SU(2) and SO(3) Homomorphism

In our previous lesson, we delved into the group of 3D rotations, SO(3), and derived its Lie algebra, $so (3)$ . We made the crucial identification of this algebra with the quantum mechanical operators for angular momentum, establishing that $[J^_{i}, J^_{j}] = i ℏ ϵ_{ij k} J^_{k}$ . This connection revealed that the abstract structure of rotations is physically embodied by the conservation of angular momentum.

Today, we will turn our attention to another vital Lie group: SU(2), the group of 2x2 special unitary matrices. Our goal is to derive its Lie algebra, $su (2)$ , and then uncover its profound and subtle relationship with SO(3). This exploration is not just a mathematical exercise; it lies at the very heart of understanding intrinsic angular momentum, or spin, and the nature of particles like the electron.

1. The SU(2) Group and its Lie Algebra

The special unitary group of degree 2, denoted SU(2), is defined as the set of all 2x2 complex matrices $U$ that are unitary and have a determinant of +1.

S U (2) = {U \in M_{2} (C) ∣ U^{†} U = I, det (U) = 1}

Let's determine the general form of a matrix in SU(2). If we write a general 2x2 complex matrix as $U = (α γ β δ)$ , the conditions impose specific constraints on its elements.

Understanding Quantum Mechanical Systems with Spherical ...

This document by Greg Greif provides a concise derivation of the general form of an SU(2) matrix.

Please read the section titled Proposition 3.2. The proof shows how the unitarity and determinant conditions constrain the four complex entries to just two, related by the condition |\alpha|^2 + |\beta|^2 = 1.

As the reading shows, any matrix $U \in S U (2)$ can be written in the form:

U = (α - β^{*} β α^{*}), where α, β \in C and ∣ α ∣^{2} + ∣ β ∣^{2} = 1

The condition $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ defines the surface of a sphere in a four-dimensional space (with coordinates being the real and imaginary parts of $α$ and $β$ ). This space is known as the 3-sphere, $S^{3}$ . Topologically, SU(2) is equivalent to $S^{3}$ .

Now, let's derive the Lie algebra $su (2)$ . As before, we consider a path through the identity, $U (t) = e^{tX}$ , where $X$ is an element of the algebra.

Unitarity: $U^{†} U = I ⟹ (e^{tX})^{†} e^{tX} = I ⟹ e^{t X^{†}} e^{tX} = I$ . Differentiating at $t = 0$ gives $X^{†} + X = 0$ , so $X$ must be skew-Hermitian.
Special Determinant: $det (U) = 1 ⟹ det (e^{tX}) = e^{Tr (tX)} = 1$ . This requires $Tr (X) = 0$ , so $X$ must be traceless.

Therefore, the Lie algebra $su (2)$ is the space of all 2x2 traceless, skew-Hermitian matrices.

2. The Pauli Matrices as Generators

A familiar basis for 2x2 Hermitian matrices is the set of Pauli matrices, which you have encountered in your quantum mechanics courses. Any 2x2 traceless, skew-Hermitian matrix can be written as a linear combination of the Pauli matrices multiplied by $i$ .

The Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ form a basis for the generators of SU(2). An arbitrary SU(2) transformation can be formed by exponentiating a linear combination of these generators, representing a rotation in an abstract "spin space".

Let's quickly review the properties of the Pauli matrices.

Spin 1/2 and Pauli Matrices

This video from the channel "Let's Get Complex" provides a solid physical context for the Pauli matrices, deriving them from the spin-1/2 angular momentum operators and outlining their algebraic properties.

First, watch from this segment, which shows how the Pauli matrices arise as the matrix representation of the spin operators \hat{S}_x, \hat{S}_y, \hat{S}_z in the spin-up/spin-down basis. Then, watch the next part from here, which details their key multiplication, commutation, and anti-commutation relations.

