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\noindent PHY--396~K/L.\qquad
Problem set \#22.\quad
Due April 18, 2001.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider a generic simple non-abelian Lie group $G$ and its
generators $T^a$.
The (quadratic)
 Casimir operator $C_2=\sum_a T^aT^a$ commutes with all the generators
and hence for any irreducible representation $(r)$ of the group,
$C_2$ restricted to $(r)$ is simply a unit matrix times a number $C(r)$.
In other words, if $T^a_{(r)}$ is a matrix of the generator $T^a$ in the
representation $(r)$, then $\sum_a T^a_{(r)}T^a_{(r)}=C(r)
\times\bf 1$.
For example, for the isospin group $SU(2)$, the irreps are
characterized by the isospin $I$ and $C(I)=I(I+1)$.

\spointbegin
By symmetry, for any complete representation $(r)$ of the group,
$$
\tr_{(r)}(T^aT^b)\ \equiv\ \tr\left(T^a_{(r)}T^b_{(r)}\right)\
=\ R(r)\delta^{ab}
\eqno\eq
$$
for some coefficient $R(r)$.
Show that for any irreducible representation,
$$
{R(r)\over C(r)}\ =\ {\dim(r)\over\dim(G)}\,.
\eqn\RC
$$
In particular, for the $SU(2)$ group, this formula gives
$R(I)={1\over3}I(I+1)(2I+1)$.

\spoint
Suppose the first three generators of $G$ generate an $SU(2)$ subgroup.
Show that if a representation $(r)$ of $G$ decomposes into several
$SU(2)$ multiplets of isospins $I_1,I_2,\ldots,I_n$, then
$$
R(r)\ =\sum_{i=1}^n\,\coeff13 I_i(I_i+1)(2I_i+1).
\eqn\decomp
$$

\spoint
Now consider the $SU(N)$ group with an obvious $SU(2)$ subgroup
of matrices acting on the first two components of a complex $N$-vector.
The fundamental representation $(N)$ of the $SU(N)$ decomposes into
one doublet and $(N-2)$ singlets of the $SU(2)$ subgroup,
hence
$$
R(N)\ =\ \half\qquad{\rm and}\quad C(N)\ =\ {N^2-1\over 2N}\,.
\eqno\eq
$$
The {\it adjoint} representation of the $SU(N)$ group consists of
traceless hermitian $N\times N$ matrices $\Phi_j^{\,k}$ which transform
according to
$$
\Phi'\ =\ U\Phi U^\dagger\qquad\ie,\quad
\Phi_j^{\prime\,k}\ =\ U_j^{\,i}\Phi_i^{\,\ell}U^{*k}_{\,\,\ell}\ ,
\eqn\Adjoint
$$
thus $({\rm adj})+(1)=(N)\times(\overline N)$
(where the singlet (1) corresponds to the missing trace part of $\Phi_j^{\,k}$).
\subpar
Show that the adjoint representation of the $SU(N)$ decomposes into
one $SU(2)$ triplet, $2(N-2)$ doublets and $(N-2)^2$ singlets and hence
$$
R({\rm adj})\ =\ C({\rm adj})\ \equiv\ C(G)\ =\ N.
\eqno\eq
$$
%Hint: $(N)\times(\overline N)=({\rm adj})+(1)$.

\spoint
The symmetric and the anti-symmetric 2--index tensors from irreducible
representations of the $SU(N)$ group.
Find out the decomposition of these irreps under an $SU(2)\subset SU(N)$
and calculate their respective $R$ factors.

\medskip
\point
For a field $\Phi^a(x)$ in the adjoint representation of an $SU(N)$ gauge
symmetry, the covariant derivative
$$
D_\mu\Phi^a(x)\ =\ \partial_\mu\Phi^a(x)\
-\ iA_\mu^c(x) [T_{\rm adj}^c]^{ab}\Phi^b(x)
\eqno\eq
$$
can be written in matrix notations as
$$
D_\mu\Phi(x)\ =\ \partial_\mu\Phi(x)\ -\ i[A_\mu(x),\Phi(x)] .
\eqn\CDA
$$
\spointbegin
Verify the covariance of this derivative.

\spoint %b
Show that $[{\cal D}_\mu,{\cal D}_\nu]\Phi= -i[F_{\mu\nu},\Phi]$.

\pointcon
The non-abelian gauge field tension $F^a_{\mu\nu}(x)$ itself transforms in the
adjoint representation of the gauge group, hence $D_\lambda F_{\mu\nu}
=\partial_\lambda F_{\mu\nu}-i[A_\lambda,F_{\mu\nu}]$, \etc, \etc\space
\spoint %c
Prove the non-abelian Bianchi identity
$$
D_\lambda F_{\mu\nu}\ +\ D_\mu F_{\nu\lambda}\ +\ D_\nu F_{\lambda\mu}\
=\ 0.
\eqn\Bianchi
$$

\pointcon
Now consider the classical Yang--Mills theory of the non-abelian gauge field
$A^\mu(x)$ governed by the Lagrangian
$$
{\cal L}_{\rm YM}\ =\ {-1\over 2g^2}\,\tr\bigl(F_{\mu\nu}F^{\mu\nu}\bigr).
\eqn\Lagrange
$$
\spoint %d
Show that $\delta F^{\mu\nu}(x)=D^\mu\delta A^\nu(x)-D^\nu\delta A^\mu(x)$
and use this observation to write the classical YM field equations of motion as
$D^\mu F_{\mu\nu}=0$.

\pointcon
Next, let us add the fermionic fields and have
$$
{\cal L}\ =\ {-1\over 2g^2}\,\tr\bigl(F_{\mu\nu}F^{\mu\nu}\bigr)\
+\ \overline\Psi(i\!\not\!\! D-m)\Psi .
\eqno\eq
$$
\spoint %e
Show that in this case the classical YM field equations of motion become
$$
D^\mu F_{\mu\nu}\ =\ -g^2 J_\nu
\eqn\EoM
$$
where
$$
J_\nu^a\ =\ \overline\Psi\gamma_\nu{\lambda^a\over2}\Psi .
\eqn\Current
$$
Together, eqs.~\Bianchi\ and \EoM\ serve as the non-abelian analogue
of the Maxwell equations.

\spoint %f
Show that eq.~\EoM\ requires the fermionic current $J_\nu$ to be
{\sl covariantly conserved}, $D^\nu J_\nu=0$.

\spoint %g
Use Dirac equations for the fermionic fields to
verify that the current~\Current\ is indeed covariantly conserved.

\spoint %h
Finally, consider the {\sl second} variation of the Yang-Mills action
expanded around some non-trivial solution
$A^\mu(x)$ of the YM equations
and show that
$$
\delta_2\left[\!\int\!\!d^4x{\cal L}_{\rm YM}\right]
=\ {1\over g^2}\int\!\!d^4x\biggl[ \tr\left( \delta A^\mu D^2 \delta A_\mu\right)\
+\ \tr\left((D_\mu\delta A^\mu)^2\right)\
+\ 4i\tr\left(F_{\mu\nu}\delta A^\mu\delta A^\nu\right) \biggr].
$$

\bye
