Pion $SU(2)$ representation with two light flavors of quark I am working through Srednicki's Quantum Field Theory.  I'm in chapter 94 and, perhaps I missed something, but I'm wondering how he arrived at a result.
He has the pion fields in his Lagrangian written as
$$\pi^aT^a\qquad(1)$$
This would be in $SU(2)$ and so when Srednicki later explains that $T^a=\frac{1}{2}\sigma^a$. I took that to mean the $\sigma^a$ here refers to the Pauli matrices.
Later in the chapter, however, Srednicki shows:
$$\pi^a\sigma^a=\begin{pmatrix}
\pi^0 & \sqrt{2}\pi^+\\
\sqrt{2}\pi^- & -\pi^0\\
\end{pmatrix}\qquad(2),$$
which is not what I'd expect if I did that sum over the index $a$ (which I presume runs from $1,2,3$ in order of the Pauli matrices).
Did I miss a detail?  Am I wrong in assuming $\sigma^a$ here refers to the Pauli matrices?  Is it that the $\pi^{\pm}$ here are actually combinations of two other fields?
 A: This is just the same thing rewritten in a different basis—the basis of charge eigenstates, rather than the $SU(2)$ eigenstates. The physical charged pion fields are complex linear combinations of the $a=1,2$ components of the isotriplet, $\pi^{\pm}=\frac{1}{\sqrt{2}}(\pi^{1}\mp i\pi^{2})$. The other physical pion state, the neutral state, is $\pi^{0}=\pi^{3}$. Thus the expansion $\pi^{a}\sigma^{a}$ may be written
$$\pi^{a}\sigma^{a}=\pi^{1}\left[
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right]+\pi^{2}\left[
\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right]+\pi^{3}\left[
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\\
=\left[
\begin{array}{cc}
\pi^{3} & \pi^{1}-i\pi^{2} \\
\pi^{1}+i\pi^{2} & -\pi^{3}
\end{array}\right]=\left[
\begin{array}{cc}
\pi^{0} & \sqrt{2}\pi^{+} \\
\sqrt{2}\pi^{-} & -\pi^{0}
\end{array}\right].$$
This is similar to the decomposition of the Pauli matrices in terms of $\sigma^{3}$ and the raising and lowering matrices $\sigma^{\pm}=\frac{1}{2}\left(\sigma^{1}\pm i\sigma^{2}\right)$, which are
$$\sigma^{+}=\left[
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}\right],\quad \sigma^{-}=\left[
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}\right].$$
