Irreducible Representations of su(2) via Highest Weight

Welcome to the first lesson of our new module on Representation Theory. In the previous module, we delved into the structure of Lie groups and algebras, culminating in the Baker-Campbell-Hausdorff formula. We saw how the commutation relations of the $su (2)$ algebra—the algebra of quantum spin and rotations—fully determine the rules for composing finite rotations. This established a deep connection between the algebra's structure and the group's properties.

Now, we will leverage that algebraic structure for a powerful purpose: classification. The goal of this lesson is to answer a fundamental question in quantum mechanics: what are all the possible types of angular momentum allowed in nature? In the language of group theory, this translates to classifying all the finite-dimensional irreducible representations of the $su (2)$ algebra. We will accomplish this using an elegant and powerful algebraic technique known as the method of highest weight. This method provides a complete blueprint for constructing the building blocks from which any representation of SU(2) can be built.

What is a Representation?

Before diving into the specifics of $su (2)$ , let's formalize what we mean by a representation in this context. In short, a representation of a Lie algebra is a mapping from the abstract algebra elements to a set of matrices (linear operators) that preserve the commutation relations. An irreducible representation (or "irrep") is one that has no non-trivial invariant subspaces; it cannot be broken down into smaller, independent representations. These irreps are the fundamental "building blocks" of the theory.

The following video provides a concise and clear introduction to these foundational concepts.

Spinors for Beginners 18: Irreducible Representations of SU(2) (Ladder Operators)

The video "Spinors for Beginners 18: Irreducible Representations of SU(2)" by eigenchris begins with an excellent formal definition of group and Lie algebra representations. It introduces the crucial distinction between reducible and irreducible representations using the concept of a direct sum.

Please watch the segment from this point. Focus on understanding the formal definition of a representation (a map that preserves the structure) and the visual idea of a reducible representation as a block-diagonal matrix.

The Algebraic Toolkit of $su (2)$

As we've established, the Lie algebra $su (2)$ is defined by generators $J_{1}, J_{2}, J_{3}$ satisfying the commutation relations:

[J_{i}, J_{j}] = i ϵ_{ij k} J_{k}

(Here we use the standard physics convention where the generators are Hermitian, e.g., $J_{k} = σ_{k} /2$ , and differ by a factor of $i$ from the anti-Hermitian generators we sometimes used for the formal Lie algebra definition).

While this basis is intuitive (representing rotations about three orthogonal axes), it's not the most convenient for classification. The key insight is to diagonalize one generator and define operators that move between its eigenspaces. This is achieved by moving to the complexification of the algebra, $sl (2, C)$ .

We choose $J_{3}$ as the generator we will represent with a diagonal matrix. This is analogous to choosing a specific axis (the z-axis) to measure a component of angular momentum. Its eigenvalues are called weights.

Next, we define the ladder operators:

J_{+} = J_{1} + i J_{2} and J_{-} = J_{1} - i J_{2}

Notice that $J_{+}^{†} = J_{-}$ . This new basis ${J_{3}, J_{+}, J_{-}}$ is far more useful for our purposes because of its commutation relations. The following video segment derives these critical relations.

Spinors for Beginners 18: Irreducible Representations of SU(2) (Ladder Operators)

The same eigenchris video now introduces the ladder operators and derives their commutation relations. The video uses the notation g_+, g_-, g_z, which correspond to our J_+, J_-, J_3 (up to some scaling factors that don't affect the final algebraic structure).

Please watch from this segment. The main goal is to see the derivation and understand the final, crucial commutation relations: [J_3, J_\pm] = \pm J_\pm [J_+, J_-] = 2J_3 Pay close attention to the bilinear property of the commutator, which is used to simplify the calculation.

The Method of Highest Weight

With these commutation relations, we can now execute the classification. Let's assume we have a finite-dimensional irreducible representation of $su (2)$ acting on a vector space $V$ .

Eigenspaces of $J_{3}$ : Since the representation is on a complex vector space, $J_{3}$ must have at least one eigenvector. Let's call it $v_{m}$ , with eigenvalue (weight) $m$ : $J_{3} v_{m} = m v_{m}$ .
Action of Ladder Operators: What happens when we apply $J_{+}$ or $J_{-}$ to this eigenvector? Let's check:
\begin{align*}
J_3 (J_+ v_m) &= (J_+ J_3 + [J_3, J_+]) v_m \
&= (J_+ J_3 + J_+) v_m \
&= J_+ (J_3 v_m) + J_+ v_m \
&= J_+ (m v_m) + J_+ v_m \
&= (m+1) J_+ v_m
\end{align*}
This shows that $J_{+} v_{m}$ is an eigenvector of $J_{3}$ with eigenvalue $m + 1$ . Similarly, one can show $J_{3} (J_{-} v_{m}) = (m - 1) J_{-} v_{m}$ . The ladder operators raise or lower the weight of a state by one unit.
The Highest Weight State: Since we assumed the representation is finite-dimensional, there can only be a finite number of distinct eigenvalues. This means the ladder of states must terminate. There must be a "top rung"—a state with the highest weight, which we will call $v_{j}$ . This state is defined by the property that it is annihilated by the raising operator:
$J_{+} v_{j} = 0$
The weight $j$ of this state is called the highest weight. By definition, $J_{3} v_{j} = j v_{j}$ .

