2.2 The group U(1) and its representations

对照

One might think that the simplest Lie group is the one dimensional additive group $\mathbf{R}$, a group that we will study together with its representations beginning in chapter 10. It turns out that one gets a much easier to analyze Lie group by adding a periodicity condition (which removes the problem of what happens as you go to $\pm \infty$), getting the “circle group” of points on a unit circle. Each such point is characterized by an angle, and the group law is addition of angles.

The circle group can be identified with the group of rotations of the plane ${\mathbf{R}}^2$ , in which case it is called $SO(2)$ , for reasons discussed in chapter 4. It is quite convenient however to identify ${\mathbf{R}}^2$ with the complex plane $\mathbf{C}$ and work with the following group (which is isomorphic to $SO(2))$ ):

Definition · The group U(1)

Definition (The group $U(1)$). The elements of the group $U(1)$ are points on the unit circle, which can be labeled by a unit complex number $e^{i\theta }$, or an angle $\theta \in \mathbf{R}$ with $\theta$ and $\theta +N2\pi$ labeling the same group element for $N\in \mathbf{Z}$. Multiplication of group elements is complex multiplication, which by the properties of the exponential satisfies

$$e^{i{\theta }_1}{e}^{i{\theta }_2}=e^{i({\theta }_1+{\theta }_2)}$$

so in terms of angles the group law is addition (mod 2$\pi$).

The name “$U(1)$” is used since complex numbers $e^{i\theta }$ are 1 by 1 unitary matrices.

Figure 2.1: $U(1)$ viewed as the unit circle in the complex plane $\mathbf{C}$.

By theorem 2.2, since $U(1)$ is a commutative group, all irreducible representations will be one dimensional. Such an irreducible representation will be given by a differentiable map

$$\pi :U(1)\to GL(1,\mathbf{C})$$

$GL(1,\mathbf{C})$ is the group of invertible complex numbers, also called ${\mathbf{C}}^{\ast }$ . A differentiable map $\pi$ that is a representation of $U(1)$ must satisfy homomorphism and periodicity properties which can be used to show:

Theorem

Theorem 2.3. All irreducible representations of the group $U(1)$ are unitary, and given by

$${\pi }_k:e^{i\theta }\in U(1)\to {\pi }_k(\theta )=e^{ik\theta }\in U(1)\subset GL(1,\mathbf{C})\simeq {\mathbf{C}}^{\ast }$$

for $k\in \mathbf{Z}.$

Proof

Proof. We will write the ${\pi }_k$ as a function of an angle $\theta \in \mathbf{R}$ , so satisfying the periodicity property

$${\pi }_k(2\pi )={\pi }_k(0)=1$$

Since it is a representation, $\pi$ will satisfy the homomorphism property

$${\pi }_k({\theta }_1+{\theta }_2)={\pi }_k({\theta }_1){\pi }_k({\theta }_2)$$

We need to show that any differentiable map

$$f:U(1)\to {\mathbf{C}}^{\ast }$$

satisfying the homomorphism and periodicity properties is of the form $f={\pi }_k$. Computing the derivative $f^{\prime }(\theta )=\frac{df}{d\theta }$ we find

$$\begin{array}{rl}{f}^{\prime }(\theta )& ={\lim }_{\Delta \theta \to 0}\frac{f(\theta +\Delta \theta )-f(\theta )}{\Delta \theta }\\ & =f(\theta ){\lim }_{\Delta \theta \to 0}\frac{(f(\Delta \theta )-1)}{\Delta \theta }\text{(using the homomorphism property)}\\ & =f(\theta )f^{\prime }(0)\end{array}$$

Denoting the constant $f^{\prime }(0)$ by $c$, the only solutions to this differential equation satisfying $f(0)=1$ are

$$f(\theta )=e^{c\theta }$$

Requiring periodicity we find

$$f(2\pi )=e^{c2\pi }=f(0)=1$$

which implies $c=ik$ for $k\in \mathbf{Z}$ , and $f={\pi }_k$ for some integer $k$. □

The representations we have found are all unitary, with ${\pi }_k$ taking values in $U(1)\subset {\mathbf{C}}^{\ast }$ . The complex numbers $e^{ik\theta }$ satisfy the condition to be a unitary 1 by 1 matrix, since

$$(e^{ik\theta }{)}^{-1}=e^{-ik\theta }=\overline{e^{ik\theta }}$$

These representations are restrictions to the unit circle $U(1)$ of irreducible representations of the group ${\mathbf{C}}^{\ast }$ , which are given by

$${\pi }_k:z\in {\mathbf{C}}^{\ast }\to {\pi }_k(z)=z^k\in {\mathbf{C}}^{\ast }$$

Such representations are not unitary, but they have an extremely simple form, so it sometimes is convenient to work with them, later restricting to the unit circle, where the representation is unitary.

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上一节：2.1 Some representation theory · 下一节：2.3 The charge operator

附录：约定与记号

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正文：英文 · 原著转录

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原书 PDF · 印刷页 17、18、19

来源版本：2025-10-20

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