Pauli Matrices
Here we summarize some properties of the Pauli matrices:
\[\begin{gather*}
\mat{\sigma}_1 = \mat{\sigma}_x = \mat{X} = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix},\\
\mat{\sigma}_2 = \mat{\sigma}_y = \mat{Y} = \begin{pmatrix}
0 & -\I\\
\I & 0
\end{pmatrix},\\
\mat{\sigma}_3 = \mat{\sigma}_z = \mat{Z} = \begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}.
\end{gather*}\]
We will often work with these as a “vector” with three components \(\mat{\sigma}_{i}\)
which will always have a Roman index \(i \in \{x, y, z\}\):
\[\begin{gather*}
\vec{\mat{\sigma}} = \mat{\sigma}_{i} = (\mat{\sigma}_x, \mat{\sigma}_y, \mat{\sigma}_z)
\end{gather*}\]
so we can write things like:
\[\begin{gather*}
\vec{a}\cdot\vec{\mat{\sigma}} = a_x\mat{\sigma}_x + a_y\mat{\sigma}_y + a_z\mat{\sigma}_z.
\end{gather*}\]
These “vectors” with an arrow have nothing to do with “kets” in the Hilbert space. They
typically represent the corresponding directions in 3D which considering the
Bloch Sphere.
To these, it will be convenient to add the identity:
\[\begin{gather*}
\mat{\sigma}_0 = \mat{1} = \begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}.
\end{gather*}\]
We can then refer to the Pauli 4-vector, which will always have a Greek index $\mu \in
{0,x,y,z}.
\[\begin{gather*}
\mat{\sigma}_{\mu} = (\mat{\sigma}_0, \mat{\sigma}_x, \mat{\sigma}_y, \mat{\sigma}_z),
\end{gather*}.
## Key Properties
In the following, Greek indices refer to all four matrices (including the identity),
while Roman indices refer only to the strict Pauli matrices. These are expressed in
terms of the Kronecker deltas (identity matrices):
\begin{gather*}
\delta_{ij} = \begin{cases}
1 & i = j \in \{x, y, z\}\\
0 & \text{otherwise}
\end{cases}, \qquad
\delta_{\mu\nu} = \begin{cases}
1 & \mu = \nu \in \{0, x, y, z\}\\
0 & \text{otherwise}
\end{cases},
\end{gather*}\]
and the Levi-Civita symbol which is \(\pm 1\) if the indices are an even or odd
permutation of (1, 2, 3) respectively and \(0\) if any index is repeated.
\[\begin{gather*}
\epsilon_{ijk} = \begin{cases}
+1 & (i, j, k) \in \{(1,2,3), (2,3,1), (3,1,2)\},\\
-1 & (i, j, k) \in \{(3,2,1), (2,1,3), (1,3,2)\},\\
0 & \text{otherwise}.
\end{cases}
\end{gather*}\]
This appears in the cross product:
\[\begin{gather*}
[\vec{a}\times\vec{b}]_{l} = \sum_{jk}\epsilon_{jkl}a_jb_k
\equiv \epsilon_{jkl}a_jb_k.
\end{gather*}\]
\[\begin{gather*}
\mat{\sigma}_{\mu} = \mat{\sigma}_{\mu}^\dagger, \qquad
\mat{\sigma}_{\mu}^2 = \mat{1}, \\
\Tr \mat{\sigma}_{\mu} = 2\delta_{\mu 0}, \qquad
\Tr \mat{\sigma}_{i} = 0,\\
\end{gather*}\]
One of the most useful relationships is:
\[\begin{gather*}
\mat{\sigma}_j \mat{\sigma}_k = \delta_{jk} \mat{1} + \I \epsilon_{jkl} \mat{\sigma}_{l}.
\end{gather*}\]
Thus:
\[\begin{gather*}
(\vec{a}\cdot\vec{\mat{\sigma}})(\vec{b}\cdot\vec{\mat{\sigma}})
= (a_j\mat{\sigma}_j)(b_k\mat{\sigma}_k)
= a_jb_k(\mat{\sigma}_j\mat{\sigma}_k)
= a_jb_k(\delta_{jk} \mat{1} + \I \epsilon_{jkl} \mat{\sigma}_{l})\\
= \vec{a}\cdot\vec{b}\mat{1} + \I (\vec{a}\times\vec{b})\cdot\vec{\mat{\sigma}}.
\end{gather*}\]
We also have
