Casimir Operators of SU(3)

Welcome back. In our previous lesson, you mastered the powerful technique of Young Tableaux to decompose tensor products of SU(3) representations. This allowed us to predict the multiplet structure of composite systems, such as finding the octet and singlet in the combination of a quark and an anti-quark ( $3 \otimes 3 ˉ = 8 \oplus 1$ ).

This raises a natural question: while we can label these multiplets by their dimension ( $8$ ) or their Dynkin labels ( $(1, 1)$ ), is there a more fundamental, invariant quantity that characterizes them? For SU(2), we know the answer is the eigenvalue of the total angular momentum operator $J^{2}$ , which is $j (j + 1)$ . This operator commutes with all the generators, and its eigenvalue is the same for all states within a given multiplet. Such operators are called Casimir operators.

In this lesson, we will identify the Casimir operators for SU(3) and develop a systematic method to calculate their eigenvalues for any irreducible representation. This will provide us with the invariant numerical labels needed to classify the SU(3) multiplets, completing the theoretical toolkit required for our final discussion on the Eightfold Way.

Invariant Labels for Multiplets

Let's begin by formalizing the concept of a Casimir operator. As you'll recall from your study of quantum mechanics and angular momentum, $J^{2}$ is special because it commutes with all three components of $J$ . This property is the essence of a Casimir operator.

The following reading from a script on symmetries in quantum mechanics provides a clear, physics-oriented definition and explores its immediate consequences.

[PDF] Symmetries in Quantum Mechanics and Particle Physics

This text introduces the concept of Casimir operators from a physical perspective, connecting them to multiplets and degeneracy.

Please read the section titled "3.4 Casimir operators, multiplets, Schur’s lemma". Start from the beginning of the section and read until the end of the "Example" for Racah's theorem (the definition and SU(2) example). Then, read the paragraph defining the Cartan subalgebra and rank of SU(3). Finally, read the paragraph identifying the two Casimir operators of SU(3).

From this reading, we can extract the key ideas:

Definition: A Casimir operator $C^$ is an operator that commutes with all generators $X^_{a}$ of a Lie algebra: $[C^, X^_{a}] = 0$ for all $a$ .
Significance: Because it commutes with all generators, Schur's Lemma implies that within an irreducible representation (a multiplet), a Casimir operator must be proportional to the identity matrix, $C^= c_{R} 1$ . The constant $c_{R}$ is the eigenvalue of the Casimir operator for that representation $R$ . This eigenvalue is a unique, invariant label for the entire multiplet.
Number of Casimirs: Racah's theorem states that a semi-simple Lie algebra of rank $r$ has exactly $r$ independent Casimir operators.
- For SU(2), the rank is 1, and the single Casimir is $C^_{2} \propto J^{2}$ .
- For SU(3), the rank is 2. Therefore, it has two Casimir operators.

The SU(3) generators $T^_{a} = \frac{1}{2} λ_{a}$ (where $λ_{a}$ are the Gell-Mann matrices) satisfy the commutation relation $[T^_{a}, T^_{b}] = i f_{ab c} T^_{c}$ and the anti-commutation relation ${T^_{a}, T^_{b}} = \frac{1}{3} δ_{ab} 1 + d_{ab c} T^_{c}$ . The two Casimir operators for SU(3) can be constructed from the generators using the totally anti-symmetric structure constants $f_{ab c}$ and the totally symmetric constants $d_{ab c}$ :

The Quadratic Casimir Operator: $C^_{2} = \sum_{a = 1}^{8} T^_{a} T^_{a}$
The Cubic Casimir Operator: $C^_{3} = \sum_{a, b, c = 1}^{8} d_{ab c} T^_{a} T^_{b} T^_{c}$

While both are important, the quadratic Casimir $C^_{2}$ is the most commonly used, and we will now focus on developing a general method for calculating its eigenvalue.

Calculating the Eigenvalue of the Quadratic Casimir

To develop a formula for the eigenvalue $c_{R}$ , we first need a more formal definition of $C^_{2}$ . This involves the Killing form, which provides a natural metric on the Lie algebra.

Let's turn to a more technical resource that walks through the rigorous derivation.

[PDF] The eigenvalues of the quadratic Casimir operator and second ...

This paper by Howard Haber provides a rigorous derivation of the eigenvalues of the quadratic Casimir operator. We will use it to develop a general formula. First, let's establish the formal definition.

Please read the first page of the document. Focus on the definitions of the Killing form g_{ab} in equation (2), the quadratic Casimir operator C_2 in equation (5), and the subsequent proof that [\hat{C}_2, \hat{T}_a]=0. Note that this resource uses a convention with anti-hermitian generators, but the algebraic structure and the results are the same.

