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Standard Model and Beyond
· Problem 3.18
Gauge Theories
Physics — Particle Physics · su3-octet-products
Problem
Let
A
j
A_j
A
j
and
B
k
B_k
B
k
transform as
S
U
(
3
)
SU(3)
S
U
(
3
)
octets. Prove that
F
i
≡
i
f
i
j
k
A
j
B
k
F_i \equiv if_{ijk}A_j B_k
F
i
≡
i
f
ij
k
A
j
B
k
and
D
i
≡
d
i
j
k
A
j
B
k
D_i \equiv d_{ijk}A_j B_k
D
i
≡
d
ij
k
A
j
B
k
also transform as octets. Note that the result for
F
i
F_i
F
i
generalizes to any simple Lie algebra, while that for
D
i
D_i
D
i
generalizes to
S
U
(
m
)
SU(m)
S
U
(
m
)
.
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