Standard Model and Beyond · Problem 4.4

The Strong Interactions and QCD

Physics — Particle Physics · so-m-gauge-theory-covariant-derivative

Problem

From (3.19) the defining (vector) representation of

SO(m)

consists of the

m(m-1)/2

antisymmetric imaginary

m \times m

matrices. These are conveniently labeled by two indices, with

\left(L^{ij}\right)_{ab} = -i(\delta^i_a\delta^j_b - \delta^i_b\delta^j_a),

where

i, j, a

, and

b

all range from 1 to

m

. Clearly,

L^{ij} = -L^{ji}

and

L^{ii} = 0

. One could restrict the indices so that

i < j

, but it is convenient not to do so, provided one is careful about double counting. The trace is

\mathrm{Tr}\left(L^{ij}L^{kl}\right) = 2(\delta^{ik}\delta^{jl} - \delta^{il}\delta^{jk}).

From the vector representation, one has the Lie algebra

\left[T^{ij}, T^{kl}\right] = i\left(-\delta^{jk}T^{il} - \delta^{il}T^{jk} + \delta^{ik}T^{jl} + \delta^{jl}T^{ik}\right),

where the generators

T^{ij}

have the same labelling convention as

L^{ij}

. (It is instructive to specialize these relations to

m = 3

.) (a) Calculate the gauge covariant derivative

(D^\mu\Phi)_a

for a scalar or fermion field

\Phi_a

a = 1 \cdots m

, transforming as a vector under

SO(m)

, labeling the gauge bosons by

A^{ij}_\mu = -A^{ji}_\mu

and the gauge coupling as

g

. (b) Calculate the field strength tensor

F^{ij}_{\mu\nu}

for

A^{ij}_\mu