Pauli Matrices

The Pauli matrices are

$$\sigma_1 = \left( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{arra... ... \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right),$$ (2.21)

where

$$\sigma_i\sigma_j = \delta_{ij} + i\epsilon_{ijk}\sigma_k .$$ (2.22)

where $\epsilon_{ijk}$ is the usual antisymmetric tensor

$$\epsilon_{ijk} = \left\{ \begin{array}{rl} +1 & \textrm{for... ...xtrm{if two or more indices are the same} \end{array} \right.$$ (2.23)

The following identity is satisfied.

$$(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma}) = \vec{a}\cdot\vec{b} I + i\vec{\sigma}\cdot(\vec{a}\times\vec{b}) ,$$ (2.24)

where $I$ is the $2\times 2$ identity matrix.