ \partial_\mu \partial^\mu + \left( \frac{\ m\ c\ }{ \hbar } \right)^2\ \right]B^\nu = 0 </math>

and with <math>\ m = 0\ </math> (the massless case) reduces to

:<math>\ \partial_\nu B^\nu = 0\ ,</math>

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

:<math>c\ \mu_0\ j^\nu \;=\; \left[\ g^{\mu \nu } \left( \partial_\sigma \partial^\sigma + \frac{\ m^2\ c^2\ }{\ \hbar^2 } \right) - \partial^\nu \partial^\mu\ \right] B_\mu\ </math>

When <math>\ m = 0\ ,</math> 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:

:<math>\ \Box\ \phi - \frac{\ \partial }{\partial t} \left(\frac{ 1 }{\ c^2} \frac{\ \partial \phi\ }{ \partial t } + \nabla\cdot\mathbf{A} \right) ~=~ -\left(\frac{\ m\ c\ }{\hbar}\right)^2 \phi\ </math>
:<math>\ \Box\ \mathbf{A} + \nabla \left( \frac{ 1 }{\ c^2 }\ \frac{\ \partial \phi\ }{\partial t} + \nabla \cdot \mathbf{A} \right) ~=~ -\left(\frac{\ m\ c\ }{ \hbar }\right)^2 \mathbf{A}\ </math>

and <math>\ \Box\ </math> is the [[D'Alembert operator]].