J
Dirac Algebra and Trace Identities
J.1 Dirac algebra
J.1.1 γ matrices
The fundamental anticommutator
{γ
μ , γ
ν
} = 2g
μν
may be used to prove the following results.
(J.1)
γ μ γ
μ
γ μ /
aγ
μ
γ μ /
a/ bγ
μ
γ μ /
a/ b/ cγ
μ
/
a/ b
= 4
= −2/ a
= 4a · b
= −2/ c/ b/ a
= −/ b/ a + 2a · b.
(J.2)
(J.3)
(J.4)
(J.5)
(J.6)
As an example, we prove this last result:
/
a/ b =
=
=
a μ b ν γ
μ γ
ν
a μ b ν (−γ
ν γ
μ + 2g
μν )
−/ b/ a + 2a · b.
J.1.2 γ 5 identities
Define
γ 5 = iγ
0 γ
1 γ
2 γ
3 .
(J.7)
In the usual representation with
γ
0 =
(
1
0
γ 5 is the matrix
0
−1
)
and
γ =
(
0
−σ
γ 5 =
(
0 1
1 0
)
.
σ
0
)
(J.8)
(J.9)
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