Quadratic Casimir operator ( $\sum_{a} t_{a} t_{a} = C_{2} (R) I$ )

For a Lie algebra representation with generators referred to an orthonormal invariant form, the sum

\sum_{a} t_{a} t_{a}

commutes with every generator. Indeed, the commutator is proportional to a contraction of antisymmetric structure constants of a Lie algebra with the symmetric product

t_{a} t_{b} + t_{b} t_{a}

. The Schur lemma makes it scalar in an irreducible complex representation; that scalar is its quadratic Casimir eigenvalue. With trace normalization

tr (t_{a} t_{b}) = C (R) δ_{ab}

, taking the trace gives

C (R) dim g = C_{2} (R) dim R

Table of contents

SU(3) Casimir in Chevalley generators ( $C = 3 C_{2}$ )

 0  0

For

e_{3} = [e_{1}, e_{2}]

f_{3} = - [f_{1}, f_{2}]

, the operator

C = h_{1}^{2} + h_{2}^{2} + h_{1} h_{2} + 3 (h_{1} + h_{2}) + 3 \sum_{i = 1}^{3} f_{i} e_{i}

commutes with all generators. In its commutator with

e_{1}

, the Cartan polynomial contributes

3 h_{1} e_{1}

, while the last sum contributes

- 3 h_{1} e_{1}

; the two mixed terms involving

f_{2} e_{3}

cancel. On highest weight

(p, q)

its eigenvalue is

p^{2} + q^{2} + pq + 3 p + 3 q

. This normalization is three times the conventional SU(3) quadratic Casimir eigenvalue.

SU(3) quadratic Casimir eigenvalue ( $C_{2} (a, b) = (a^{2} + ab + b^{2} + 3 a + 3 b) /3$ )

 0  0

For generators normalized by

tr_{3} (t_{a} t_{b}) = δ_{ab} /2

, the quadratic Casimir operator on the irreducible SU(3) representation of Dynkin labels

(a, b)

has the displayed scalar value. In SU(3) highest-weight coordinates

(p, q)

it is

p^{2} + q^{2} + p + 3 q

To derive this, write the operator as

H_{1}^{2} + H_{2}^{2} + \sum_{i = 1}^{3} (E_{+}^{i} E_{-}^{i} + E_{-}^{i} E_{+}^{i})

. On a highest-weight vector, the raising operators vanish, and the three root commutators sum to

H_{1} + 3 H_{2}

. Centrality and the Schur lemma extend this value to the representation. The singlet, fundamental, adjoint and decuplet values are

0, 4/3, 3, 6

, respectively. Tensor labels and multiplet dimensions are summarized in the Particle Data Group's SU(n) multiplet guide.

Tensor-product Casimir trace identity ( $\sum_{i} d (R_{i}) C_{2} (R_{i}) = d (R_{1}) d (R_{2}) [C_{2} (R_{1}) + C_{2} (R_{2})]$ )

 0  0

For traceless generators, the tensor product of Lie algebra representations has generators

t_{a} \otimes I + I \otimes s_{a}

. The trace of their summed squares is the sum of the two factor Casimir traces; the cross term vanishes because

tr t_{a} = tr s_{a} = 0

. Decomposition into irreducible representations, with multiplicities counted, gives the displayed identity. Without tracelessness an extra term

2 \sum_{a} tr t_{a} tr s_{a}

is present. A one-dimensional abelian representation of charge one tensored with itself has Casimir four, not the sum two; semisimple Lie algebras avoid this problem because every generator is a sum of commutators.