The generators of the Lorentz algebra in this basis are givenby
S 0 i =1
2
γ 0 γi=−1
2
γ^0 γi=1
2
(
−σi 0
0 σi)Sij = Sij=1
4
(
[σi,σj] 0
0 [σi,σj])
=
i
2
εijk(
σk 0
0 σk)
(22.21)We may use these expressions to find also the representationsof the generatorsJkandKkin this
basis,
Jk=1
2
(
σk 0
0 σk)
Kk=
i
2(
σk 0
0 −σk)
(22.22)and from these we get
Ak=1
2
(
σk 0
0 0)
Bk=1
2
(
0 0
0 σk)
(22.23)These expressions reveal that the subalgebrasAandBindeed correspond to the reducibility of the
spinor representation.
22.3 Action of Lorentz transformations onγ-matrices
The Lorentz algebra in the Dirac spinor representation actsby the following infinitesimal transfor-
mations,
Λμν = δμν+ωμν+O(ω^2 )
D(Λ) = I+1
2
ωμνSμν+O(ω^2 ) (22.24)which allows us to compute the action of the Lorentz algebra on theγ-matrices,
D(Λ)γκD(Λ)−^1 =(
I+1
2
ωμνSμν)
γκ(
I−1
2
ωμνSμν)
+O(ω^2 )= γκ+1
2
ωμν[Sμ,ν,γκ] +O(ω^2 ) (22.25)The commutator may be evaluated using only the Clifford algebra relations, and we find,
[Sμ,ν,γκ] =ηκνγμ−ηκμγν (22.26)Asa result, we have
D(Λ)γκD(Λ)−^1 = γκ+ωμκγμ+O(ω^2 )
= Λμκγμ (22.27)