Pauli matrices

The Pauli matrices are a set of traceless involutive hermitian matrices with a number of nice properties

𝜎 1 = [0110], 𝜎 2 = [0 - 𝑖 𝑖 0], 𝜎 3 = [10 0 - 1]

Notably these form a basis for the real vector space $𝔰 𝔲 (2)$ , and with the addition of $𝜎 0 = 𝐈$ , they form a basis for the complex vector space $ℂ 2 \times 2$ .

Properties

Here we use Einstein summation convention.

Linear commutator: $[𝜎 𝑗, 𝜎 𝑘] = 𝜎 𝑗 𝜎 𝑘 - 𝜎 𝑘 𝜎 𝑗 = 2 𝑖 𝜖 𝑗 𝑘 ℓ 𝜎 ℓ$ with Levi-Civita symbol
Matrix anticommutator: ${𝜎 𝑗, 𝜎 𝑘} = 𝜎 𝑗 𝜎 𝑘 + 𝜎 𝑘 𝜎 𝑗 = 2 𝛿 𝑗 𝑘 𝐈$
Product: $𝜎 𝑗 𝜎 𝑘 = 12 [𝜎 𝑗, 𝜎 𝑘] + 12 {𝜎 𝑗, 𝜎 𝑘} = 𝛿 𝑗 𝑘 𝐈 + 𝑖 𝜖 𝑗 𝑘 ℓ 𝜎 ℓ$

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