This logic—that a finite-dimensional representation must contain a highest weight state which is annihilated by the raising operator—is the cornerstone of the entire method.

The video segment below visualizes this process and reinforces the concept of the highest weight vector.

Spinors for Beginners 18: Irreducible Representations of SU(2) (Ladder Operators)

This part of the eigenchris video explains precisely how the commutation relations imply that the ladder operators shift the eigenvalues. It then introduces the crucial concept of the highest weight vector (or eigenvector with the highest weight).

Watch from this section. Focus on the argument that for any finite-dimensional representation, repeatedly applying the raising operator must eventually result in zero.

The Classification Theorem and its Proof

We now have all the pieces to formally classify the irreps. The procedure is to start from the highest weight state $v_{j}$ and apply the lowering operator $J_{-}$ repeatedly. This generates a sequence of states: $v_{j}, v_{j - 1}, \dots$ . This chain must also terminate at a lowest weight state $v_{m_{min}}$ such that $J_{-} v_{m_{min}} = 0$ .

The central result, which we will now prove, is that for a given highest weight $j$ , the lowest weight must be $- j$ , and $2 j$ must be a non-negative integer. This quantizes the possible values of spin.

The following document provides a concise and rigorous proof of this classification. It uses the notation $L_{3}, L_{+}, L_{-}$ which you can map directly to our $J_{3}, J_{+}, J_{-}$ .

The Irreducible Representations of the Lie algebra su(2)

This document presents the core proof for classifying the irreducible representations of \mathfrak{su}(2) (or, isomorphically, \mathfrak{so}(3)). It formalizes the logic we've developed.

Read the section titled "Proof of Theorem 2" (pages 16-17). The proof proceeds in these key steps: It begins with an arbitrary eigenvector v of L_3. It applies the raising operator L_+ repeatedly until it finds the highest weight vector, which it calls v_0 with eigenvalue \mu. It defines a basis of vectors v_j = (L_-)^j v_0. It shows that this sequence of vectors must terminate at some vector v_N, meaning v_{N+1} = (L_-)^{N+1}v_0 = 0. Using an identity for L_+ v_j (which it asks you to prove by induction), it shows that the termination condition implies 2\mu - N = 0, or \mu = N/2. This final result, \mu = N/2, is the punchline. It demonstrates that the highest weight (j in our notation) must be an integer or half-integer, since N is an integer. It also shows that the dimension of the representation is N+1 = 2\mu+1 = 2j+1. The weights m must therefore run from j down to -j in integer steps.

The proof you just read through solidifies the entire classification. To summarize the conclusion:

Classification of $su (2)$ Irreducible Representations:

The irreducible representations are labeled by a number $j$ , which can be a non-negative integer or half-integer: $j = 0, \frac{1}{2}, 1, \frac{3}{2}, \dots$ . This number $j$ is the "spin" or total angular momentum quantum number.
For each $j$ , there is a unique (up to equivalence) irreducible representation of dimension $2 j + 1$ .
The basis states for this representation can be labeled by a second quantum number, $m$ , which takes on the $2 j + 1$ values: $m = j, j - 1, \dots, - j + 1, - j$ . This is the magnetic quantum number.
The representation space is spanned by the set of orthonormal vectors ${∣ j, m ⟩}$ where $m$ ranges from $- j$ to $+ j$ .

Conclusion

In this lesson, we have used the algebraic structure of $su (2)$ to achieve a complete classification of its fundamental building blocks—the irreducible representations.

Key Takeaways:

An irreducible representation (irrep) of a Lie algebra is a mapping to matrices that preserves the commutator structure and cannot be simplified into smaller, block-diagonal forms.
By complexifying the $su (2)$ algebra and changing to the basis of a diagonal operator ( $J_{3}$ ) and ladder operators ( $J_{\pm}$ ), we can systematically explore the structure of its representations.
The method of highest weight relies on the fact that any finite-dimensional representation must contain a state of maximum eigenvalue ("weight") that is annihilated by the raising operator $J_{+}$ .
By generating a "ladder" of states from this highest weight state, and requiring the ladder to terminate, we proved that the highest weight $j$ must be an integer or half-integer.
For each allowed value of $j$ , there exists a unique irreducible representation of dimension $2 j + 1$ . This gives us the complete set of possible angular momentum multiplets in quantum mechanics.

We have now answered the question of what representations exist. In our next lesson, we will move from classification to construction. We will derive the explicit matrix elements for the generators $J_{3}, J_{+}, J_{-}$ for any given spin $j$ , giving us the concrete matrices that realize these abstract algebraic structures.

Recommended resources

What is a Representation?

The Algebraic Toolkit of su(2)

The Method of Highest Weight

The Classification Theorem and its Proof

Conclusion

The Algebraic Toolkit of $su (2)$