1.3 Unitary group representations

对照

The mathematical framework of quantum mechanics is closely related to what mathematicians describe as the theory of “unitary group representations”. We will be examining this notion in great detail and working through many examples in coming chapters, but here is a quick summary of the general theory.

1.3.1 Lie groups

A fundamental notion that appears throughout different fields of mathematics is that of a group:

Definition · Group

Definition (Group). A group $G$ is a set with an associative multiplication, such that the set contains an identity element, as well as the multiplicative inverse of each element.

If the set has a finite number of elements, this is called a “finite group”. The theory of these and their use in quantum mechanics is a well-developed subject, but one we mostly will bypass in favor of the study of “Lie groups”, which have an infinite number of elements. The elements of a Lie group make up a geometrical space of some dimension, and choosing local coordinates on the space, the group operations are given by differentiable functions. Most of the Lie groups we will consider are “matrix groups”, meaning subgroups of the group of $n$ by $n$ invertible matrices (with real or complex matrix entries). The group multiplication in this case is matrix multiplication. An example we will consider in great detail is the group of all rotations about a point in three dimensional space, in which case such rotations can be identified with 3 by 3 matrices, with composition of rotations corresponding to multiplication of matrices.

Digression. A standard definition of a Lie group is as a smooth manifold, with group laws given by smooth (infinitely differentiable) maps. More generally, one might consider topological manifolds and continuous maps, but this gives nothing new (by the solution to Hilbert’s Fifth problem). Most of the finite dimensional Lie groups of interest are matrix Lie groups, which can be defined as closed subgroups of the group of invertible matrices of some fixed dimension. One particular group of importance in quantum mechanics (the metaplectic group, see chapter 20) is not a matrix group, so the more general definition is needed to include this case.

1.3.2 Group representations

Groups often occur as “transformation groups”, meaning groups of elements acting as transformations of some particular geometric object. In the example mentioned above of the group of three dimensional rotations, such rotations are linear transformations of ${\mathbf{R}}^3$ . In general:

Definition · Group action on a set

Definition (Group action on a set). An action of a group $G$ on a set $M$ is given by a map

$$(g,x)\in G\times M\to g\cdot x\in M$$

that takes a pair $(g,x)$ of a group element $g\in G$ and an element $x\in M$ to another element $g\cdot x\in M$ such that

$$g_1\cdot (g_2\cdot x)=(g_1{g}_2)\cdot x\text{\hspace{1em}(1.2)}$$

and

$$e\cdot x=x$$

where $e$ is the identity element of $G$

A good example to keep in mind is that of three dimensional space $M={\mathbf{R}}^3$ with the standard inner product. This comes with two different group actions preserving the inner product

• An action of the group $G_1={\mathbf{R}}^3$ on ${\mathbf{R}}^3$ by translations.

• An action of the group $G_2=O(3)$ of three dimensional orthogonal transformations of ${\mathbf{R}}^3$ . These are the rotations about the origin (possibly combined with a reflection). Note that in this case order matters: for non-commutative groups like $O(3)$ one has $g_1{g}_2\ne g_2{g}_1$ for some group elements $g_1,g_2$

A fundamental principle of modern mathematics is that the way to understand a space $M,$ given as some set of points, is to look at $F(M)$ , the set of functions on this space. This “linearizes” the problem, since the function space is a vector space, no matter what the geometrical structure of the original set is. If the set has a finite number of elements, the function space will be a finite dimensional vector space. In general though it will be infinite dimensional and one will need to further specify the space of functions (e.g. , continuous functions, differentiable functions, functions with finite norm, etc.) under consideration.

Given a group action of $G$ on $M,$ functions on $M$ come with an action of $G$ by linear transformations, given by

$$(g\cdot f)(x)=f(g^{-1}\cdot x)\text{\hspace{1em}(1.3)}$$

where $f$ is some function on $M$ .

The order in which elements of the group act may matter, so the inverse is needed to get the group action property 1.2, since

$$\begin{array}{rl}{g}_1\cdot (g_2\cdot f)(x)& =(g_2\cdot f)(g_1^{-1}\cdot x)\\ & =f(g_2^{-1}\cdot (g_1^{-1}\cdot x))\\ & =f((g_2^{-1}{g}_1^{-1})\cdot x)\\ & =f((g_1{g}_2{)}^{-1}\cdot x)\\ & =(g_1{g}_2)\cdot f(x)\end{array}$$

This calculation would not work out properly for non-commutative $G$ if one defined $(g\cdot f)(x)=f(g\cdot x)$

One can abstract from this situation and define as follows a representation as an action of a group by linear transformations on a vector space:

Definition · Representation

Definition (Representation). A representation $(\pi ,V)$ of a group $G$ is a homomorphism

$$\pi :g\in G\to \pi (g)\in GL(V)$$

where $GL(V)$ is the group of invertible linear maps $V\to V$ , with $V$ a vector space.

