Next, we determine the structure relations of the Lorentz algebra. To this end, we consider an
arbitrary representationD(Λ) of the Lorentz group, satisfyingD(Λ 1 Λ 2 ) =D(Λ 1 )D(Λ 2 ) by defini-
tion of a representation. LetLμνbe the representation matrices of the corresponding Lie algebra,
related toD(Λ) for Λ close to the identity in the following way,
D(Λ) =I+
1
2
ωμνLμν+O(ω^2 ) (21.70)By construction, we haveLνμ=−Lμν. The structure relations of the Lorentz algebra are deduced
by considering the following combination,
D(Λ)D(I+ω 1 )D(Λ)−^1 = D(I+ ̃ω 1 ) +O(ω^2 )
ω ̃μν 1 = ΛμρΛνσωρσ 1 (21.71)Identifying terms of orderω, for all finite values of Λ, we find the transformation law for the
generatorsLμνunder a Lorentz transformation Λ,
D(Λ)LμνD(Λ)−^1 = ΛρμΛσνLρσ (21.72)Taking now the special case where Λ is infinitesimally close to the identity, Λρμ=δρμ+ωρμ+O(ω^2 ),
and retaining the terms to first order inω, we derive the structure constants ofLμν,
[Lκλ,Lμν] = +ηλμLκν−ηκμLλν+ηλνLμκ−ηκνLμλ (21.73)Alternatively, we may express the algebra in terms of rotationsJk, and boostsKk, fori= 1, 2 ,3,
whereLijcorresponds to rotationsJk, provided we include a factor ofiin its definition, andLk 0
corresponds to boostsKk, for which we shall also include a factor ofi,
J 1 =−iL 23 K 1 =iL 10
J 2 =−iL 31 K 2 =iL 20
J 3 =−iL 12 K 3 =iL 30 (21.74)The commutation relation [L 23 ,L 31 ] =−L 12 corresponds to the following commutator of theJk,
[J 1 ,J 2 ] = iJ 3 , giving indeed the standard normalization of the commutation relations. Putting
together all structure relations in terms ofJkandKk, we have,
[Ji,Jj] = iεijkJk
[Ji,Kj] = iεijkKk
[Ki,Kj] = −iεijkJk (21.75)A further change of variables of the basic generators reveals that the Lorentz algebra is actually a
direct sum of two algebras, provided wecomplexify the Lorentz algebra. Complexifing means that
we consider the same generators as we had in the original algebra, but we allow linear combinations
withcomplexcoefficients. The resulting algebra no longer acts on real objects such as the real