The Pauli matrices are:

σ_{1} = σ_{x} = (0110), σ_{2} = σ_{y} = (0 i - i 0), σ_{3} = σ_{z} = (10 0 - 1)

They are Hermitian and traceless. Their fundamental commutation relation is:

[σ_{i}, σ_{j}] = 2 i k = 1 \sum 3 ϵ_{ij k} σ_{k}

Since the elements of $su (2)$ must be skew-Hermitian, a convenient basis for the algebra is $T_{k} = - \frac{i}{2} σ_{k}$ . The factor of $- \frac{i}{2}$ is a standard convention that simplifies the algebra. Let's verify:

Skew-Hermitian: $T_{k}^{†} = (- \frac{i}{2} σ_{k})^{†} = \frac{i}{2} σ_{k}^{†} = \frac{i}{2} σ_{k} = - T_{k}$ . Correct.
Traceless: $Tr (T_{k}) = - \frac{i}{2} Tr (σ_{k}) = 0$ . Correct.

Now, let's find the Lie algebra by computing the commutators of these basis generators:

[T_{i}, T_{j}] = [- \frac{i}{2} σ_{i}, - \frac{i}{2} σ_{j}] = (- \frac{i}{2})^{2} [σ_{i}, σ_{j}] = - \frac{1}{4} (2 i ϵ_{ij k} σ_{k}) = - \frac{i}{2} ϵ_{ij k} σ_{k} = k = 1 \sum 3 ϵ_{ij k} T_{k}

So, the Lie algebra $su (2)$ is defined by the commutation relation:

[T_{i}, T_{j}] = ϵ_{ij k} T_{k}

This should look strikingly familiar. It is exactly the same commutation relation we found for the basis generators $L_{k}$ of $so (3)$ in the last lesson!

3. The SU(2) to SO(3) Homomorphism

The fact that $su (2)$ and $so (3)$ have identical commutation relations means they are isomorphic as Lie algebras: $su (2) ≅ so (3)$ . This implies that the groups SU(2) and SO(3) are at least locally identical. However, their global properties are different. They are not isomorphic groups. Instead, there is a 2-to-1 surjective homomorphism from SU(2) onto SO(3). This is also described by saying SU(2) is the double cover of SO(3).

Topological Intuition

A fantastic way to build intuition for this relationship is by looking at the topology of the two groups. As we noted, SU(2) is topologically equivalent to a 3-sphere ( $S^{3}$ ). SO(3), it turns out, is equivalent to a 3-sphere with antipodal points identified, a space known as real projective 3-space ( $R P^{3}$ ).

Spinors for Beginners 10: SU(2) double covers SO(3) [ SL(2,C) double covers SO+(1,3) ]

This video by eigenchris provides an excellent topological explanation of the double cover relationship.

Watch the video from this point. It begins by stating the algebraic fact that two SU(2) matrices, +U and -U, correspond to one SO(3) rotation. It then masterfully builds intuition by exploring lower-dimensional analogies (S¹ covering RP¹, S² covering RP²) before demonstrating that SU(2) is topologically an S³ and SO(3) is an RP³.

The key insight is that if we have a transformation of the form $V^{'} = U V U^{†}$ , replacing $U$ with $- U$ gives $V^{'} = (- U) V (- U)^{†} = U V U^{†}$ . The result is identical. This means two distinct elements in SU(2), $U$ and $- U$ , perform the same function that a single element in SO(3) does. This is the essence of the 2-to-1 mapping.

Explicit Construction of the Map

We can make this mapping explicit. Any vector $x \in R^{3}$ can be bijectively mapped to a 2x2 traceless Hermitian matrix by forming the combination $X = x \cdot σ$ .

