Rotation Group and SU(2) Representations

Hello! Welcome to your second lesson.

In our first lesson, we established the profound connection between continuous symmetries and conserved quantities through Noether's theorem. We saw that the conserved quantity is the Hermitian generator of the symmetry transformation. Today, we'll dive deep into one of the most important continuous symmetries in physics: rotation. Its generator is the angular momentum operator, and its mathematical properties have far-reaching consequences, including the quantization of spin.

This lesson directly addresses the learning outcome: Analyze the algebraic structure of the rotation group and SU(2) representations. We will start by deriving the commutation relations that define the "Lie algebra" of the rotation group, SO(3). Then, in a purely algebraic way, we will derive the quantization of angular momentum. Finally, we'll uncover the deep and beautiful relationship between the rotation group SO(3) and the special unitary group SU(2), which is the mathematical key to understanding spin-1/2 particles and other fermions.

1. The Algebra of Rotations: SO(3) and its Generators

Rotations in three-dimensional space form a mathematical group called the Special Orthogonal group in 3 dimensions, or SO(3). "Special" means the determinant of the rotation matrix is +1 (preserving orientation), and "Orthogonal" means the matrix preserves lengths and angles. As you know from rotating an object in your hands, the order of rotations matters—the group is non-Abelian.

In quantum mechanics, these rotations are generated by the components of the angular momentum operator, ${\hat{J}}_x,{\hat{J}}_y,{\hat{J}}_z$. The non-Abelian nature of the group is encoded in the fact that these generators do not commute with each other. The following video derives their commutation relations starting from the classical definition of orbital angular momentum, $\vec{L}=\vec{r}\times \vec{p}$.

The central result from the video is that the angular momentum operators obey the cyclic commutation relations:

This can be written compactly using the Levi-Civita symbol ${ϵ}_{ijk}$:

This set of commutation relations is known as the Lie algebra of SO(3) (or, more precisely, $\mathfrak{so}(3)$). As the lecturer emphasizes, this algebraic structure is more profound than the specific definition of orbital angular momentum. Any quantity that acts as a generator of rotations in quantum mechanics—including intrinsic properties like spin—must have components that satisfy this algebra.

2. From Algebra to Quantization

We will now perform one of the most elegant derivations in quantum mechanics: showing that the commutation relations we just established inevitably lead to the quantization of angular momentum. We will not assume anything about wavefunctions or spatial coordinates, relying only on the algebra itself.

This derivation is foundational. It introduces the "ladder operator" technique, which you will encounter again in many areas of quantum theory, particularly in second quantization.

The following reading provides a rigorous, step-by-step derivation.

Let's break down the critical steps of the proof presented in the text:

1. Commuting Operators: The operator ${\hat{J}}^2={\hat{J}}_x^2+{\hat{J}}_y^2+{\hat{J}}_z^2$, known as a Casimir operator, commutes with all three components: $[{\hat{J}}^2,{\hat{J}}_i]=0$. This means we can find simultaneous eigenstates of ${\hat{J}}^2$ and one component, conventionally ${\hat{J}}_z$. Let's label these states $|j,m⟩$, where:

   $${\hat{J}}^2|j,m⟩={\hslash }^2{\lambda }_j|j,m⟩\text{and}{\hat{J}}_z|j,m⟩=\hslash m|j,m⟩$$

   (The text uses ${\hslash }^{2}j(j+1)$ for the eigenvalue of ${\hat{J}}^2$ from the start; we'll see why shortly).

2. Ladder Operators: We define the non-Hermitian ladder operators ${\hat{J}}_{\pm }={\hat{J}}_x\pm i{\hat{J}}_y$. Using the fundamental commutation relations, one can show:

   $$[{\hat{J}}_z,{\hat{J}}_{\pm }]=\pm \hslash {\hat{J}}_{\pm }$$

   This implies that if $|j,m⟩$ is an eigenstate of ${\hat{J}}_z$ with eigenvalue $\hslash m$, then ${\hat{J}}_{\pm }|j,m⟩$ is an eigenstate of ${\hat{J}}_z$ with eigenvalue $\hslash (m\pm 1)$. They act as rungs on a ladder of $m$ values. Since $[{\hat{J}}^2,{\hat{J}}_{\pm }]=0$, they do not change the $j$ value.

3. Bounds on $m$: Since ${\hat{J}}^2-{\hat{J}}_z^2={\hat{J}}_x^2+{\hat{J}}_y^2$, the expectation value of the right-hand side must be non-negative. This leads to the constraint ${\lambda }_j\ge m^2$. This means for a given $j$, there must be a maximum and minimum value of $m$.

