1.4 Representations and quantum mechanics

对照

The fundamental relationship between quantum mechanics and representation theory is that whenever we have a physical quantum system with a group $G$ acting on it, the space of states $\mathcal{H}$ will carry a unitary representation of $G$ (at least up to a phase factor ambiguity). For physicists working with quantum mechanics, this implies that representation theory provides information about quantum mechanical state spaces when $G$ acts on the system. For mathematicians studying representation theory, this means that physics is a very fruitful source of unitary representations to study: any physical system with a group $G$ acting on it will provide one.

For a representation $\pi$ and group elements $g$ that are close to the identity, exponentiation can be used to write $\pi (g)\in GL(n,\mathbf{C})$ as

$$\pi (g)=e^A$$

where $A$ is also a matrix, close to the zero matrix. We will study this situation in much more detail and work extensively with examples, showing in particular that if $\pi (g)$ is unitary (i.e. , in the subgroup $U(n)\subset GL(n,\mathbf{C}))$ ), then $A$ will be skew-adjoint:

$$A^{†}=-A$$

where $A^{†}$ is the conjugate-transpose matrix. Defining $B=iA$ , we find that $B$ is self-adjoint

$$B^{†}=B$$

We thus see that, at least in the case of finite dimensional $\mathcal{H}$, the unitary representation $\pi$ of $G$ on $\mathcal{H}$ coming from an action of $G$ on our physical system gives us not just unitary matrices $\pi (g)$ , but also corresponding self-adjoint operators $B$ on $\mathcal{H}.$ Lie group actions thus provide us with a class of quantum mechanical observables, with the self-adjointness property of these operators corresponding to the unitarity of the representation on state space. It is a remarkable fact that for many physical systems the class of observables that arise in this way include the ones of most physical interest.

In the following chapters we’ll see many examples of this phenomenon. A fundamental example that we will study in detail is that of action by translation in time. Here the group is $G=\mathbf{R}$ (with the additive group law) and we get a unitary representation of $\mathbf{R}$ on the space of states $\mathcal{H}$. The corresponding self-adjoint operator is the Hamiltonian operator $H$ (divided by $\hslash$) and the representation is given by

$$t\in \mathbf{R}\to \pi (t)=e^{-\frac{i}{\hslash }Ht}$$

which one can check is a group homomorphism from the additive group $\mathbf{R}$ to a group of unitary operators. This unitary representation gives the dynamics of the theory, with the Schrödinger equation 1.1 just the statement that $-\frac{i}{\hslash }H\Delta t$ is the skew-adjoint operator that gets exponentiated to give the unitary transformation that moves states $\psi (t)$ ahead in time by an amount $\Delta t$

One way to construct quantum mechanical state spaces $\mathcal{H}$ is as “wavefunctions”, meaning complex-valued functions on space-time. Given any group action on space-time, we get a representation $\pi$ on the state space $\mathcal{H}$ of such wavefunctions by the construction of equation 1.3. Many of the representations of interest will however not come from this construction, and we will begin our study of the subject in the next few chapters with such examples, which are simpler because they are finite dimensional. In later chapters we will turn to representations induced from group actions on space-time, which will be infinite dimensional.

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上一节：1.3 Unitary group representations · 下一节：1.5 Groups and symmetries

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 10、11

来源版本：2025-10-20

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