- #1
Pouramat
- 28
- 1
- Homework Statement
- The transformation law for Weyl spinors is as following (3.37); these transformation laws are connected by complex conjugation; using the identity (3.38)
- Relevant Equations
- (3.37) and (3.38) peskin
\begin{align}
\psi_L \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} - \vec\beta . \frac{\vec\sigma}{2}) \psi_L \\
\psi_R \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} + \vec\beta . \frac{\vec\sigma}{2}) \psi_R
\end{align}
I really cannot evaluate these from boost and rotation generator which was introduced in (3.26) and (3.27) Peskin.
Although the main porblem is the identity introduced below:
$$
\sigma^2 \vec\sigma^* = -\vec \sigma \sigma^2
$$
My attempt:
$$
\sigma^1 =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix};
\sigma^2 =
\begin{pmatrix}
0 & -i \\
i & 0 \\
\end{pmatrix};
\sigma^3 =
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
$$
the h.c. of the Pauli Sigma matrices is as following:
$$
(\sigma^1)^* =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix};
(\sigma^2)^* =
\begin{pmatrix}
0 & i \\
-i & 0 \\
\end{pmatrix};
(\sigma^3)^* =
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
$$
so ##\sigma^2 = diag(1,1)##,right?
I think my problem is that I cannot write the identity in components form.
\psi_L \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} - \vec\beta . \frac{\vec\sigma}{2}) \psi_L \\
\psi_R \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} + \vec\beta . \frac{\vec\sigma}{2}) \psi_R
\end{align}
I really cannot evaluate these from boost and rotation generator which was introduced in (3.26) and (3.27) Peskin.
Although the main porblem is the identity introduced below:
$$
\sigma^2 \vec\sigma^* = -\vec \sigma \sigma^2
$$
My attempt:
$$
\sigma^1 =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix};
\sigma^2 =
\begin{pmatrix}
0 & -i \\
i & 0 \\
\end{pmatrix};
\sigma^3 =
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
$$
the h.c. of the Pauli Sigma matrices is as following:
$$
(\sigma^1)^* =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix};
(\sigma^2)^* =
\begin{pmatrix}
0 & i \\
-i & 0 \\
\end{pmatrix};
(\sigma^3)^* =
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
$$
so ##\sigma^2 = diag(1,1)##,right?
I think my problem is that I cannot write the identity in components form.
Last edited: