2.7 Symmetry groups

For physicists, the real interest in groups lies in their connection with the symmetries of a space of interest or some important function such as the Lagrangian. Here the concept of a space X will be taken in its broadest terms to mean a set X with a ‘structure’ imposed on it, as discussed in Section 1.6. The definitions of such spaces may involve combinations of algebraic and geometric structures, but the key thing is that their definitions invariably involve the specification of certain functions on the space. For example, algebraic structures such as groups require laws of composition, which are functions defined on cartesian products of the underlying sets. Geometric structures such as topology usually involve a selection of subsets of X – this can also be defined as a characteristic function on the power set of X. For the present purposes let us simply regard a space as being a set X together with one or more functions $F:XY$ to another set Y defined on it. This concept will be general enough to encapsulate the basic idea of a ‘space’.

If F is a Y-valued function on $X,$ we say a transformation $g:X\to X$ leaves F invariant if

$$F(x)=F(gx)\text{ for all }x\in X,$$

where, as in Section 2.6, we denote the left action by $gx\equiv g(x)$

Theorem 2.6

The set ofall transformations ofX leaving F invariantform a group.

Proof

We show the usual three things:

Closure: if g and h leave F invariant then $F(x)=F(hx)$ for all $x\in X$ and $F(y)=F(gy)$ for all $y\in X$ . Hence $gh\equiv g\circ h$ leaves F invariant since $F(ghx)=F(g(hx))=F(hx)=$ $F(x)$

Identity: obviously $F(x)=F({\mathrm{id}}_X(x))$ ; that is, ${\mathrm{id}}_X$ leaves $F$ invariant.

Inverse: if g is a transformation then there exists an inverse map $g^{-1}$ such that $g{g}^{-1}={\mathrm{id}}_X$ The map $g^{-1}$ leaves F invariant if $g$ does, since

$$F(g^{-1}x)=F(g(g^{-1}x))=F(x).$$

It is a straightforward matter to extend the above theorem to an arbitrary set $\mathcal{F}$ offunctions on X. The group oftransformations leaving all functions $F\in \mathcal{F}$ invariant will be called the invariance group or symmetry group of $\mathcal{F}$ . The following are some important examples of symmetry groups in mathematical physics.

Example 2.27

The rotation group SO(3). As in Example 2.11, let ${\mathbb{R}}^3$ be the set of all $3\times 1$ column vectors

$$\mathbf{r}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\text{ such that }x,y,z\in \mathbb{R}.$$

Consider the set ofall linear transformations $\mathbf{r}\mapsto {\mathbf{r}}^{\prime }=\mathbf{Ar}\mathsf{on}{\mathbb{R}}^3$ , where A is a $3\times 3$ matrix, which leave the distance ofpoints from the origin $r=|\mathbf{r}|=\sqrt{x^2+y^2+z^2}$ invariant. Since $r^2={\mathbf{r}}^T\mathbf{r}.$ , we have

$$r^{\prime 2}={\mathbf{r}}^{\prime T}{\mathbf{r}}^{\prime }={\mathbf{r}}^T{\mathsf{A}}^T\mathsf{Ar}=r^2={\mathbf{r}}^T\mathbf{r},$$

which holds for arbitrary vectors r if and only if A is an orthogonal matrix, $\mathsf{A}{\mathsf{A}}^T=\mathsf{I}$. As shown in Example 2.11, orthogonal transformations all have determinant $\pm 1$. Those with determinant 1 are called rotations, while transformations of determinant −1 must involve a reflection with respect to some plane; for example, the transformation $x^{\prime }=x,y^{\prime }=y,z^{\prime }=-z$.

In a similar manner $O(n)$ is the group of symmetries of the distance function in $n-$ dimensions,

$$r=\sqrt{x_1^2+x_2^2+\cdots +x_n^2}$$

and those with positive determinant are denoted $SO(n)$ , called the group of rotations in n-dimensions. There is no loss of generality in our assumption of linear transformations for this group since it can be shown that any transformation of ${\mathbb{R}}^n$ leaving $r^2$ invariant must be linear (see Chapter 18).