X = x_{1} σ_{1} + x_{2} σ_{2} + x_{3} σ_{3} = (x_{3} x_{1} + i x_{2} x_{1} - i x_{2} - x_{3})

A rotation of this vector in $R^{3}$ can be performed by acting on the matrix $X$ with a similarity transformation by an SU(2) matrix $U$ :

X^{'} = U X U^{†}

One can show that $X^{'}$ is also a traceless Hermitian matrix, and thus corresponds to a new vector $x^{'}$ . The map from $x$ to $x^{'}$ is a linear, norm-preserving transformation—a rotation. Therefore, for every $U \in S U (2)$ , there is a corresponding rotation matrix $R (U) \in S O (3)$ such that $x^{'} = R (U) x$ . This defines the homomorphism $Φ : S U (2) \to S O (3)$ where $Φ (U) = R (U)$ .

The following document provides a clear, step-by-step derivation of this homomorphism.

[PDF] SU(2) to SO(3) homomorphism

This paper walks through the entire derivation of the homomorphism, proving all the necessary properties along the way.

Please read the whole document. It's concise and covers all the key points. Section 1 states the goal. Section 2 sets up the mapping from \mathbb{R}^3 vectors to matrices and shows how the transformation UXU^\dagger preserves the inner product, establishing the map into O(3). Section 3 proves that the map is a homomorphism and that it lands in SO(3) (the connected component of O(3)). Section 4 is crucial: it explicitly calculates the kernel of the map, showing that U = I and U = -I are the only two SU(2) elements that map to the identity rotation in SO(3). Section 5 derives the explicit formula for the SO(3) matrix elements in terms of the SU(2) matrix.

The conclusion is that the kernel of the homomorphism is the two-element group ${I, - I}$ . This rigorously proves that the map is 2-to-1.

Physical Interpretation: The Belt Trick

This topological property has a famous physical analogy known as the "Dirac belt trick" or "plate trick". Imagine holding a plate flat on your hand. A 360° ( $2 π$ ) rotation of the plate does not return your arm to its original configuration; your arm is twisted. However, a second 360° rotation (for a total of 720° or $4 π$ ) untwists your arm and returns the system to its initial state.

The Dirac belt trick illustrates the topology of SU(2). A full $2\pi$ rotation of the end of the belt introduces a twist that cannot be removed. A second $2\pi$ rotation (total $4\pi$) allows the belt to be untangled, returning it to the original state. This is analogous to a spinor returning to its original state only after a $4\pi$ rotation.

This is precisely the behavior of spin-1/2 particles like electrons. Their state is described not by a simple vector but by a spinor, which transforms under SU(2). A $2 π$ rotation corresponds to applying the matrix $- I \in S U (2)$ , which multiplies the spinor's wavefunction by -1. Only a full $4 π$ rotation corresponds to the identity element $I \in S U (2)$ and leaves the wavefunction unchanged. This sign change is a real, measurable quantum phase.

Conclusion

In this lesson, we have uncovered a deep and fundamental connection in the theory of symmetry.

Key Takeaways:

The Lie algebra $su (2)$ consists of 2x2 traceless, skew-Hermitian matrices. Its generators can be represented by the Pauli matrices.
The Lie algebras $su (2)$ and $so (3)$ are isomorphic, sharing the commutation relation $[T_{i}, T_{j}] = ϵ_{ij k} T_{k}$ . This is why angular momentum and spin have the same algebraic structure.
The groups SU(2) and SO(3) are not isomorphic. SU(2) is the universal double cover of SO(3). There exists a 2-to-1 surjective homomorphism from SU(2) to SO(3).
Two distinct elements, $U$ and $- U$ , in SU(2) map to the same rotation in SO(3).
This property explains the behavior of spin-1/2 particles (spinors), whose wavefunctions acquire a minus sign under a $2 π$ rotation and are restored only after a $4 π$ rotation.

We have seen that while $su (2)$ and $so (3)$ are identical at the infinitesimal level, their global structures are different. This difference has profound physical consequences.

In our next lesson, we will explore a powerful tool for working with Lie algebras: the Baker-Campbell-Hausdorff formula. This formula provides a way to express the product of two group elements, $e^{X} e^{Y}$ , in terms of a single exponential, $e^{Z}$ , where $Z$ is given by a series of commutators of $X$ and $Y$ . This will give us a concrete way to compose transformations directly at the algebra level.

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