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[tex] \Psi = ( \sqrt{p \cdot \sigma} \chi, \sqrt{p \cdot \bar{\sigma}} \chi)^T e^{-i p \cdot x}[/tex]

For any two component spinor Chi. But the dot product with the four vector p and the sigma vector is a matrix, so here one is taking the square root of a matrix. What do we mean by that? Or am I interpreting this wrong?