3.1. BASIC CONCEPTS 119
By the definition of spinors in Subsection2.2.6, under the Lorentz transformation of
(3.1.42), we have
(3.1.46)
Gaμ→ ̃LGaμ for 1≤a≤N^2 − 1 ,
ψi→Rψi for 1≤i≤N,and ̃L= (LT)−^1 , andRis the spinor representation matrix determined by the Lorentz matrix
L, as given by (2.2.67).
Hence, forSU(N)gauge fields (3.1.45), if we take the linear transformations for the
bundle spaces of (3.1.45) as
̃L:TpM→TpM,R:C^4 p→C^4 p,then the gauge fields (3.1.45) transform as (3.1.46).
Remark 3.4.The Lorentz transformations (3.1.42) and (3.1.43) are independent ofp∈M.
Hence, the associated vector bundles in (3.1.41) are trivial. In other words, the vector bundles
corresponding only to the Lorentz transformations are trivial. But, other vector bundles given
above are in general nontrivial.
3.1.4 Connections and covariant derivatives
In the last subsection, we see that a field
(3.1.47) F:M→M⊗pEN
undergoes a transformation:
(3.1.48) F ̃=TpF, p∈M,
if the bundle spaceENpundergoes a linear transformation
(3.1.49) Tp:ENp→ENp.
By PLD, for a physical fieldFas defined by (3.1.47), there is a Lagrangian action
(3.1.50) L=
∫
L(F,DF,···,DmF)dx.Physical symmetry principles amount to saying that the action (3.1.50) is invariant under
the transformation (3.1.48). This requires that the derivativeDFin (3.1.50) be covariant.
Namely, for (3.1.48) we have
(3.1.51) D ̃F ̃=NDF, Nis a matrix depends onTp.
If the transformationTpin (3.1.48) is independent ofp∈M, then
Dμ=∂μ (∂μ=∂/∂xμ).