The Pauli Vector and Lie Groups - Part 1

Introduction

This is part one of two in a series of posts where I elaborate on Pauli matrices, the Pauli vector, Lie groups, and Lie algebras. I have found that most resources on the subjects of Lie groups and Lie algebras present the material in an overly formal way, using notation that masks the simplicity of these concepts. Learning about Lie algebras and Lie groups is much easier if one first develops an intuition about what is actually happening, and then later dives into complicated notation.

We will start our discussion with the properties of the Pauli matrices and the Pauli vector.

The Pauli Z Matrix

\[\begin{equation} \sigma_z = \begin{bmatrix} 1 & 0 \\[0.1cm] 0 & -1 \end{bmatrix} \end{equation}\]

Eigenvectors and Eigenvalues

\[\begin{equation} \begin{bmatrix} 1 & 0 \\[0.1cm] 0 & -1 \end{bmatrix} \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} = \lambda \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} \end{equation}\] \[\begin{equation} \begin{bmatrix} 1 - \lambda & 0 \\[0.1cm] 0 & -1 - \lambda \end{bmatrix} \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} = 0 \end{equation}\] \[\begin{equation} det \begin{bmatrix} 1 - \lambda & 0 \\[0.1cm] 0 & -1 - \lambda \end{bmatrix} = 0 \end{equation}\] \[\begin{equation} (1 - \lambda)(-1 - \lambda) = 0 \end{equation}\] \[\begin{equation} \lambda = \pm 1 \end{equation}\] \[\begin{equation} \begin{bmatrix} 1 & 0 \\[0.1cm] 0 & -1 \end{bmatrix} \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} = (+1) \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} \end{equation}\] \[\begin{equation} \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} = \begin{bmatrix} 1 \\[0.1cm] 0 \end{bmatrix} \end{equation}\] \[\begin{equation} \begin{bmatrix} 1 & 0 \\[0.1cm] 0 & -1 \end{bmatrix} \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} = (-1) \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} \end{equation}\] \[\begin{equation} \begin{bmatrix} x \\[0.1cm] y \end{bmatrix} = \begin{bmatrix} 0 \\[0.1cm] 1 \end{bmatrix} \end{equation}\]

Perturbation Theory

Final Thoughts