409
J.2. Trace theorems
Note that these forms are specific to the normalizations
¯
¯ = −2m
(J.22)
uu = 2m
vv
for the spinors.
J.2 Trace theorems
Tr1 = 4
(theorem 1)
(J.23)

Trγ 5 = 0
(theorem 2)
(J.24)

Tr(odd number of γ’s) = 0
(theorem 3).
(J.25)

Proof
Consider
T ≡ Tr(a / a / 2 . . . a / )
(J.26)
1
n
where n is odd. Now insert 1 = (γ 5 )
2 into T , so that
T = Tr(a / a / 2 . . . a / γ 5 γ 5 ).
(J.27)
1
n
Move the first γ 5 to the front of T by repeatedly using the result
aγ / 5 =
a.
−γ 5 /
(J.28)
We therefore pick up n minus signs:
T = Tr(a / 1 . . . a / ) = (−1)
n Tr(γ 5 a / 1 . . . a / γ 5 )
n
n
= (−1)
n Tr(a / 1 . . . a / γ 5 γ 5 )
(cyclic property of trace)
n
= −Tr(a / 1 . . . a / )
for n odd.
(J.29)
n
Thus, for n odd, T must vanish:
Tr(a / / b) = 4a · b
(theorem 4).
(J.30)
Proof
1
Tr(a / / b) =
Tr(a / / b + / ba /)
2
1
μν )
=
a μ b ν Tr(1.2g
2
= 4a · b.
Tr(a / / b/ cd /) = 4[(a · b)(c · d) + (a · d)(b · c) − (a · c)(b · d)].
(theorem 5)
(J.31)