Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three ${\displaystyle 2\times 2}$ complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter ${\displaystyle \sigma }$ (sigma), and occasionally by ${\displaystyle \tau }$ (tau) when used in connection with isospin symmetries.$${\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\\\end{aligned}}}$$

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is Hermitian, and together with the identity matrix ${\displaystyle \mathbb {I} }$ (sometimes considered as the zeroth Pauli matrix ${\displaystyle \sigma _{0}}$), the Pauli matrices form a basis of the vector space of ${\displaystyle 2\times 2}$ Hermitian matrices over the real numbers, under addition. This means that any ${\displaystyle 2\times 2}$ Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

The Pauli matrices satisfy the useful product relation:

$${\displaystyle {\begin{aligned}\sigma _{i}\ \sigma _{j}=\delta _{ij}\ \mathbb {I} +i\ \varepsilon _{ijk}\ \sigma _{k}\ ,\end{aligned}}}$$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta, which equals ${\displaystyle +1}$ if ${\displaystyle i=j}$ otherwise ${\displaystyle 0}$, and the Levi-Civita symbol ${\displaystyle \varepsilon _{ijk}}$is used.

Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, ${\displaystyle \sigma _{k}}$ represents the observable corresponding to spin along the ${\displaystyle k}$th coordinate axis in three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$.

The Pauli matrices (after multiplication by ${\displaystyle i}$ to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: The matrices ${\displaystyle i\sigma _{1}}$, ${\displaystyle i\sigma _{2}}$, and ${\displaystyle i\sigma _{3}}$ form a basis for the real Lie algebra ${\displaystyle {\mathfrak {su}}(2)}$, which exponentiates to the special unitary group SU(2). The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of ${\displaystyle \ \mathbb {R} ^{3}}$ and the (unital) associative algebra generated by ${\displaystyle i\sigma _{1}}$, ${\displaystyle i\sigma _{2}}$, and ${\displaystyle i\sigma _{3}}$ functions identically (is isomorphic) to that of quaternions (${\displaystyle \mathbb {H} }$).