Pauli matrices

The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. They are:

σ1 = [0  1]
     [1  0]

σ2 = [0 -i]
     [i  0]

σ3 = [1  0]
     [0 -1]

Together with the identity matrix I, they form a basis for the set of 2 × 2 complex Hermitian matrices (as a result the identity is sometimes written as σ₀). These three in particular form a basis for the traceless Hermitian matrices, which form the Lie algebra su(2).

σ₁, σ₂, and σ₃ obey commutation and anticommutation relations:

[σ_i , σ_j] = 2 i ε_{i j k} σ_k

{σ_i , σ_j} = 2 δ_{i j}

Their determinant and trace are, respectively:

det(σ_i) = -1

Tr (σ_i) = 0

The Lie algebra su(2) is important because it is the algebra associated with the group of rotations in three-dimensional space. The unique simple group associated with this algebra is not the group of rotations, however, but a double cover of the same. This gives rise to an extra set of representations called spinor representations, which are used to describe non-relativistic spin 1/2 particles (in the relativistic case, one needs to consider the Lorentz-Poincare group).