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| == Zenith Port Royal Rectangle 02.0251.684-22.M251 Black Dial a == | | {{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]]. |
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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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| | :<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> |
| | :<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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| | :<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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| Guide Written content:
| | ==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 |
| | * Supersymmetry P. Labelle, Demystified, 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(10-) 0-07-145546 9 |
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| | ==See also== |
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| == Relief For Ril & Ongc Govt., Likely To Approve Higher Gas Pr ==
| | * [[Maxwell's equations]] |
| | * [[Photon]] |
| | * [[Vector boson]] |
| | * [[Electromagnetic field]] |
| | * [[Quantum electrodynamics]] |
| | * [[Quantum gravity]] |
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| Ease To have Ril & Ongc Federal government., Quite likely going to Authorize High Fumes Valuations
| | {{DEFAULTSORT:Proca Action}} |
| | | [[Category:Quantum field theory]] |
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In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation.[1] The Proca action and equation are named after Romanian physicist Alexandru Proca.
This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.
Lagrangian density
The field involved is the 4-potential Aμ = (φ/c, A), where φ is the electric potential and A is the magnetic potential. The Lagrangian density is given by:
where c is the speed of light, ħ is the reduced Planck constant, and ∂μ is the 4-gradient.
Equation
The Euler–Lagrange equation of motion for this case, also called the Proca equation, is:
which is equivalent to the conjunction of[2]
with
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.
In the more familiar vector calculus notation, the equations are:
and 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.
They are not invariant under the electromagnetic gauge transformations
where f is an arbitrary function, except for when m = 0.
References
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Textbooks
- W. Greiner, "Relativistic quantum mechanics", Springer, p. 359, ISBN 3-540-67457-8
- Supersymmetry P. Labelle, Demystified, 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(10-) 0-07-145546 9
See also
- ↑ Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
- ↑ McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3