Standard Model and Beyond · Problem 4.3

The Strong Interactions and QCD

Physics — Particle Physics · adjoint-representation-covariant-derivative

Problem

Let

\Phi^i

i = 1 \cdots m^2-1

, be

m

Hermitian scalars transforming according to the adjoint representation of

SU(m)

. Define the

m \times m

matrix

\Phi = \sum_i \Phi^i L^i

as in (3.31), where

L^i

are the fundamental representation matrices

L^i_m

. Show that the gauge covariant derivative is

D_\mu\Phi = \partial_\mu\Phi + ig\left[\vec{A}_\mu \cdot \vec{L},\, \Phi\right].