Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter σ {\displaystyle \sigma } (sigma), and occasionally by τ {\displaystyle \tau } (tau) when used in connection with isospin symmetries. σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 1 ) . {\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 I {\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrix σ 0 {\displaystyle \sigma _{0}} ), the Pauli matrices form a basis of the vector space of 2 × 2 {\displaystyle 2\times 2} Hermitian matrices over the real numbers, under addition. This means that any 2 × 2 {\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: σ i σ j = δ i j I + i ε i j k σ k , {\displaystyle {\begin{aligned}\sigma _{i}\ \sigma _{j}=\delta _{ij}\ \mathbb {I} +i\ \varepsilon _{ijk}\ \sigma _{k}\ ,\end{aligned}}} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta, which equals...