The following derivation is the outcome of my own personal journey in trying to learn more group theory (and representation theory) which appears in many areas of physics. Despite the importance of groups in physics, I don’t believe physicists are well-educated at the university level on this topic (or at least I wasn’t) unless they go out of their way.

All too often physics texts throw out statements like “spin \(1/2\) particles transform under elements of the \(\text{SU}(2)\) group” without taking the time to establish what any of this machinary means. Mathematics texts, on the other hand, are often written with a level of rigor and language that can be opaque to the average physicist. As a result, there’s a plethora of books out there with titles like “Group Theory for Physicists” to varying levels of usefulness.

So the purpose of this post is to use a simple proof about the properties of the \(\text{SU}(2)\) group as a Trojan horse to introduce concepts in group theory. I won’t be starting from total scratch however, so some familiarity with mathematical notation (just undergraduate level?) will be needed to decipher what comes next. With that said, I’ll try and stop to define vocabulary as it arises using footnotes.

What is compactness?

Compactness of a group conveys that the set is “small” and bounded to some region of a larger space. In Euclidean space \(\mathbb{R}^n\), the Heine-Borel theorem tells us that a set is compact if and only if it is closed and bounded. Therefore we need to verify two things about \(\text{SU}(2)\):

  1. Closed: A set is closed if it contains all its limit points.
  2. Bounded: All elements in the set are within some finite distance of a fixed point.

If you’ve learned any quantum mechanics, then you’ll know that \(\text{SU}(2)\) defines matrices which are used to rotate the two-component spinors \(\chi=(\chi_{1},\chi_{2})^{T}\) for spin \(1/2\) particles such as the electron. If we consider Euclidean rotations1 in \(\mathbb{R}^3\), we intuitively understand that rotations are compact compared to translations which can move vectors arbitrary distances. Therefore, we expect such rotation in spinor-space to also be “compact” as described above. Let’s more closely look at how \(\text{SU}(2)\) is defined.

What is \(\text{SU}(2)\)?

The special unitary group (of degree 2)2 \(\text{SU}(2)\) is defined as the set of all \(2 \times 2\) unitary matrices \(U\) with determinant equal to 1:

\[\text{SU}(2) = \{ U \in \text{GL}(2, \mathbb{C}) \mid U^\dagger U = I \text{ and } \det(U) = 1 \}.\]

Let’s break down the properties outlined above:

  • \(\text{GL}(2, \mathbb{C})\) is the general linear group of degree 2 which is all \(2\times2\) complex matrices \(U\) which are invertible \(U^{-1}U=I\).
  • Unitarity (\(U^\dagger U = I\))3 means \(U\) preserves inner products and thus is an isometry4, which implies its columns are orthonormal5.
  • \(\det(U) = I\) further restricts \(U\) to matrices that can be continuously deformed6 to the identity matrix within the set of unitary matrices.

To show that \(\text{SU}(2)\) is a compact Lie group, we need to demonstrate that:

  1. \(\text{SU}(2)\) is a Lie group.
  2. \(\text{SU}(2)\) is compact.

\(\text{SU}(2)\) as a Lie group

A Lie group is a group that is also a smooth manifold7. The structure of \(\text{SU}(2)\) makes it a subgroup of \(\text{GL}(2, \mathbb{C})\), the group of invertible \(2 \times 2\) complex matrices. Since the unitary matrices \(\text{U}(n)\) form a smooth manifold, and the condition \(\det(U) = 1\) is a smooth constraint8, \(\text{SU}(2)\) inherits a smooth manifold structure, making it a Lie group. This is a very abbreviated description of Lie groups, but ultimately we want a group whose elements describe a geometry we can do calculus on.

\(\text{SU}(2)\) is compact

To show compactness, we can parametrize \(\text{SU}(2)\) explicitly. Any \(U \in \text{SU}(2)\) can be written as

\[U = \begin{pmatrix} \alpha & -\beta^* \\ \beta & \alpha^* \end{pmatrix},\]

where \(\alpha, \beta \in \mathbb{C}\) and \(\alpha\alpha^{*} + \beta\beta^{*} = 1\) (a consequence of unitarity and the determinant condition). This parametrization reveals that \(U\) depends on four \(n^2 = 2^2 = 4\) real parameters given by the real and imaginary parts of \(\alpha\) and \(\beta\). Letting \(\alpha = x_1 + i x_2\) and \(\beta = x_3 + i x_4\) where \(x_1, x_2, x_3, x_4 \in \mathbb{R}\), we can write

\[|\alpha|^2 = x_1 ^2 + x_2 ^2\ \mathrm{and}\ |\beta|^2 = x_3 ^2 + x_4 ^2,\]

allowing us recast the constraint as

\[x_1 ^2 + x_2 ^2 + x_3 ^2 + x_4 ^2 = 1,\]

which is the 3-sphere in \(\mathbb{R}^4\). Therefore, \(\text{SU}(2)\) is diffeomorphic9 to the 3-dimensional sphere \(S^3\) in \(\mathbb{R}^4\). Since \(S^3\) is a compact manifold, it follows that \(\text{SU}(2)\) is also compact.

How do we know the 3-sphere manifold \(S^3\) is compact?

To see why the 3-sphere \(S^3\) is compact, let’s go through a couple of fundamental points about compactness and spheres in general10. The 3-sphere \(S^3\) in \(\mathbb{R}^4\) is defined as the set of points in \(\mathbb{R}^4\) at a fixed distance (radius) from the origin:

\[S^3 = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1 \}.\]

This is the 3-dimensional analog of the 2-sphere \(S^2\) in \(\mathbb{R}^3\). To show that \(S^3\) is compact, we need to verify as discussed earlier according to the Heine-Borel theorem:

  1. Closedness: \(S^3\) is a closed subset of \(\mathbb{R}^4\).
  2. Boundedness: \(S^3\) is bounded in \(\mathbb{R}^4\).

\(S^3\) is closed

The 3-sphere is defined by the equation \(x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1\), which represents a continuous function \(f:\mathbb{R}^4\rightarrow\mathbb{R}\) defined by

\[f(x_1, x_2, x_3, x_4) = x_1^2 + x_2^2 + x_3^2 + x_4^2.\]

Since this function is continuous and \(S^3\) is the preimage11 of the closed set \(\{1\}\) under \(f\)

\[S^3 = f^{-1}(\{1\}),\]

therefore \(S^3\) is closed in \(\mathbb{R}^4\). I hope it is easy to see why \(\{1\}\) is a closed set! Due to the continuous nature of \(f\), that closed property of \(\{1\}\) transfers to \(S^3\).

\(S^3\) is bounded

Since every point \((x_1, x_2, x_3, x_4) \in S^3\) satisfies \(x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1\), all points on \(S^3\) are within a distance of 1 from the origin. Therefore, \(S^3\) is bounded in \(\mathbb{R}^4\).

Thus, \(\text{SU}(2)\) is compact. QED.

Further discussion of this blog post can be found on reddit and tildes.

Footnotes

  1. It turns out that \(\text{SU}(2)\) and the Euclidean 3-rotation group \(\text{SO}(3)\) are related via what we call “double cover.” This means that every element of \(\text{SO}(3)\) corresponds to two elements of \(\text{SU}(2)\). ↩

  2. The degree of a group tells us the size of its matrices. E.g. \(\text{SU}(n)\) can be represented by \(n\times n\) matrices. A specific element of the group however has \(d=n^2 - 1\) dimensions or degrees of freedom. This makes \(\text{SU}(2)\) have dimension \(d = 2^2 - 1 = 3\) corresponding to three real parameters \(\mathbb{R}^3\). ↩

  3. The dagger notation \(U^\dagger\) represents the complex conjugate transpose \(U^\dagger = (U^*)^T\). For unitary matrices, the conjugate transpose is also the inverse \(U^\dagger = U^{-1}\). ↩

  4. An isometry is an operation \(U\) which preserves distances and angles in a vector space. If we consider a pair of vectors \(\mathbf{u}\) and \(\mathbf{v}\), the inner product is then \(\mathbf{u}^\dagger\mathbf{v} = \sum_{k}^{N}u_{k}^{*}v_{k}\). The operation \(\mathbf{v}\rightarrow U\mathbf{v}\) therefore preserves the inner product as \(\mathbf{u}^\dagger U^\dagger U\mathbf{v} = \mathbf{u}^\dagger\mathbf{v}\). ↩

  5. If we denote the columns of \(U\) as \(\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_n\), then \(U^\dagger U = I\) means that \(U^\dagger\) times \(U\) gives the identity matrix: \((U^\dagger U)_{ij} = \mathbf{u}_i^\dagger \mathbf{u}_j = \delta_{ij}\). ↩

  6. A complex \(2\times2\) unitary matrix can be parameterized by two complex phases \((\phi,\psi)\) and an angle \(\theta\). For a suitable choice of complex phases and angle, this can be continuously deformed to the identity matrix within the group of unitary matrices. This property is also known as path-connectedness. ↩

  7. A smooth manifold (the underlying geometry) has the following properties: (a) We can assign coordinates (e.g. charts \((x,y,z)\)) that define the space, (b) it locally resembles a Euclidean space \(\mathbb{R}^{n}\), and (c) it is smooth letting us do calculus and differentiate. ↩

  8. The determinant constraint doesn’t stop us from doing calculus. ↩

  9. Diffeomorphic means we can write a (bijective) map \(f\) between two manifolds \(f:M\rightarrow N\) such that \(f\) is smooth (preserves calculus) and smoothly invertible \(f^{-1}\). This implies \(M\) and \(N\) are smoothly equivalent. ↩

  10. An n-sphere is defined by the condition \(\sum_i ^N x_i ^2 = 1\) on \(\mathbb{R}^n\). ↩

  11. The preimage \(f^{-1}(\{1\})\) refers to all points in \(\mathbb{R}^4\) that \(f\) maps to 1. It’s the set of solutions to \(x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1\), which defines \(S^3\). ↩