408
J. Dirac Algebra and Trace Identities
Either from the definition or using this explicit form it is easy to prove that
γ
2
5 = 1
(J.10)
and
{γ 5 , γ
μ
} = 0
(J.11)
i.e. γ 5 anticommutes with the other γ-matrices. Defining the totally antisymmetric tensor
( +1 for an even permutation of 0, 1, 2, 3
∈ μνρσ = −1 for an odd permutation of 0, 1, 2, 3
(J.12)
0
if two or more indices are the same
we may write
γ 5 =
i ∈ μνρσ γ
μ γ
ν γ
ρ γ
σ .
(J.13)
4!
With this form it is possible to prove
γ 5 γ σ =
i ∈ μνρσ γ
μ γ
ν γ
ρ
(J.14)
3!
and the identity
γ
μ γ
ν γ
ρ
μν γ
ρ
− g
μρ γ
ν
νρ γ
μ + iγ 5 ∈
μνρσ γ σ .
= g
+ g
(J.15)
J.1.3 Hermitian conjugate of spinor matrix elements
′
′
[¯ u(p , s
′ )Γu(p, s)]
† = ¯
u(p, s)Γ ¯ u(p , s
′ )
(J.16)
where Γ is any collection of γ matrices and
Γ ¯ ≡ γ
0 Γ
† γ
0 .
(J.17)
For example
γ μ = γ
μ
(J.18)
and
γ μ γ 5 = γ
μ γ 5 .
(J.19)
J.1.4 Spin sums and projection operators
Positive-energy projection operator:
∑
[Λ + (p)] αβ ≡
u α (p, s)¯ u β (p, s) = ( / p + m) αβ .
(J.20)
s
Negative-energy projection operator:
∑
[Λ − (p)] αβ ≡ −
v α (p, s)¯ v β (p, s) = (− / p + m) αβ .
(J.21)
s