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Pauli matrices

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

𝜎1=[0110],𝜎2=[0βˆ’π‘–π‘–0],𝜎3=[100βˆ’1]

Notably these form a basis for the real vector space 𝔰𝔲(2), and with the addition of 𝜎0 =𝐈, they form a basis for the complex vector space β„‚2Γ—2.

Properties

Here we use Einstein summation convention.

  1. Linear commutator: [πœŽπ‘—,πœŽπ‘˜] =πœŽπ‘—πœŽπ‘˜ βˆ’πœŽπ‘˜πœŽπ‘— =2π‘–πœ–π‘—π‘˜β„“πœŽβ„“ with Levi-Civita symbol
  2. Matrix anticommutator: {πœŽπ‘—,πœŽπ‘˜} =πœŽπ‘—πœŽπ‘˜ +πœŽπ‘˜πœŽπ‘— =2π›Ώπ‘—π‘˜πˆ
  3. Product: πœŽπ‘—πœŽπ‘˜ =12[πœŽπ‘—,πœŽπ‘˜] +12{πœŽπ‘—,πœŽπ‘˜} =π›Ώπ‘—π‘˜πˆ +π‘–πœ–π‘—π‘˜β„“πœŽβ„“

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