Editing Proca action

{{Short description|Action of a massive abelian gauge field}}
{{Quantum field theory|cTopic=Equations}}

In [[physics]], specifically [[field theory (physics)|field theory]] and [[particle physics]], the '''Proca action''' describes a [[mass]]ive [[Spin (physics)|spin]]-1 [[quantum field|field]] of mass ''m'' in [[Minkowski spacetime]]. The corresponding equation is a [[relativistic wave equation]] called the '''Proca equation'''.<ref>Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, {{ISBN|978-0-470-03294-7}}</ref> The Proca action and equation are named after Romanian physicist [[Alexandru Proca]].

The Proca equation is involved in the [[Standard Model]] and describes there the three massive [[vector boson]]s, i.e. the Z and W bosons.

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

==Lagrangian density==

The field involved is a complex [[4-potential]] <math> B^\mu = \left (\frac{\phi}{c}, \mathbf{A} \right)</math>, where <math> \phi </math> is a kind of generalized [[electric potential]] and <math> \mathbf{A} </math> is a generalized [[Magnetic vector potential|magnetic potential]]. The field <math>B^\mu</math> transforms like a complex [[four-vector]].

The [[Lagrangian (field theory)|Lagrangian density]] is given by:<ref>W. Greiner, "Relativistic quantum mechanics", Springer, p.&nbsp;359, {{ISBN|3-540-67457-8}}</ref>

:<math>\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.</math>

where <math> c </math> is the [[speed of light|speed of light in vacuum]], <math> \hbar </math> is the [[reduced Planck constant]], and <math> \partial_{\mu}</math> is the [[4-gradient]].

==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]].

==Gauge fixing==

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

If <math>\ m \neq 0\ ,</math> they are not invariant under the gauge transformations of electromagnetism

:<math>\ B^\mu \mapsto B^\mu - \partial^\mu f\ </math>

where <math>\ f\ </math> is an arbitrary function.

==See also==
* [[Electromagnetic field]]
* [[Photon]]
* [[Quantum electrodynamics]]
* [[Quantum gravity]]
* [[Vector boson]]
* [[Relativistic wave equations]]
* [[Klein-Gordon equation]] (spin 0)
* [[Dirac equation]] (spin 1/2)

==References==

{{reflist}}

== Further reading ==

* 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}}

{{Quantum field theories}}

{{DEFAULTSORT:Proca Action}}
[[Category:Gauge theories]]