18
G. Altarelli and S. Forte
For A, B, C . . . . up or down indices make no difference: T A = T A etc. The structure
constants C ABC are completely antisymmetric in their indices, as can be easily seen.
In the following, for each quantity f A we define
f =
A
T
A f
A .
(2.12)
For example, we can rewrite Eq. (2.9) in the form:
U(θ
A ) = exp [igθ ] ∼ 1 + igθ + . . . ,
(2.13)
If we now make the parameters θ A depend on the space–time coordinates θ A =
θ A (x μ ), L[φ, ∂ μ φ] is in general no longer invariant under the gauge transformations
U [θ A (x μ )], because of the derivative terms: indeed ∂ μ φ = ∂ μ (U φ) = U∂ μ φ.
Gauge invariance is recovered if the ordinary derivative is replaced by the covariant
derivative:
D μ = ∂ μ + igV μ ,
(2.14)
where V A
μ are a set of D gauge vector fields (in one-to-one correspondence with the
group generators) with the transformation law
V
μ = U V μ U
−1
− (1/ig)(∂ μ U)U
−1 .
(2.15)
For constant θ A , V reduces to a tensor of the adjoint (or regular) representation of
the group:
V
μ = U V μ U
−1
V μ + ig[θ , V μ ] . . . ,
(2.16)
which implies that
V
C
μ = V
C
μ − gC ABC θ
A V
B
μ . . . ,
(2.17)
where repeated indices are summed up.
As a consequence of Eqs. (2.14) and (2.15), D μ φ has the same transformation
properties as φ:
(D μ φ)
= U(D μ φ) .
(2.18)
In fact
(D μ φ)
= (∂ μ + igV
μ )φ
= (∂ μ U)φ + U∂ μ φ + igU V μ φ − (∂ μ U)
φ = U(D μ φ) .
(2.19)