The Lie algebra \mathfrak{so}(3) is the Lie algebra of SO(3), the circling accumulation for three-dimensional Euclidean space. It is semisimple of rank 1, and so it has a distinct absolute Casimir. The Killing anatomy for the circling accumulation is aloof the Kronecker delta, and so the Casimir invariant is artlessly the sum of the squares of the generators L_x,\, L_y,\, L_z of the algebra. That is, the Casimir invariant is accustomed by
L^2=L_x^2+L_y^2+L_z^2.
In an irreducible representation, the invariance of the Casimir abettor implies that it is a assorted of the character aspect e of the algebra, so that
L^2=L_x^2+L_y^2+L_z^2=\ell(\ell+1)e.
In breakthrough mechanics, the scalar amount \ell is referred to as the absolute angular momentum. For finite-dimensional matrix-valued representations of the circling group, \ell consistently takes on accumulation ethics (for bosonic representations) or half-integer ethics (for fermionic representations).
For a accustomed amount of \ell, the cast representation is (2\ell+1)-dimensional. Thus, for example, the three-dimensional representation for so(3) corresponds to \ell\,=\,1, and is accustomed by the generators
L_x= \begin{pmatrix} 0& 0& 0\\ 0& 0& -1\\ 0& 1& 0 \end{pmatrix}, L_y= \begin{pmatrix} 0& 0& 1\\ 0& 0& 0\\ -1& 0& 0 \end{pmatrix}, L_z= \begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 0 \end{pmatrix}.
The boxlike 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 \ell(\ell+1)\,=\,2 back \ell\,=\,1. Similarly, the two dimensional representation has a base accustomed by the Pauli matrices, which accord to circuit 1/2.
L^2=L_x^2+L_y^2+L_z^2.
In an irreducible representation, the invariance of the Casimir abettor implies that it is a assorted of the character aspect e of the algebra, so that
L^2=L_x^2+L_y^2+L_z^2=\ell(\ell+1)e.
In breakthrough mechanics, the scalar amount \ell is referred to as the absolute angular momentum. For finite-dimensional matrix-valued representations of the circling group, \ell consistently takes on accumulation ethics (for bosonic representations) or half-integer ethics (for fermionic representations).
For a accustomed amount of \ell, the cast representation is (2\ell+1)-dimensional. Thus, for example, the three-dimensional representation for so(3) corresponds to \ell\,=\,1, and is accustomed by the generators
L_x= \begin{pmatrix} 0& 0& 0\\ 0& 0& -1\\ 0& 1& 0 \end{pmatrix}, L_y= \begin{pmatrix} 0& 0& 1\\ 0& 0& 0\\ -1& 0& 0 \end{pmatrix}, L_z= \begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 0 \end{pmatrix}.
The boxlike 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 \ell(\ell+1)\,=\,2 back \ell\,=\,1. Similarly, the two dimensional representation has a base accustomed by the Pauli matrices, which accord to circuit 1/2.
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