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. 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 bosons, i.e. the Z and W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.
Lagrangian density The field involved is a complex 4-potential B μ = ( ϕ c , A ) {\displaystyle B^{\mu }=\left({\frac {\phi }{c}},\mathbf {A} \right)} , where ϕ {\displaystyle \phi } is a kind of generalized electric potential and A {\displaystyle \mathbf {A} } is a generalized magnetic potential. The field B μ {\displaystyle B^{\mu }} transforms like a complex four-vector. The Lagrangian density is given by:
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 },}
where c {\displaystyle c} is the speed of light in vacuum, ℏ {\displaystyle \hbar } is the reduced Planck constant, and ∂ μ {\displaystyle \partial _{\mu }} is the 4-gradient.
Equation The Euler–Lagrange equation of motion for this case, also called the Proca 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
[ ∂ 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 generalized Lorenz 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:
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