Action of a massive abelian gauge field
Inphysics , specificallyfield theory andparticle physics , theProca action describes amassive spin -1field of massm inMinkowski spacetime . The corresponding equation is arelativistic wave equation called theProca equation .[ 1] Alexandru Proca .
The Proca equation is involved in theStandard Model and describes there the three massivevector bosons , i.e. the Z and W bosons.
This article uses the (+−−−)metric signature andtensor index notation in the language of4-vectors .
The field involved is a complex4-potential B μ = ( ϕ c , A ) {\displaystyle B^{\mu }=\left({\frac {\phi }{c}},\mathbf {A} \right)} ϕ {\displaystyle \phi } electric potential andA {\displaystyle \mathbf {A} } magnetic potential . The fieldB μ {\displaystyle B^{\mu }} four-vector .
TheLagrangian density is given by:[ 2]
L = − 1 2 ( ∂ μ B ν ∗ − ∂ ν B μ ∗ ) ( ∂ μ B ν − ∂ ν B μ ) + m 2 c 2 ℏ 2 B ν ∗ B ν . {\displaystyle {\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 }.} wherec {\displaystyle c} speed of light in vacuum ,ℏ {\displaystyle \hbar } reduced Planck constant , and∂ μ {\displaystyle \partial _{\mu }} 4-gradient .
TheEuler–Lagrange equation of motion for this case, also called theProca equation , is:
∂ μ ( ∂ μ B ν − ∂ ν B μ ) + ( m c ℏ ) 2 B ν = 0 {\displaystyle \partial _{\mu }{\Bigl (}\ \partial ^{\mu }B^{\nu }-\partial ^{\nu }B^{\mu }\ {\Bigr )}+\left({\frac {\ m\ c\ }{\hbar }}\right)^{2}B^{\nu }=0} which is conjugate equivalent to[ 3]
[ ∂ μ ∂ μ + ( m c ℏ ) 2 ] B ν = 0 {\displaystyle \left[\ \partial _{\mu }\partial ^{\mu }+\left({\frac {\ m\ c\ }{\hbar }}\right)^{2}\ \right]B^{\nu }=0} and with m = 0 {\displaystyle \ m=0\ }
∂ ν B ν = 0 , {\displaystyle \ \partial _{\nu }B^{\nu }=0\ ,} which may be called a generalizedLorenz gauge condition . For non-zero sources, with all fundamental constants included, the field equation is:
c μ 0 j ν = [ g μ ν ( ∂ σ ∂ σ + m 2 c 2 ℏ 2 ) − ∂ ν ∂ μ ] B μ {\displaystyle 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 }\ } When m = 0 , {\displaystyle \ m=0\ ,} Maxwell's equations without charge or current, and the above reduces to Maxwell's charge equation. This Proca field equation is closely related to theKlein–Gordon equation , because it is second order in space and time.
In thevector calculus notation, the source free equations are:
◻ ϕ − ∂ ∂ t ( 1 c 2 ∂ ϕ ∂ t + ∇ ⋅ A ) = − ( m c ℏ ) 2 ϕ {\displaystyle \ \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 \ } ◻ A + ∇ ( 1 c 2 ∂ ϕ ∂ t + ∇ ⋅ A ) = − ( m c ℏ ) 2 A {\displaystyle \ \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} \ } and ◻ {\displaystyle \ \Box \ } D'Alembert operator .
The Proca action is thegauge-fixed version of theStueckelberg action via theHiggs mechanism . Quantizing the Proca action requires the use ofsecond class constraints .
If m ≠ 0 , {\displaystyle \ m\neq 0\ ,}
B μ ↦ B μ − ∂ μ f {\displaystyle \ B^{\mu }\mapsto B^{\mu }-\partial ^{\mu }f\ } where f {\displaystyle \ f\ }
^ Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008,ISBN 978-0-470-03294-7 ^ W. Greiner, "Relativistic quantum mechanics", Springer, p. 359,ISBN 3-540-67457-8 ^ Parker, C.B., ed. (1994). "conjugate equivalence".McGraw Hill Encyclopaedia of Physics (2nd ed.). New York, NY: McGraw Hill.ISBN 0-07-051400-3 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
Theories Models
Regular Low dimensional Conformal Supersymmetric Superconformal Supergravity Topological Particle theory
Related
Template:Quantum mechanics topics