# Split Pauli Four-vector as quadratic terms of spinors

If I have the Pauli Four-vector $$x_{\mu}\sigma^{\mu} = \left(\begin{array}{cc} t+z & x-i y \\ x+i y & t-z \end{array}\right)$$ with $$\sigma^0$$ as Identity Matrix. Is there some way to write it as quadratic terms of spinors? In this case, spinors would encapsulate coordinates as entries of their components.

• Please double check the correctness of your equation: $x_{\mu}\sigma^{\mu} = \left(\begin{array}{cc} -t+z & x-i y \\ x+i y & t-z \end{array}\right)$. Are you sure about the changing sign of $t$? Commented Jul 3 at 17:24
• In diagonal, t+z and t-z It has this format. I edited correctly now Commented Jul 3 at 17:59
• Is the WP detail on Weyl's surjective homomorphism not clear? 4-vectors map to pairs of spinor indices. Commented Jul 4 at 10:06
• Near duplicate and linked questions. Commented Jul 4 at 10:15
• Why would you "want" that? Hitting the matrix $\sigma^\mu$ with a Dirac spinor on each side yields an object transforming as a four-vector. It is not clear what you have not mastered about the spinor map. Have you slugged through a standard QFT text? Commented Jul 4 at 19:52