4. Terminating the Ladder: Let $m_{max}$ be the top rung. Then ${\hat{J}}_{+}|j,m_{max}⟩=0$. Let $m_{min}$ be the bottom rung. Then ${\hat{J}}_{-}|j,m_{min}⟩=0$. By acting on these states with ${\hat{J}}_{-}{\hat{J}}_{+}$ and ${\hat{J}}_{+}{\hat{J}}_{-}$ respectively, the derivation shows:

   $${\lambda }_j=m_{max}(m_{max}+1)\text{and}{\lambda }_j=m_{min}(m_{min}-1)$$

   This implies $m_{min}=-m_{max}$. Let's define $j\equiv m_{max}$, so the eigenvalue of ${\hat{J}}^2$ is indeed ${\hslash }^{2}j(j+1)$.

5. Quantization Condition: The ladder of states must go from $m_{min}=-j$ to $m_{max}=j$ in integer steps. This means the total number of steps, $m_{max}-m_{min}=2j$, must be a non-negative integer.

This final point is the punchline: $j$ must be a non-negative integer or a half-integer ($0,1/2,1,3/2,\dots$). The abstract algebra of rotations forces its quantum mechanical realization to be quantized in this specific way. For each allowed $j$, there is a multiplet of $2j+1$ states, corresponding to $m=-j,-j+1,\dots ,j$.

3. The SU(2) Group and its Connection to SO(3)

The existence of half-integer values for $j$ is mysterious from the perspective of classical rotations. A rotation of $2\pi$ brings any classical object back to its starting orientation. How can a quantum system be different? The answer lies in the group SU(2), the group of 2x2 unitary matrices with determinant +1.

The following document provides a concise and clear explanation of the relationship between SU(2) and SO(3).

Let's synthesize the main points:

• Isomorphic Lie Algebras: The generators of SU(2) are $T_i={\sigma }_i/2$, where ${\sigma }_i$ are the Pauli matrices. A direct calculation shows they satisfy $[{\sigma }_i/2,{\sigma }_j/2]=i{ϵ}_{ijk}({\sigma }_k/2)$. This is the exact same algebraic structure as the ${\hat{J}}_i$ operators (up to a factor of $\hslash$). This means SU(2) and SO(3) are locally isomorphic; they look the same for infinitesimal transformations.

• Double Cover: The relationship is not a full isomorphism. As the document shows, for any rotation in SO(3), there are two corresponding matrices in SU(2), $U$ and $-U$. We say that SU(2) is the double cover of SO(3).

• Half-Angles and Physical Rotations: An SU(2) transformation corresponding to a rotation by an angle $\varphi$ around an axis $\hat{n}$ is given by:

  $$U(\hat{n},\varphi )=\exp \left(-i\frac{\varphi }{2}\hat{n}\cdot \vec{\sigma }\right)=\cos (\varphi /2)I-i\sin (\varphi /2)(\hat{n}\cdot \vec{\sigma })$$

  Notice the $\varphi /2$! This "half-angle" is the key. A rotation by $\varphi =2\pi$ in physical space gives $U=\cos (\pi )I=-I$. The SU(2) matrix is not the identity! A full $4\pi$ rotation is needed to bring the SU(2) matrix back to the identity, $I$.

This structure perfectly explains our findings from the algebraic derivation.

• Integer $j$ representations (bosons): These correspond to single-valued representations of SO(3). A $2\pi$ rotation is the identity.
• Half-integer $j$ representations (fermions): These are not true representations of SO(3), but are single-valued representations of its double cover, SU(2). A $2\pi$ rotation results in multiplying the state by -1.

This sign flip is not just a mathematical curiosity; it has been experimentally verified in neutron interferometry experiments, where a $2\pi$ rotation of a neutron beam (a spin-1/2 particle) changes the interference pattern from constructive to destructive.

Visualizing SU(2) Rotations

The state of a spin-1/2 particle (a qubit) can be visualized as a vector pointing to a location on the surface of a sphere, known as the Bloch sphere. Rotations of the particle in physical space correspond to rotations of its state vector on this sphere. These transformations are precisely the elements of SU(2).

Conclusion

This lesson has taken us from the concrete idea of rotation to the abstract algebraic structures that govern angular momentum in the quantum world.

Key Takeaways:

• The generators of 3D rotations, ${\hat{J}}_i$, form the Lie algebra of SO(3), defined by the commutation relation $[{\hat{J}}_i,{\hat{J}}_j]=i\hslash {ϵ}_{ijk}{\hat{J}}_k$.
• This algebraic structure alone, through the ladder operator method, forces the total angular momentum quantum number $j$ to be a non-negative integer or half-integer.
• The group SU(2) shares the same Lie algebra as SO(3) but is its double cover. This means it provides a richer structure that can naturally describe both integer and half-integer angular momentum.
• The existence of half-integer spin (fermions) is a direct consequence of this SU(2) structure. A $2\pi$ rotation of a half-integer spin state results in a phase factor of -1, a purely quantum mechanical phenomenon with observable consequences.

Preview of the next lesson:
Having explored the crucial continuous symmetry of rotation, we will turn our attention to two fundamental discrete symmetries: parity (space inversion) and time reversal. We will define their corresponding quantum operators and analyze their impact on quantum systems, which has important implications, particularly in particle and condensed matter physics.