Casimir Invariant - Example: So(3)

Example: So(3)

The Lie algebra so(3) is the Lie algebra of SO(3), the rotation group for three-dimensional Euclidean space. It is semisimple of rank 1, and so it has a single independent Casimir. The Killing form for the rotation group is just the Kronecker delta, and so the Casimir invariant is simply the sum of the squares of the generators L_x, L_y, L_z of the algebra. That is, the Casimir invariant is given by

In an irreducible representation, the invariance of the Casimir operator implies that it is a multiple of the identity element e of the algebra, so that

In quantum mechanics, the scalar value ℓ is referred to as the total angular momentum. For finite-dimensional matrix-valued representations of the rotation group, ℓ always takes on integer values (for bosonic representations) or half-integer values (for fermionic representations).

For a given value of ℓ, the matrix representation is (2ℓ + 1)–dimensional. Thus, for example, the three-dimensional representation for so(3) corresponds to ℓ = 1, and is given by the generators

$L_x= \begin{pmatrix} 0& 0& 0\\ 0& 0& -1\\ 0& 1& 0 \end{pmatrix}, \quad L_y= \begin{pmatrix} 0& 0& 1\\ 0& 0& 0\\ -1& 0& 0 \end{pmatrix}, \quad L_z= \begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 0 \end{pmatrix}.$

The quadratic Casimir invariant is then

$L^2=L_x^2+L_y^2+L_z^2= 2 \begin{pmatrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix}$

as ℓ(ℓ + 1) = 2 when ℓ = 1. Similarly, the two dimensional representation has a basis given by the Pauli matrices, which correspond to spin 1/2.

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