2.3 The charge operator

对照

Recall from chapter 1 the claim of a general principle that, when the state space $\mathcal{H}$ is a unitary representation of a Lie group, we get an associated self-adjoint operator on $\mathcal{H}$. We’ll now illustrate this for the simple case of $G=U(1)$ , where the self-adjoint operator we construct will be called the charge operator and denoted $Q.$

If the representation of $U(1)$ on $\mathcal{H}$ is irreducible, by theorem 2.2 it must be one dimensional with $\mathcal{H}=\mathbf{C}$ . By theorem 2.3 it must be of the form $({\pi }_q,\mathbf{C})$ for some $q\in \mathbf{Z}$ . In this case the self-adjoint operator $Q$ is multiplication of elements of $\mathcal{H}$ by the integer $q.$ Note that the integrality condition on $q$ is needed because of the periodicity condition on $\theta ,$ corresponding to the fact that we are working with the group $U(1)$ , not the group $\mathbf{R}$.

For a general $U(1)$ representation, by theorems 2.1 and 2.3 we have

$$\mathcal{H}={\mathcal{H}}_{q_1}\oplus {\mathcal{H}}_{q_2}\oplus \cdots \oplus {\mathcal{H}}_{q_n}$$

for some set of integers $q_1,q_2,\dots ,q_n$ ($n$ is the dimension of $\mathcal{H},$ , the $q_j$ may not be distinct), where ${\mathcal{H}}_{q_j}$ is a copy of $\mathbf{C}$, with $U(1)$ acting by the ${\pi }_{q_j}$ representation. One can then define

Definition · Charge operator

Definition. The charge operator $Q$ for the $U(1)$ representation $(\pi ,\mathcal{H})$ is the self-adjoint linear operator on $\mathcal{H}$ that acts by multiplication by $q_j$ on the irreducible sub-representation ${\mathcal{H}}_{q_j}$ . Taking basis elements in ${\mathcal{H}}_{q_j}$ it acts on $\mathcal{H}$ as the matrix

$$Q=\left(\begin{array}{cccc}{q}_1& 0& \dots & 0\\ 0& q_2& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & q_n\end{array}\right)$$

Thinking of $\mathcal{H}$ as a quantum mechanical state space, $Q$ is our first example of a quantum mechanical observable, a self-adjoint operator on $\mathcal{H}.$ . States in the subspaces ${\mathcal{H}}_{q_j}$ will be eigenvectors for $Q$ and will have a well-defined numerical value for this observable, the integer $q_j$ . A general state will be a linear superposition of state vectors from different ${\mathcal{H}}_{q_j}$ and there will not be a well-defined numerical value for the observable $Q$ on such a state.

The representation can be recovered from the action of $Q$ on $\mathcal{H},$ with the action of the group $U(1)$ on $\mathcal{H}$ given by multiplying by $i$ and exponentiating, to get

$$\pi (e^{i\theta })=e^{iQ\theta }=\left(\begin{array}{cccc}{e}^{i{q}_1\theta }& 0& \dots & 0\\ 0& e^{i{q}_2\theta }& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & e^{i{q}_n\theta }\end{array}\right)\in U(n)\subset GL(n,\mathbf{C})$$

The standard physics terminology is that “$Q$ is the generator of the $U(1)$ action by unitary transformations on the state space $\mathcal{H}$”.

The general abstract mathematical point of view (which we will discuss in much more detail in chapter 5) is that a representation $\pi$ is a map between manifolds, from the Lie group $U(1)$ to the Lie group $GL(n,\mathbf{C})$ , that takes the identity of $U(1)$ to the identity of $GL(n,\mathbf{C})$ . As such it has a differential ${\pi }^{\prime }$ which is a linear map from the tangent space at the identity of $U(1)$ (which here is $i\mathbf{R}$) to the tangent space at the identity of $GL(n,\mathbf{C})$ (which is the space $M(n,\mathbf{C})$ of $n$ by $n$ complex matrices). The tangent space at the identity of a Lie group is called a “Lie algebra”. In later chapters we will study many different examples of such Lie algebras and such maps ${\pi }^{\prime }$ , with the linear map ${\pi }^{\prime }$ often determining the representation $\pi .$

In the $U(1)$ case, the relation between the differential of $\pi$ and the operator $Q$ is

$${\pi }^{\prime }:i\theta \in i\mathbf{R}\to {\pi }^{\prime }(i\theta )=iQ\theta$$

The following drawing illustrates the situation:

Figure 2.2: Visualizing a representation $\pi :U(1)\to U(n)$, along with its differential.

The spherical figure in the right-hand side of the picture is supposed to indicate the space $U(n)\subset GL(n,\mathbf{C})(GL(n,\mathbf{C})$ is the $n$ by $n$ complex matrices, ${\mathbf{C}}^{n^2}$ , minus the locus of matrices with zero determinant, which are those that can’t be inverted). It has a distinguished point, the identity. The representation $\pi$ takes the circle $U(1)$ to a circle inside $U(n)$ . Its derivative ${\pi }^{\prime }$ is a linear map taking the tangent space $i\mathbf{R}$ to the circle at the identity to a line in the tangent space to $U(n)$ at the identity.

In the very simple example $G=U(1)$ , this abstract picture is over-kill and likely confusing. We will see the same picture though occurring in many other much more complicated examples in later chapters. Just like in this $U(1)$ case, for finite dimensional representations the linear maps ${\pi }^{\prime }$ will be matrices, and the representation matrices $\pi$ can be found by exponentiating the ${\pi }^{\prime }$

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上一节：2.2 The group U(1) and its representations · 下一节：2.4 Conservation of charge and U(1) symmetry

附录：约定与记号

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正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 19、20、21

来源版本：2025-10-20

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