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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 \begin{equation} \sqrt{p \cdot \sigma}. \end{equation} How is the 'dot product' evaluated? is it pointwise multiplication: \begin{equation} \sqrt{p \cdot \sigma}=\sqrt{IE+p^{i}\sigma^{i}}? \end{equation} or, is it with Minkowskian signitaure: \begin{equation} \sqrt{p \cdot \sigma}=\sqrt{p_{\mu}\sigma^{\mu}}=\sqrt{IE-p^{i}\sigma^{i}}? \end{equation} The second option yields the right answer. So let's hope it's that one.

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

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  • $\begingroup$ Thanks, that's all I wanted to know. The confusion arises because we are strictly speaking multiplying matrix entries. But apparently, the entries multiplication is pointwise with Minkowskian signature. $\endgroup$ – M91 Jun 7 at 13:55

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