The key definition is $C_{2} \equiv g^{ab} T_{a} T_{b}$ , where $g^{ab}$ is the inverse of the Killing form metric $g_{ab} = f_{a c}^{d} f_{b d}^{c}$ . For a simple Lie algebra like su(3), this metric is proportional to the identity matrix, $g_{ab} \propto δ_{ab}$ , so our earlier definition $C^_{2} = \sum_{a} T^_{a}^{2}$ is valid up to a normalization constant. The crucial part is that this construction guarantees $[C^_{2}, T^_{c}] = 0$ .

Now, how do we find the eigenvalue $c_{R}$ for an irreducible representation $R$ ? The most elegant formula connects $c_{R}$ to the geometry of the weight lattice. A specific representation $R$ is uniquely identified by its highest weight $M$ . The eigenvalue of the quadratic Casimir is then given by:

c_{R} = (M, M + 2 δ)

where $(,)$ is the inner product in weight space, and $δ = \frac{1}{2} \sum_{α > 0} α$ is the Weyl vector, defined as half the sum of all positive roots. This formula, while beautiful, is not immediately practical. A more useful version expresses $c_{R}$ in terms of the Dynkin labels $(n_{1}, n_{2}, \dots, n_{ℓ})$ that you learned to use to classify irreps.

The following reading provides the derivation of this practical formula.

[PDF] The eigenvalues of the quadratic Casimir operator and second ...

Now that we have the definition, let's see how to calculate its eigenvalue for a given irreducible representation, which is uniquely specified by its highest weight or, equivalently, its Dynkin labels (n_1, n_2, \dots).

Read section III, focusing on the definition of Dynkin labels in equation (26). Then, read the derivation in section IV. Pay close attention to how the eigenvalue is first expressed in terms of the highest weight M and the Weyl vector \delta in equation (30), and then re-expressed in terms of Dynkin labels in equation (31).

The final result, Equation (31) from the paper, gives us the tool we need. For a representation $R$ with Dynkin labels $(n_{1}, \dots, n_{ℓ})$ , the eigenvalue of the quadratic Casimir is:

c_{R} = \frac{1}{2} j, k = 1 \sum ℓ (α_{k}, α_{k}) (n_{k} + 2) (A^{- 1})_{k j} n_{j}

Here, $(α_{k}, α_{k})$ are the squared lengths of the simple roots, and $A^{- 1}$ is the inverse of the Cartan matrix.

Application to SU(3)

Let's apply this formula to the case of SU(3). The Lie algebra is su(3) (also known as $A_{2}$ ), which has rank $ℓ = 2$ . An irreducible representation is specified by two Dynkin labels $(n_{1}, n_{2})$ . The necessary ingredients for su(3) are:

Inverse Cartan Matrix: For su(n), the Cartan matrix $A_{ij}$ is $2$ on the diagonal, $- 1$ on the super/sub-diagonals, and 0 otherwise. For su(3), this gives $A = (2 - 1 - 1 2)$ , and its inverse is: $A^{- 1} = \frac{1}{3} (2112)$
Root Lengths: For su(n), all roots have the same length. In the standard normalization used in the paper, the simple roots have squared length $(α_{k}, α_{k}) = \frac{1}{n}$ . For su(3), this is $(α_{1}, α_{1}) = (α_{2}, α_{2}) = \frac{1}{3}$ .

Now we can calculate the eigenvalue $c_{R}$ for any SU(3) irrep $(n_{1}, n_{2})$ .

Example 1: The Fundamental Representation $3$

The fundamental representation $3$ has Dynkin labels $(1, 0)$ . So, $n_{1} = 1$ and $n_{2} = 0$ . Plugging this into the general formula:

c_{3} = \frac{1}{2} j, k = 1 \sum 2 (α_{k}, α_{k}) (n_{k} + 2) (A^{- 1})_{k j} n_{j}

Since $n_{2} = 0$ , the only non-zero terms are when $j = 1$ .

c_{3} = \frac{1}{2} [(α_{1}, α_{1}) (n_{1} + 2) (A^{- 1})_{11} n_{1} + (α_{2}, α_{2}) (n_{2} + 2) (A^{- 1})_{21} n_{1}]

Substituting the values:

c_{3} = \frac{1}{2} [(\frac{1}{3}) (1 + 2) (\frac{2}{3}) (1) + (\frac{1}{3}) (0 + 2) (\frac{1}{3}) (1)]

c_{3} = \frac{1}{2} [(\frac{1}{3}) (3) (\frac{2}{3}) + (\frac{1}{3}) (2) (\frac{1}{3})] = \frac{1}{2} [\frac{2}{3} + \frac{2}{9}] = \frac{1}{2} [\frac{6 + 2}{9}] = \frac{4}{9}

Example 2: The Adjoint Representation $8$

The adjoint representation $8$ has Dynkin labels $(1, 1)$ . So, $n_{1} = 1$ and $n_{2} = 1$ . The calculation is more involved, but systematic.

c_{8} = \frac{1}{2} j, k = 1 \sum 2 (α_{k}, α_{k}) (n_{k} + 2) (A^{- 1})_{k j} n_{j}

For a simple Lie group, the eigenvalue of the quadratic Casimir in the adjoint representation is always 1, i.e., $c_{A} = 1$ , in this normalization. You can verify this by carrying out the summation, which provides a good check on your understanding of the formula.

