A variationδAμofAμthen yields the following variation of the action,
δS[A] =∫
d^4 x(
−1
2
FμνδFμν+δAμjμ)=
∫
d^4 x(Fμν∂νδAμ+δAμjμ) (21.63)Integration by part of the first term gives (surface terms vanish for variationsδAμwith compact
support, and will be omitted here),
δS[A] =∫
d^4 xδAμ(−∂νFμν+jμ) (21.64)Its vanishing for allδAμrequires the second group of Maxwell equations.
Maxwell’s equations may also be viewed directly as differential equations forAμwith source
jμ. This is achieved by eliminatingFμνin terms ofAμ, so that
∂μ(∂μAν−∂νAμ) =−jν (21.65)In a relativistic framework, it is more convenient to choosea relativistic gauge condition than it
would be to choose the transverse gauge∇·A= 0, which is not Lorentz invariant. A convenient
gauge is theLandau gauge,∂μAμ= 0, in terms of which the field equation forAμbecomes the
wave equation with a source,
∂μ∂μAν=−jν (21.66)and current density conservation now holds in view of the gauge choice.
21.9 Structure of the Poincar ́e and Lorentz algebras
The Lorentz algebra forms a 6-dimensional Lie algebra, parametrized by 3 rotations and 3 boosts.
The Poincar ́e algebra is the semi-direct sum of the Lorentz algebra with the 4 translations of time
and space, so that the Poincar ́e algebra is 10-dimensional.We begin by determining the structure
of the Lorentz algebra, by expanding the defining relation offinite Lorentz transformations,
ημν= ΛμρΛνσηρσ (21.67)around the identity,
λμρ=δμρ+ωμρ+O(ω^2 ) (21.68)This relation implies the requirement that
ωνμ=−ωμν (21.69)