#sigma matrix

[peeterjoot]

Explicit form of the square root of p . sigma.

Explicit form of the square root of p . sigma.

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With the help of Mathematica, a fairly compact form was found for the root of \( p \cdot \sigma \) \begin{equation}\label{eqn:DiracUVmatricesExplicit:120} \sqrt{ p \cdot \sigma } = \inv{ \sqrt{ \omega_\Bp- \Norm{\Bp} } + \sqrt{ \omega_\Bp+ \Norm{\Bp} } } \begin{bmatrix} \omega_\Bp- p^3 + \sqrt{ \omega_\Bp^2 – \Bp^2 } & – p^1 + i p^2 \\ –…

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[peeterjoot]

Explicit expansion of the Dirac u,v matrices

Explicit expansion of the Dirac u,v matrices

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We found that the solution of the \( u(p), v(p) \) matrices were \begin{equation}\label{eqn:DiracUVmatricesExplicit:20} \begin{aligned} u(p) &= \begin{bmatrix} \sqrt{p \cdot \sigma} \zeta \\ \sqrt{p \cdot \overline{\sigma}} \zeta \\ \end{bmatrix} \\ v(p) &= \begin{bmatrix} \sqrt{p \cdot \sigma} \eta \\ -\sqrt{p \cdot \overline{\sigma}} \eta…

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[peeterjoot]

Reflection using Pauli matrices.

Reflection using Pauli matrices.

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In class yesterday (lecture 19, notes not yet posted) we used \( \Bsigma^\T = -\sigma_2 \Bsigma \sigma_2 \), which implicitly shows that \( (\Bsigma \cdot \Bx)^\T \) is a reflection about the y-axis. This form of reflection will be familiar to a student of geometric algebra (see [1] — a great book, one copy of which is in the physics…

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