4.6 Orthogonal and unitary transformations

对照

A special class of linear transformations will be invertible transformations that preserve the inner product, i.e., satisfying

$$⟨Lv,Lw⟩=⟨v,w⟩$$

for all $v,w\in V$ . Such transformations take orthonormal bases to orthonormal bases, so one role in which they appear is as a change of basis between two orthonormal bases.

In terms of adjoints, this condition becomes

$$⟨Lv,Lw⟩=⟨v,L^{†}Lw⟩=⟨v,w⟩$$

so

$$L^{†}L=1$$

or equivalently

$$L^{†}=L^{-1}$$

In matrix notation this first condition becomes

$$\sum _{k=1}^n(L^{†}{)}_{jk}{L}_{kl}=\sum _{k=1}^n\overline{L_{kj}}{L}_{kl}={\delta }_{jl}$$

which says that the column vectors of the matrix for $L$ are orthonormal vectors. Using instead the equivalent condition

$$L{L}^{†}=1$$

we find that the row vectors of the matrix for $L$ are also orthonormal. Since such linear transformations preserving the inner product can be composed and are invertible, they form a group, and some of the basic examples of Lie groups are given by these groups for the cases of real and complex vector spaces.

4.6.1 Orthogonal groups

We’ll begin with the real case, where these groups are called orthogonal groups:

Definition · Orthogonal group

Definition (Orthogonal group). The orthogonal group $O(n)$ in $n$ dimensions is the group of invertible transformations preserving an inner product on a real $n$ dimensional vector space $V$ . This is isomorphic to the group of $n$ by $n$ real invertible matrices $L$ satisfying

$$L^{-1}=L^T$$

The subgroup of $O(n)$ of matrices with determinant 1 (equivalently, the subgroup preserving orientation of orthonormal bases) is called $SO(n)$.

Recall that for a representation $\pi$ of a group $G$ on $V$, there is a dual representation on $V^{\ast }$ given by taking the transpose-inverse of $\pi$. If $G$ is an orthogonal group, then $\pi$ and its dual are the same matrices, with $V$ identified by $V^{\ast }$ by the inner product.

Since the determinant of the transpose of a matrix is the same as the determinant of the matrix, we have

$$L^{-1}L=1⟹\det (L^{-1})\det (L)=\det (L^T)\det (L)=(\det (L){)}^2=1$$

so

$$\det (L)=\pm 1$$

$O(n)$ is a continuous Lie group, with two components distinguished by the sign of the determinant: $SO(n)$ , the subgroup of orientation-preserving transformations, which include the identity, and a component of orientation-changing transformations.

The simplest non-trivial example is for $n=2$ , where all elements of $SO(2)$ are given by matrices of the form

$$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

These matrices give counter-clockwise rotations in ${\mathbf{R}}^2$ by an angle $\theta$. The other component of $O(2)$ will be given by matrices of the form

$$\left(\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right)$$

which describe a reflection followed by a rotation. Note that the group $SO(2)$ is isomorphic to the group $U(1)$ by

$$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\leftrightarrow e^{i\theta }$$

so the representation theory of $SO(2)$ is just as for $U(1)$ , with irreducible complex representations one dimensional and classified by an integer.

In chapter 6 we will consider in detail the case of $SO(3)$ , which is crucial for physical applications because it is the group of rotations in the physical three dimensional space.

4.6.2 Unitary groups

In the complex case, groups of invertible transformations preserving the Hermitian inner product are called unitary groups:

Definition · Unitary group

Definition (Unitary group). The unitary group $U(n)$ in $n$ dimensions is the group of invertible transformations preserving a Hermitian inner product on a complex $n$ dimensional vector space $V$ . This is isomorphic to the group of $n$ by $n$ complex invertible matrices satisfying

$$L^{-1}={\overline{L}}^T=L^{†}$$

The subgroup of $U(n)$ of matrices with determinant 1 is called $SU(n)$.

In the unitary case, the dual of a representation $\pi$ has representation matrices that are transpose-inverses of those for $\pi ,$ but

$$(\pi (g{)}^T{)}^{-1}=\overline{\pi (g)}$$

so the dual representation is given by conjugating all elements of the matrix.

The same calculation as in the real case here gives

$$\det (L^{-1})\det (L)=\det (L^{†})\det (L)=\overline{\det (L)}\det (L)=|\det (L){|}^2=1$$

so $\det (L)$ is a complex number of modulus one. The map

$$L\in U(n)\to \det (L)\in U(1)$$

is a group homomorphism.

We have already seen the examples $U(1),U(2)$ and $SU(2)$ . For general values of $n,$ the study of $U(n)$ can be split into that of its determinant, which lies in $U(1)$ so is easy to deal with, followed by the subgroup $SU(n)$ , which is a much more complicated story.

Digression. Note that it is not quite true that the group $U(n)$ is the product group $SU(n)\times U(1)$ . If one tries to identify the $U(1)$ as the subgroup of $U(n)$ of elements of the form $e^{i\theta }1$ , then matrices of the form

$$e^{i\frac{m}{n}2\pi }1$$

for $m$ an integer will lie in both $SU(n)$ and $U(1)$ , so $U(n)$ is not a product of those two groups (it is an example of a semi-direct product, these will be discussed in chapter 18).

We saw at the end of section 3.1.2 that $SU(2)$ can be identified with the three-sphere $S^3$ , since an arbitrary group element can be constructed by specifying one row (or one column), which must be a vector of length one in ${\mathbf{C}}^2$ . For the case $n=3$ , the same sort of construction starts by picking a row of length one in ${\mathbf{C}}^3$, which will be a point in $S^5$ . The second row must be orthonormal, and it can be shown that the possibilities lie in a three-sphere $S^3$ . Once the first two rows are specified, the third row is uniquely determined. So as a manifold, $SU(3)$ is eight dimensional, and one might think it could be identified with $S^5\times S^3$ . It turns out that this is not the case, since the $S^3$ varies in a topologically non-trivial way as one varies the point in $S^5$ . As spaces, the $SU(n)$ are topologically “twisted” products of odd dimensional spheres, providing some of the basic examples of quite non-trivial topological manifolds.

---

上一节：4.5 Adjoint operators · 下一节：4.7 Eigenvalues and eigenvectors

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 41、42、43、44

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837