I have a question in Peskin and Schroeder's book (Quantum field theory) page 46. Why
$$\big[ \sqrt{E+p^3}(\frac{1-\sigma^3}{2})+\sqrt{E-p^3}(\frac{1+\sigma^3}{2})\Big]=\sqrt{p.\sigma}~?$$
and why $$(p\cdot \sigma )(p\cdot \bar{\sigma})=p^2~?$$
I don't understand the meaning of $\sqrt{p\cdot \sigma}$. I'd really appreciate if someone could explain me.
What I've tried: Because we have a boost along the 3-direction, $p=(E,0,0,p^3)$. So $p\cdot \sigma^{\mu}=(E,0,0,p^3)\cdot (1,\sigma)=E-p^3 \sigma^3=\begin{pmatrix}
E-p^3 & 0 \\ 0 & E+p^3
\end{pmatrix}$. Thus we have $$\sqrt{p\cdot \sigma} =\begin{pmatrix}
\sqrt{E-p^3} & 0 \\ 0 & \sqrt{E+p^3}
\end{pmatrix}.$$
Best Answer
I have read quite a large proportion of Peskin and Schroeder's book on Quantum Field Theory (P&S) and I do not recall the first equation you write. So, I will sketch a proof for the second one. If there are still questions, please let me know in the comments.
Consider two four vectors, whose spatial components are Pauli matrices
$$\sigma^{\mu}=(\mathbb{1},\sigma^i) \hspace{3em} \bar{\sigma}^{\mu}=(\mathbb{1},-\sigma^i)$$
where $1$ is the 2x2 unit matrix.
Now consider the dot products
$p\cdot\sigma=p^01-p^1\sigma^1-p^2\sigma^2-p^3\sigma^3= \begin{pmatrix} p^0 & 0 \\ 0 & p^0 \end{pmatrix}- \begin{pmatrix} 0 & p^1 \\ p^1 & 0 \end{pmatrix}- \begin{pmatrix} 0 & -ip^2 \\ ip^2 & 0 \end{pmatrix}- \begin{pmatrix} p^3 & 0 \\ 0 & -p^3 \end{pmatrix}= \begin{pmatrix} p^0-p^3 & -p^1+ip^2 \\ -p^1-ip^2 & p^0+p^3 \end{pmatrix}$
$p\cdot\bar{\sigma}=p^01+p^1\sigma^1+p^2\sigma^2+p^3\sigma^3= \begin{pmatrix} p^0 & 0 \\ 0 & p^0 \end{pmatrix}+ \begin{pmatrix} 0 & p^1 \\ p^1 & 0 \end{pmatrix}+ \begin{pmatrix} 0 & -ip^2 \\ ip^2 & 0 \end{pmatrix}+ \begin{pmatrix} p^3 & 0 \\ 0 & -p^3 \end{pmatrix}= \begin{pmatrix} p^0+p^3 & p^1-ip^2 \\ p^1+ip^2 & p^0-p^3 \end{pmatrix}$
Hence, we multiply $$(p\cdot\sigma)(p\cdot\bar{\sigma})= \begin{pmatrix} p^0-p^3 & -p^1+ip^2 \\ -p^1-ip^2 & p^0+p^3 \end{pmatrix} \begin{pmatrix} p^0+p^3 & p^1-ip^2 \\ p^1+ip^2 & p^0-p^3 \end{pmatrix}= \begin{pmatrix} (p^0)^2-(p^3)^2-(p^1)^2-(p^2)^2 & 0 \\ 0 & (p^0)^2-(p^3)^2-(p^1)^2-(p^2)^2 \end{pmatrix}=p^2\times 1$$
The meaning of $\sqrt{p\cdot\sigma\:\vphantom{tfrac{a}{b}}}$ is such that $$\sqrt{p\cdot\sigma\:\vphantom{tfrac{a}{b}}}\sqrt{p\cdot\bar{\sigma}\:\vphantom{tfrac{a}{b}}}=\sqrt{p^2}=m$$ for particles that are on-shell.