2.5 Summary

对照

To summarize the situation for $G=U(1)$ , we have found

• Irreducible representations $\pi$ are one dimensional and characterized by their derivative ${\pi }^{\prime }$ at the identity. If $G=\mathbf{R}$, ${\pi }^{\prime }$ could be any complex number. If $G=U(1)$ , periodicity requires that ${\pi }^{\prime }$ must be $iq,q\in \mathbf{Z}$ , so irreducible representations are labeled by an integer.

• An arbitrary representation $\pi$ of $U(1)$ is of the form

$$\pi (e^{i\theta })=e^{i\theta Q}$$

where $Q$ is a matrix with eigenvalues a set of integers $q_j$ . For a quantum system, $Q$ is the self-adjoint observable corresponding to the $U(1)$ group action on the system, and is said to be a “generator” of the group action.

• If $[Q,H]=0$ , the $U(1)$ group acts on the state space as “symmetries”. In this case the $q_j$ will be “conserved quantities”, numbers that characterize the quantum states, and do not change as the states evolve in time.

Note that we have so far restricted attention to finite dimensional representations. In section 11.1 we will consider an important infinite dimensional case, a representation on functions on the circle which is essentially the theory of Fourier series. This comes from the action of $U(1)$ on the circle by rotations, giving an induced representation on functions by equation 1.3.

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上一节：2.4 Conservation of charge and U(1) symmetry · 下一节：2.6 For further reading

附录：约定与记号

来源与版本

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 23

来源版本：2025-10-20

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