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)} , whereϕ {\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 {mc}{\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 {mc}{\hbar }}\right)^{2}\right]B^{\nu }=0} and for m ≠ 0 implies
∂ ν B ν = 0 , {\displaystyle \partial _{\nu }B^{\nu }=0,} equivalent to a generalizedLorenz gauge condition . For the massive case however, this is a physical constraint rather than an optional 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 }} Whenm = 0 {\displaystyle m=0} , the source-free equations reduce toMaxwell'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 {mc}{\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 {mc}{\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 .
Ifm ≠ 0 {\displaystyle m\neq 0} , they are not invariant under the gauge transformations of electromagnetism
B μ ↦ B μ − ∂ μ f {\displaystyle B^{\mu }\mapsto B^{\mu }-\partial ^{\mu }f} wheref {\displaystyle f}
^ B.R. Martin; G. Shaw (2008),Particle Physics (2nd ed.), John Wiley & Sons,ISBN 978-0-470-03294-7 ^ W. Greiner (2000),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
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