Example 2.28

The Euclidean group. The Euclidean space ${\mathbb{E}}^3$ is defined as the cartesian space ${\mathbb{R}}^3$ with a distance function between any pair of points given by

$$\Delta s^2={\left({\mathbf{r}}_2-{\mathbf{r}}_1\right)}^2=\Delta {\mathbf{r}}^T\Delta \mathbf{r}.$$

A transformation of ${\mathbb{E}}^3$ that leaves the distance between any pair of points invariant will be called a Euclidean transformation. As for the rotation group, a Euclidean transformation $\mathbf{r}\mapsto {\mathbf{r}}^{\prime }$ has

$$\Delta {\mathbf{r}}^{\prime }=\mathsf{A}\Delta \mathbf{r},{\mathsf{AA}}^T=\mathsf{I}.$$

For any pair of points ${\mathbf{r}}_2^{\prime }-{\mathbf{r}}_1^{\prime }=\mathsf{A}({\mathbf{r}}_2-{\mathbf{r}}_1)$ , and if we set ${\mathbf{r}}_1=0$ to be the origin and $0^{\prime }=\mathbf{a}$ , then ${\mathbf{r}}_2^{\prime }-\mathbf{a}=\mathsf{A}{\mathbf{r}}_2$ . Since ${\mathbf{r}}_2$ is an arbitrary point in ${\mathbb{E}}^3$ , the general Euclidean transformations have the form

$${\mathbf{r}}^{\prime }=\mathbf{Ar}+\mathbf{a}\text{ where }{\mathbf{A}}^T\mathbf{A}=\mathbf{I},\mathbf{a}=\text{ const. }\text{\hspace{1em}(2.18)}$$

Transformations of this form are frequently called affine or inhomogeneous linear transformations.

Exercise

Check directly that these transformations form a group – do not use Theorem 2.6.

The group ofEuclidean transformations, called the Euclidean group, can also be written as a matrix group by replacing r with the $4\times 1$ column matrix $(x,y,z,1{)}^T$ and writing

$$\left(\genfrac{}{}{0ex}{}{{\mathbf{r}}^{\prime }}{1}\right)=\left(\begin{array}{cc}\mathsf{A}& \mathbf{a}\\ 0^T& 1\end{array}\right)\left(\genfrac{}{}{0ex}{}{\mathbf{r}}{1}\right)=\left(\genfrac{}{}{0ex}{}{A\mathbf{r}+\mathbf{a}}{1}\right).$$

This may seem an odd trick, but its value lies in demonstrating that the Euclidean group is isomorphic to a matrix group – the Euclidean transformations are affine, not linear, on ${\mathbb{R}}^3$ and thus cannot be written as $3\times 3$ matrices.

Example 2.29

The Galilean group. To find the set of transformations of space and time that preserve the laws of Newtonian mechanics we follow the lead of special relativity (see Chapter 9) and define an event to be a point of ${\mathbb{R}}^4$ characterized by four coordinates $(x,y,z,t)$ . Define Galilean space ${\mathbb{G}}^4$ to be the space ofevents with a structure consisting of three elements:

1. Time intervals $\Delta t=t_2-t_1$

2. The spatial distance $\Delta s=|{\mathbf{r}}_2-{\mathbf{r}}_1|$ between any pair of simultaneous events (events having the same time coordinate, $t_1=t_2)$ .

3. Motions of inertial (free) particles, otherwise known as rectilinear motions,

$$\mathbf{r}(t)=\mathbf{u}t+{\mathbf{r}}_0,\text{\hspace{1em}(2.19)}$$

where u and ${\mathbf{r}}_0$ are arbitrary constant vectors.

Note that only the distance between simultaneous events is relevant. A simple example should make this clear. Consider a train travelling with uniform velocity v between two stations A and B. In the frame of an observer who stays at A the distance between the (non-simultaneous) events $E_1="$ ‘train leaving $A^{\prime }$ and $E_2="$ ‘train arriving at $B^{\prime }$ is clearly $d=vt$ , where t is the time of the journey. However, in the rest frame of the train it hasn’ moved at all and the distance between these two events is zero! Assuming no accelerations at the start and end of the journey, both frames are equally valid Galilean frames of reference.

Note that $\Delta t$ is a function on all of ${\mathbb{G}}^4\times {\mathbb{G}}^4$ , while $\Delta s$ is a function on the subset of ${\mathbb{G}}^4\times {\mathbb{G}}^4$ consisting ofsimultaneous pairs ofevents, $\{(\mathbf{r},t),({\mathbf{r}}^{\prime },t^{\prime }))|\Delta t=t^{\prime }-t=0\}$ . We define a Galilean transformation as a transformation $\phi :{\mathbb{G}}^4\to {\mathbb{G}}^4$ that preserves the three given structural elements. All Galilean transformations have the form

$$t^{\prime }=t+a(a=\mathrm{const}),\text{\hspace{1em}(2.20)}$$

$${\mathbf{r}}^{\prime }=\mathsf{Ar}-\mathbf{v}t+\mathbf{b}({\mathsf{A}}^T\mathsf{A}=\mathsf{I},\mathbf{v},\mathbf{b}=\text{ consts }).\text{\hspace{1em}(2.21)}$$

Proof

From the time difference equation $t^{\prime }-0^{\prime }=t-0$ we obtain Eq. (2.20) where $a=0^{\prime }$ Invariance of Property 2. gives, by a similar argument to that used to deduce Euclidean transformations.

$${\mathbf{r}}^{\prime }=\mathsf{A}(t)\mathbf{r}+\mathbf{a}(t),{\mathsf{A}}^T\mathsf{A}=\mathsf{I}\text{\hspace{1em}(2.22)}$$

where $\mathsf{A}(t)$ is a time-dependent orthogonal matrix and $\mathbf{a}(t)$ is an arbitrary vector function of time. These transformations allow for rotating and accelerating frames of reference and are certainly too general to preserve Newton’s laws.