The paper you have been referencing tabulates these results and provides simplified formulas for specific cases, which can be very useful.

This is the root diagram for the su(3) algebra. The origin is at the center, the six vectors pointing from the origin are the root vectors, and the six outer vertices form the weight diagram of the adjoint representation $\mathbf{8}$. The Weyl vector $\delta$ for su(3) is the sum of the two simple roots, $\delta = \alpha_1 + \alpha_2$, which points to the weight at the top-right corner of the inner hexagon of the $\mathbf{27}$ representation. The eigenvalue formula $c_R = (M, M+2\delta)$ has a beautiful geometric interpretation on diagrams like this.

The Eigenvalue Formula in a Simpler Form

While the formula involving the Cartan matrix is completely general, for SU(N) there exists a simpler expression for $c_{R}$ directly in terms of the Dynkin labels $(n_{1}, \dots, n_{N - 1})$ and the highest weight $M$ . The eigenvalue of the quadratic Casimir operator for SU(N) is often given in the physics literature as:

C_{2} (R) = \frac{1}{2 N} N i = 1 \sum N - 1 (M_{i})^{2} - (i = 1 \sum N - 1 M_{i})^{2} + (M, δ)

where $M_{i}$ are components of the highest weight vector $M$ . This can be evaluated and expressed entirely in terms of the Dynkin labels $(n_{1}, n_{2})$ for SU(3). The result is:

c_{(n_{1}, n_{2})} = \frac{1}{3} (n_{1}^{2} + n_{2}^{2} + n_{1} n_{2} + 3 n_{1} + 3 n_{2})

Let's check our previous results with this much simpler formula:

For $3 = (1, 0)$ : $c_{(1, 0)} = \frac{1}{3} (1^{2} + 0^{2} + 1 \cdot 0 + 3 \cdot 1 + 3 \cdot 0) = \frac{1}{3} (1 + 3) = \frac{4}{3}$ .
For $8 = (1, 1)$ : $c_{(1, 1)} = \frac{1}{3} (1^{2} + 1^{2} + 1 \cdot 1 + 3 \cdot 1 + 3 \cdot 1) = \frac{1}{3} (1 + 1 + 1 + 3 + 3) = \frac{9}{3} = 3$ .

Notice these values are exactly 3 times the ones we calculated before ( $\frac{4}{9} \to \frac{4}{3}$ , $1 \to 3$ ). This discrepancy is extremely common and arises from different normalization conventions for the generators $T^_{a}$ . Physicists often use $Tr (T^_{a} T^_{b}) = \frac{1}{2} δ_{ab}$ , while the paper by Haber uses a convention where this trace is related to the dual Coxeter number. The simplified formula above corresponds to the common physics convention. The key takeaway is not the absolute number, but that the ratio of eigenvalues for different representations is convention-independent.

Conclusion

In this lesson, you have learned to identify and characterize the invariant operators of SU(3). These operators provide a unique numerical fingerprint for each irreducible representation, much like the quantum number $j$ does for SU(2).

Key Takeaways:

A Casimir operator commutes with all generators of a Lie algebra. Its eigenvalue is an invariant label for an irreducible representation.
The rank of a semi-simple Lie algebra equals the number of its independent Casimir operators. SU(3) has rank 2, with a quadratic ( $C^_{2}$ ) and a cubic ( $C^_{3}$ ) Casimir.
The eigenvalue of the quadratic Casimir operator $C^_{2}$ for an SU(3) representation with Dynkin labels $(n_{1}, n_{2})$ can be calculated using a standard formula.
In a common physics normalization, this eigenvalue is given by $c_{(n_{1}, n_{2})} = \frac{1}{3} (n_{1}^{2} + n_{2}^{2} + n_{1} n_{2} + 3 n_{1} + 3 n_{2})$ .
These eigenvalues, $c_{(1, 0)} = 4/3$ for quarks ( $3$ ) and $c_{(1, 1)} = 3$ for the octet ( $8$ ), are fundamental properties used to classify particles.

We have now assembled all the necessary theoretical machinery. You can construct weight diagrams, classify representations with Dynkin labels, decompose tensor products with Young Tableaux, and now label the resulting multiplets with Casimir eigenvalues. In our next lesson, we will see how Murray Gell-Mann and Yuval Ne'eman brilliantly applied these very tools to bring order to the chaos of the particle zoo, leading to the celebrated Eightfold Way and the prediction of the quark model.

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