# Four-momentum matrix dot product sigma matrix

In the QFT book of Peskin and Schroeder, they introduce the notation: \begin{align} \sigma^{\mu}=(I,\sigma^{i})\\ \bar{\sigma^{\mu}}=(I,-\sigma^{i}). \end{align} On page 46 (Eq.(3.50)), They take the dot product with four-momentum $$$$\sqrt{p \cdot \sigma}.$$$$ How is the 'dot product' evaluated? is it pointwise multiplication: $$$$\sqrt{p \cdot \sigma}=\sqrt{IE+p^{i}\sigma^{i}}?$$$$ or, is it with Minkowskian signitaure: $$$$\sqrt{p \cdot \sigma}=\sqrt{p_{\mu}\sigma^{\mu}}=\sqrt{IE-p^{i}\sigma^{i}}?$$$$ The second option yields the right answer. So let's hope it's that one.

The dot product on a Minkowski manifold is defined to have indefinite signature, so the second option is correct. P.S. uses $$\eta_{\mu\nu} = \text{diag}(1, -1, -1, -1)$$.