The simplest example of a Lie group is the group of rotations of the plane, with elements parametrized by a single number, the angle of rotation . It is useful to identify such group elements with unit vectors in the complex plane. The group is then denoted , since such complex numbers can be thought of as 1 by 1 unitary matrices . We will see in this chapter how the general picture described in chapter 1 works out in this simple case. State spaces will be unitary representations of the group , and we will see that any such representation decomposes into a sum of one dimensional representations. These one dimensional representations will be characterized by an integer and such integers are the eigenvalues of a self-adjoint operator we will call , which is an observable of the quantum theory.
One motivation for the notation is that this is the conventional physics notation for electric charge, and this is one of the places where a group occurs in physics. Examples of groups acting on physical systems include:
- Quantum particles can be described by a complex-valued “wavefunction” (see chapter 10), and acts on such wavefunctions by pointwise phase transformations of the value of the function. This phenomenon can be used to understand how particles interact with electromagnetic fields, and in this case the physical interpretation of the eigenvalue of the operator will be the electric charge of the state. We will discuss this in detail in chapter 45.
- If one chooses a particular direction in three dimensional space, then the group of rotations about that axis can be identified with the group The eigenvalues of will have a physical interpretation as the quantum version of angular momentum in the chosen direction. The fact that such eigenvalues are not continuous, but integral, shows that quantum angular momentum has quite different behavior than classical angular momentum.
- When we study the harmonic oscillator (chapter 22) we will find that it has a symmetry (rotations in the position-momentum plane), and that the Hamiltonian operator is a multiple of the operator for this case. This implies that the eigenvalues of the Hamiltonian (which give the energy of the system) will be integers times some fixed value. When one describes multi-particle systems in terms of quantum fields one finds a harmonic oscillator for each momentum mode, and then the for that mode counts the number of particles with that momentum.
We will sometimes refer to the operator as a “charge” operator, assigning a much more general meaning to the term than that of the specific example of electric charge. representations are also ubiquitous in mathematics, where often the integral eigenvalues of the operator will be called “weights”.
In a very real sense, the reason for the “quantum” in “quantum mechanics” is precisely because of the role of groups acting on the state space. Such an action implies observables that characterize states by an integer eigenvalue of an operator and it is this “quantization” of observables that motivates the name of the subject.
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 13、14
来源版本：2025-10-20
来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837