whereIis the unit matrix in the Dirac representation space. Given the Dirac matrices, which we
shall realize shortly in explicit form, one automatically constructs the Dirac spinor representation
of the Lorentz algebra,
D(Lμν)≡Sμν=1
4
[γμ,γν] (22.6)To show thatSμν indeed forms a representation of the Lorentz algebra, all weneed to do is to
show that it satisfies the structure constants of (21.73). Todo so, it will be useful to recast
Sμν =^12 γμγν−^12 ημνI, since the term proportional toIwill cancel out in the argument of the
commutator. Thus, we have
[Sκλ,Sμν] =1
4
[γκγλ,γμγν] =1
4
(γκγλγμγν−γμγνγκγλ) (22.7)=1
4
γκ(−γμγλ+ 2ημλ)γν−1
4
γμ(−γκγν+ 2ηκν)γλ=1
2
ημλγκγν−1
2
ηκνγμγλ−1
4
(2ηκμ−γμγκ)γλγν+1
4
γμγκ(2ηνλ−γλγν)The terms quartic inγ-matrices cancel on the last line, and we are left with terms quadratic inγ
only. They become,
[Sκλ,Sμν] =1
2
ημλγκγν−1
2
ηκνγμγλ−1
2
ηκμγλγν+1
2
ηνλγμγκ=1
4
ημλ[γκ,γν]−1
4
ηκν[γμ,γλ]−1
4
ηκμ[γλ,γν] +1
4
ηνλ[γμ,γκ]
= ημλSκν−ηκνSμλ−ηκμSλν+ηνλSμκ (22.8)which is precisely the structure relation of the Lorentz algebra.
The representationSμν obtained this way is always reducible. This may be established by
constructing the famous chirality, orγ^5 -matrix, which is given by the product of allγ-matrices, up
to an overall complex multiple. We shall make the choice
γ^5 ≡−iγ^0 γ^1 γ^2 γ^3 (22.9)so that its square equals the identity matrix,
(
γ^5
) 2
=−γ^0 γ^1 γ^2 γ^3 γ^0 γ^1 γ^2 γ^3 =γ^0 γ^1 γ^2 γ^0 γ^1 γ^2 =γ^0 γ^1 γ^0 γ^1 =−γ^0 γ^0 =I (22.10)Furthermore,γ^5 anti-commutes with allγμ,
{γ^5 ,γμ}= 0 μ= 0, 1 , 2 , 3 (22.11)and therefore must commute with the generators of the Lorentz algebra,
[γ^5 ,Sμν] = 0 μ,ν= 0, 1 , 2 , 3 (22.12)Since{γ^5 ,γμ}= 0, the matrixγ^5 cannot simply be proportional to the identity matrix. Thus,we
have a non-trivial matrixγ^5 commuting with the entire representationSμν, which implies that the
representationSμνof the Lorentz algebra is areducible representation. Note that, although the
representation of the Lorentz algebra is reduced byγ^5 , the Clifford algebra of theγ-matrices is
irreducible.