- #1

- 26

- 6

- 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: