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| {{Quantum field theory|cTopic=Equations}}
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| 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]].
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| This article uses the (+−−−) [[metric signature]] and [[tensor index notation]] in the language of [[4-vector]]s.
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| ==Lagrangian density==
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| The field involved is the [[4-potential]] ''A''<sup>μ</sup> = (φ/''c'', '''A'''), where φ is the [[electric potential]] and '''A''' is the [[magnetic potential]]. The [[Lagrangian density]] is given by:
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| :<math>\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.</math>
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| where ''c'' is the [[speed of light]], ''ħ'' is the [[reduced Planck constant]], and ∂<sup>μ</sup> is the [[4-gradient]].
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| ==Equation==
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| The [[Euler–Lagrange equation]] of motion for this case, also called the '''Proca equation''', is:
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| :<math>\partial_\mu(\partial^\mu A^\nu - \partial^\nu A^\mu)+\left(\frac{mc}{\hbar}\right)^2 A^\nu=0</math> | |
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| which is equivalent to the conjunction of<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
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| :<math>\left[\partial_\mu \partial^\mu+ \left(\frac{mc}{\hbar}\right)^2\right]A^\nu=0</math> | |
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| with
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| :<math>\partial_\mu A^\mu=0 \!</math>
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| 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.
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| In the more familiar [[vector calculus]] notation, the equations are:
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| :<math>\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 \!</math>
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| :<math>\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}\!</math>
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| and <math>\Box </math> is the [[D'Alembert operator]].
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| ==Gauge fixing==
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| 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]].
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| They are not invariant under the electromagnetic gauge transformations
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| :<math>A^\mu \rightarrow A^\mu - \partial^\mu f </math>
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| where ''f'' is an arbitrary function, except for when ''m'' = 0.
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| ==References==
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| {{reflist}}
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| == Textbooks ==
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| * W. Greiner, "Relativistic quantum mechanics", Springer, p. 359, ISBN 3-540-67457-8
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| * Supersymmetry P. Labelle, Demystified, McGraw–Hill (USA), 2010, ISBN 978-0-07-163641-4
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| * Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
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| * Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
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| ==See also==
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| * [[Maxwell's equations]]
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| * [[Photon]]
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| * [[Vector boson]]
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| * [[Electromagnetic field]]
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| * [[Quantum electrodynamics]]
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| * [[Quantum gravity]]
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| {{DEFAULTSORT:Proca Action}}
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| [[Category:Quantum field theory]]
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