Property 3. is essentially the invariance of Newton’s first law of motion, or equivalently Galileo’s principle of inertia. Consider a particle in uniform motion given by Eq. (2.19). Under a Galilean transformation it must become ${\mathbf{r}}^{\prime }(t^{\prime })={\mathbf{u}}^{\prime }{t}^{\prime }+{\mathbf{r}}_0^{\prime }$. Substituting $t^{\prime }=t+a$ from Eq. (2.20), the constant vector ${\mathbf{u}}^{\prime }a$ can be absorbed into the initial position. From the transformation law Eq. (2.22), using this adjusted initial position,

$${\mathbf{u}}^{\prime }t+{\mathbf{r}}_0^{\prime }=\mathsf{A}(t)(\mathbf{u}t+{\mathbf{r}}_0)+\mathbf{a}(t),$$

and taking twice time derivatives of both sides of this equation gives

$$0=(\ddot{\mathbf{A}}t+2\dot{\mathbf{A}})\mathbf{u}+\ddot{\mathbf{A}}{\mathbf{r}}_0+\ddot{\mathbf{a}}.$$

Since u and ${\mathbf{r}}_0$ are arbitrary constant vectors it follows that

$$0=\ddot{\mathbf{A}},0=\ddot{\mathbf{A}}t+2\dot{\mathbf{A}}\text{ and }0=\ddot{\mathbf{a}}.$$

Hence $\dot{\mathsf{A}}=0$ , so that A is a constant orthogonal matrix, and $\mathbf{a}=-\mathbf{v}t+\mathbf{b}$ for some constant vectors v and b. -

Exercise

Exhibit the Galilean group as a matrix group, as was done for the Euclidean group in Eq. (2.18).

Example 2.30

The Lorentz group. The Galilean transformations do not preserve the ligh cone at the origin

$$\Delta x^2+\Delta y^2+\Delta z^2=c^2\Delta t^2(\Delta x=x_2-x_1,\text{ etc. }).$$

The correct transformations that achieve this important property preserve the metric of Minkowski space,

$$\begin{array}{r}\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2\\ =\Delta {\mathbf{x}}^T\mathbf{G}\Delta \mathbf{x},\end{array}$$

where

$$\mathbf{x}=\left(\begin{array}{c}x\\ y\\ z\\ ct\end{array}\right),\Delta \mathbf{x}=\left(\begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\\ c\Delta t\end{array}\right)\text{ and }\mathsf{G}=[g_{\mu \nu }]=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right).$$

The transformations in question must have the form

$${\mathbf{x}}^{\prime }=\mathsf{Lx}+\mathbf{a},$$

and the invariance law $\Delta s^{\prime 2}=\Delta s^2$ implies

$$\Delta {\mathbf{x}}^{\prime T}\mathsf{G}\Delta {\mathbf{x}}^{\prime }=\Delta {\mathbf{x}}^T{\mathsf{L}}^T\mathsf{GL}\Delta \mathbf{x}=\Delta {\mathbf{x}}^T\mathsf{G}\Delta \mathbf{x}.$$

Since this equation holds for arbitrary x, the $4\times 4$ matrix L must satisfy the equation

$$\mathsf{G}={\mathsf{L}}^T\mathsf{GL}.\text{\hspace{1em}(2.23)}$$

The linear transformations, having $\mathbf{a}=0_{;}$ , are called Lorentz transformations while the general transformations with arbitrary a are called Poincar´e transformations. The corresponding groups are called the Lorentz group and Poincar´e group, respectively. The essence of the special theory of relativity is that all laws of physics are Poincar´e invariant.

Problems

Problem 2.20

The projective transformations of the line are defined by

$$x^{\prime }=\frac{ax+b}{cx+d}\text{ where }ad-bc=1.$$

Show that projective transformations preserve the cross-ratio

$$\frac{(x_1-x_2)(x_3-x_4)}{(x_3-x_2)(x_1-x_4)}$$

between any four points $x_1,x_2,x_3$ and $x_4.$ . Is every analytic transformation that preserves the cross ratio between any four points on the line necessarily a projective transformation? Do the projective transformations form a group?