Saying the map $\pi$ is a homomorphism means

$$\pi (g_1)\pi (g_2)=\pi (g_1{g}_2)$$

for all $g_1,g_2\in G$ , i.e., that it satisfies the property needed to get a group action. We will mostly be interested in the case of complex representations, where $V$ is a complex vector space, so one should assume from now on that a representation is complex unless otherwise specified (there will be cases where the representations are real).

When $V$ is finite dimensional and a basis of $V$ has been chosen, then linear maps and matrices can be identified (see the review of linear algebra in chapter 4). Such an identification provides an isomorphism

$$GL(V)\simeq GL(n,\mathbf{C})$$

of the group of invertible linear maps of $V$ with $GL(n,\mathbf{C})$ , the group of invertible $n$ by $n$ complex matrices. We will begin by studying representations that are finite dimensional and will try to make rigorous statements. Later on we will get to representations on function spaces, which are infinite dimensional, and will then often neglect rigor and analytical difficulties. Note that only in the case of $M$ a finite set of points will we get an action by finite dimensional matrices this way, since then $F(M)$ will be a finite dimensional vector space (${\mathbf{C}}^{\#\text{ of points in }M}$).

A good example to consider to understand this construction in the finite dimensional case is the following:

• Take $M$ to be a set of 3 elements $x_1,x_2,x_3$ . So $F(M)={\mathbf{C}}^3$ . For $f\in$ $F(M),f$ is a vector in ${\mathbf{C}}^3$ , with components $(f(x_1),f(x_2),f(x_3))$ .

• Take $G=S_3$ , the group of permutations of 3 elements. This group has $3!=6$ elements.

• Take $G$ to act on $M$ by permuting the 3 elements

$$(g,x_j)\to g\cdot x_j$$

• This group action provides a representation of $G$ on $F(M)$ by the linear maps

$$(\pi (g)f)(x_j)=f(g^{-1}\cdot x_j)$$

Taking the standard basis of $F(M)={\mathbf{C}}^3$ , the $j^{\prime }$ th basis element will correspond to the function $f$ that takes value 1 on $x_j$ , and 0 on the other two elements. With respect to this basis the $\pi (g)$ give six 3 by 3 complex matrices, which under multiplication of matrices satisfy the same relations as the elements of the group under group multiplication. In this particular case, all the entries of the matrix will be 0 or 1 , but that is special to the permutation representation.

A common source of confusion is that representations $(\pi ,V)$ are sometimes referred to by the map $\pi$, leaving implicit the vector space $V$ that the matrices $\pi (g)$ act on , but at other times referred to by specifying the vector space $V,$ leaving implicit the map $\pi .$ One reason for this is that the map $\pi$ may be the identity map: often $G$ is a matrix group, so a subgroup of $GL(n,\mathbf{C})$ , acting on $V\simeq {\mathbf{C}}^n$ by the standard action of matrices on vectors. One should keep in mind though that just specifying $V$ is generally not enough to specify the representation, since it may not be the standard one. For example, it could very well be the trivial representation on $V,$ where

$$\pi (g)=1_n$$

i.e., each element of $G$ acts on $V$ as the identity.

1.3.3 Unitary group representations

The most interesting classes of complex representations are often those for which the linear transformations $\pi (g)$ are “unitary”, preserving the notion of length given by the standard Hermitian inner product, and thus taking unit vectors to unit vectors. We have the definition:

Definition · Unitary representation

Definition (Unitary representation). A representation $(\pi ,V)$ on a complex vector space $V$ with Hermitian inner product $⟨\cdot ,\cdot ⟩$ is a unitary representation if it preserves the inner product, i.e.

$$⟨\pi (g)v_1,\pi (g)v_2⟩=⟨v_1,v_2⟩$$

for all $g\in G$ and $v_1,v_2\in V$

For a unitary representation, the matrices $\pi (g)$ take values in a subgroup $U(n)\subset GL(n,\mathbf{C})$ . In our review of linear algebra (chapter 4) we will see that $U(n)$ can be characterized as the group of $n$ by $n$ complex matrices $U$ such that

$$U^{-1}=U^{†}$$

where $U^{†}$ is the conjugate-transpose of $U$. Note that we’ll be using the notation “$†$” to mean the “adjoint” or conjugate-transpose matrix. This notation is pretty universal in physics, whereas mathematicians prefer to use “$\ast$” instead of “$†$”.

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