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:
Their determinant and trace? are, respectively:
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]?).