Problem 2.21

Show that a matrix U is unitary, satisfying Eq. (2.12), if and only if it preserves the ‘norm’

$$\Vert \mathbf{z}{\Vert }^2=\sum _{i=1}^n{z}_i{\bar{z}}_i$$

defined on column vectors $(z_1,z_2,\dots ,z_n{)}^T$ in ${\mathbb{C}}^n$ . Verify that the set of $n\times n$ complex unitary matrices $U(n)$ forms a group.

Problem 2.22

Show that two rotations belong to the same conjugacy classes of the rotation group $SO(3)$ if and only if they have the same magnitude; that is, they have the same angle of rotation but possibly a different axis of rotation.

Problem 2.23

The general Galilean transformation

$$t^{\prime }=t+a,{\mathbf{r}}^{\prime }=\mathbf{Ar}-\mathbf{v}t+\mathbf{b}\text{ where }{\mathbf{A}}^T\mathbf{A}=\mathbf{I}$$

may be denoted by the abstract symbol $(a,\mathbf{v},\mathbf{b},\mathsf{A})$ . Show that the result of performing two Galilean transformations

$$G_1=(a_1,{\mathbf{v}}_1,{\mathbf{b}}_1,{\mathsf{A}}_1)\text{ and }{G}_2=(a_2,{\mathbf{v}}_2,{\mathbf{b}}_2,{\mathsf{A}}_2)$$

in succession is

$$G=G_2{G}_1=(a,\mathbf{v},\mathbf{b},\mathsf{A})$$

where

$$a=a_1+a_2,\mathbf{v}={\mathsf{A}}_2{\mathbf{v}}_1+{\mathbf{v}}_2,\mathbf{b}={\mathbf{b}}_2-a_1{\mathbf{v}}_2+{\mathsf{A}}_2{\mathbf{b}}_1\text{ and }\mathsf{A}={\mathsf{A}}_2{\mathsf{A}}_1.$$

Show from this rule of composition that the Galilean transformations form a group. In particular verify explicitly that the associative law holds.

Problem 2.24

(a) From the matrix relation defining a Lorentz transformation L,

$$\mathsf{G}={\mathsf{L}}^T\mathsf{GL},$$

where G is the $4\times 4$ diagonal matrix whose diagonal components are $(1,1,1,-1)$ ); show that Lorentz transformations form a group.

(b) Denote the Poincar´e transformation

$${\mathbf{x}}^{\prime }=\mathsf{Lx}+\mathbf{a}$$

by $\left(\mathbf{L},\mathbf{a}\right)$ , and show that two Poincar´e transformations $({\mathsf{L}}_1,\mathsf{a})$ and $({\mathsf{L}}_2,\mathsf{b})$ performed in succession i equivalent to the Poincar´e transformation

$$({\mathsf{L}}_2{\mathsf{L}}_1,\mathbf{b}+{\mathsf{L}}_2\mathbf{a}).$$

(c) From this law of composition show that the Poincar´e transformations form a group. As in the previous problem the associative law should be shown explicitly.

Problem 2.25

Let V be an abelian group with law of composition $+$, and $G$ any group with a left action on V, denoted as usual by $g:v\mapsto gv.$ . Assume further that this action is a homomorphism o $V,$ ,

$$g(v+w)=gv+gw.$$

(a) Show that $G\times V$ is a group with respect to the law of composition

$$(g,v)(g^{\prime },v^{\prime })=(g{g}^{\prime },v+g{v}^{\prime }).$$

This group is known as the semi-direct product of $G$ and $V,$ and is denoted $G\text{S◯}V$

(b) Show that the elements of type $(g,0)$ form a subgroup of $G\text{S◯}V$ that is isomorphic with G and that V is isomorphic with the subgroup $(e,V)$ . Show that the latter is a normal subgroup.

(c) Show that every element of $G\text{S◯}V$ has a unique decomposition of the form vg, where $g\equiv (g,0)\in G\mathrm{and}v\equiv (e,v)\in V.$

Problem 2.26

The following provide examples of the concept of semi-direct product defined in Problem 2.25:

(a) Show that the full Euclidean group is the semi-direct product of the orthogonal group $O(3,\mathbb{R})$ and ${\mathbb{R}}^3$, the space of column 3-vectors. Its orientation-preserving subgroup is the semi-direct product of $SO(3,\mathbb{R})$ and ${\mathbb{R}}^3$.

(b) Show that the Poincar´e group is the semi-direct product of the Lorentz group O(3, 1) and the abelian group of four-dimensional vectors ${\mathbb{R}}^4$ under vector addition (see Problem 2.24).

(c) Display the Galilean group as the semi-direct product of two groups.

Problem 2.27

The group A of affine transformations of the line consists of transformations of the form

$$x^{\prime }=ax+b,a\ne 0.$$

Show that these form a semi-direct product on $\dot{\mathbb{R}}\times \mathbb{R}$ . Although the multiplicative group of reals R˙ and the additive group R are both abelian, demonstrate that their semi-direct product is not.