Appendix D - Group Theory

Introduction

''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''

Hermann Weyl

Symmetry arguments have been used widely in mathematics, physics, chemistry, biology, computer science, engineering, and elsewhere. Group theory can be an organizational tool which is invaluable, whether it is used explicitly or implicitly, in many areas of the sciences.

In physics, symmetry principles are often used to describe what changes and what does not in a physical system undergoing some particular transformation. For example, if a knob is turned in an experiment and nothing changes, then that is an invariant of the system and thus indicates a symmetry. (Of course, the trivial case where the knob has nothing to do with the experiment-for example the machine with the knob is unplugged-should be excluded.) The objective here is to explain group theory with this practical viewpoint in mind. and the idea is for this motivation to be kept in mind throughout these notes.

It is perhaps worth noting that very general things tend to need to be abstract. And so it with group theory. However, to reiterate, the objective here is to be as concrete as possible with the emphasis on physical applications. In this regard, it is worth mentioning that, directly or indirectly, Michael Tinkham's book on group theory very much influenced these notes.

Definitions and Examples

Definition: A group ${\mathcal {G}}\,\!$ is a set of objects, $\{A,B,C,...\}\,\!$ together with a composition rule between them, (denoted $\circ \,\!$ here and called a product or multiplication) such that the following are satisified:

1. $(A\circ B)\circ C=A\circ (B\circ C)\,\!$. ($\circ \,\!$ is associative.)
2. If $A\in {\mathcal {G}}\,\!$ and $B\in {\mathcal {G}}\,\!$, then their product $A\circ B\in {\mathcal {G}}\,\!$. (The set is closed under multiplication.)
3. There is an element $\mathbb {I} \in {\mathcal {G}}\,\!$ such that, for all $A\in {\mathcal {G}}\,\!$, $\mathbb {I} A=A=A\mathbb {I} \,\!$. (${\mathcal {G}}\,\!$ contains the identity element.)
4. For all $A\in {\mathcal {G}}\,\!$ there exists an element $A^{-1}\in {\mathcal {G}}\,\!$ such that $AA^{-1}=\mathbb {I} =A^{-1}A\,\!$.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The objective is to argue that a set of symmetries forms a group (supposing that an appropriate product rule is given) since it satisfies all the conditions in the definition.

Example 1

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at $\pm 1\,\!$ cm of the x-axis. If the line segment were rotated 180$^{o}\,\!$ about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points $\pm 1\,\!$ could be acted upon by an operator which exchanges the two. This rotation operation can be represented by multiplication by $-1\,\!$. Then, there are two elements in the set of operations to consider. The first is do nothing represented by $1\,\!$. (This, of course, is the identity operation $\mathbb {I} \,\!$ for this group.) The other element is $-1\,\!$. Thus, representing multiplication by $\circ \,\!$, we have a group with the set $\{+1,-1\}\,\!$ and operation $\circ \equiv \times \,\!$. Clearly the product is associative (it is multiplication), the set contains the identity, products are either $+1\,\!$ or $-1\,\!$ which are both in the group so it satisfies the closure property, and there are inverses, the inverse of $-1\,\!$ is $-1\,\!$. In fact this is the simplest group.

Example 2

Figure D.1

Figure D.1: An equilateral triangle with vertices in the x-y plane, $v_{1}\,\!$ at $(0,1/{\sqrt {3}})\,\!$, $v_{2}\,\!$ at $(-1/2,-1/(2{\sqrt {3}})\,\!$, and $v_{3}\,\!$ at $(1/2,-1/(2{\sqrt {3}})\,\!$.

Consider an equilateral triangle with its center at the origin of the x-y plane and vertices placed at the following points: $(0,1/{\sqrt {3}})\,\!$, $(1/2,-1/(2{\sqrt {3}})\,\!$, $(-1/2,-1/(2{\sqrt {3}})\,\!$. (See Figure D.1.) Now consider the following operations on the triangle: a rotation of $0^{o}\,\!$ (do nothing), a rotation of $120^{o}\,\!$, a rotation of $240^{o}\,\!$, and a reflection about the $x\,\!$ axis. One way to identify all possible configurations of the triangle that leave the triangle looking the same, is to identify the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise from the top, we can have $P_{0}=(1,2,3)\,\!$, $P_{2}=(3,1,2)\,\!$, $P_{4}=(2,3,1)\,\!$, $P_{1}=(1,3,2)\,\!$, $P_{3}=(3,2,1)\,\!$, $P_{5}=(2,1,3)\,\!$. If we take the first of these, $P_{0}\,\!$ to be the original configuration, shown in Fig.~(\ref{fig:triangle}), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:

	$\mathbb {I} _{2}=\left({\begin{array}{cc}1&0\\0&1\end{array}}\right),R_{1}=\left({\begin{array}{cc}-1/2&-{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;R_{2}=\left({\begin{array}{cc}-1/2&{\sqrt {3}}/2\\-{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;r=\left({\begin{array}{cc}-1&0\\0&1\end{array}}\right),\,\!$	(D.1)

where $R_{1}\,\!$ is a rotation of the x-y plane by $120^{o}\,\!$, $R_{2}\,\!$ is a rotation of the x-y plane by $240^{o}\,\!$, and $r\,\!$ is a reflection about the y-axis. In addition to these operations, two others must be included to complete the set,

	$r_{1}=rR_{1}=\left({\begin{array}{cc}1/2&{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;r_{2}=rR_{2}=\left({\begin{array}{cc}1/2&-{\sqrt {3}}/2\\-{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;\,\!$	(D.2)

where $rR_{1}\,\!$ means $r\circ R_{1}\,\!$, but the $\circ \,\!$ has been dropped since this is ordinary matrix multiplication. This group will be used as an example for several group properties and is called $S_{3}\,\!$. The products of these elements are summarized in Table D.1, which is called the multiplication table for the group. The multiplication table will be discussed repeatedly throughout this appendix due to its importance in group theory. It would be advisable to stare at it for some time to see what patterns can be identified. The meaning of these patterns will be discussed later.

Table D.1: Group Multiplication Table for $S_{3}\,\!$

$\downarrow \rightarrow \,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$r\,\!$	$r_{1}\,\!$	$r_{2}\,\!$
$\mathbb {I} _{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$r\,\!$	$r_{1}\,\!$	$r_{2}\,\!$
$R_{1}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$r_{2}\,\!$	$r\,\!$	$r_{1}\,\!$
$R_{2}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$r_{1}\,\!$	$r_{2}\,\!$	$r\,\!$
$r\,\!$	$r\,\!$	$r_{1}\,\!$	$r_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$
$r_{1}\,\!$	$r_{1}\,\!$	$r_{2}\,\!$	$r\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$
$r_{2}\,\!$	$r_{2}\,\!$	$r\,\!$	$r_{1}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$

Table D.1: Group multiplication table for the group $S_{3}\,\!$. The notation in the upper left corner ($\downarrow \rightarrow \,\!$) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.

Definition: The number of elements in a group is called the order of the group.

Example 1 has two elements and so has order two. Example 2 has six elements, and so the order of this group is six.

Definition: A group for which every element of the group commutes with every other element of the group is called abelian. If any two elements do not commute, the group is called nonabelian.

It is clear that Example 1 is an abelian group consisting of only two elements $+1\,\!$ and $-1\,\!$. However, Example 2 is clearly a nonabelian group as can be seen from the multiplication table. For example $r_{1}R_{2}=r\,\!$, but $R_{2}r_{1}=r_{2}\neq r\,\!$.

Definition: A cyclic group is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the generating element.

Example 4 provides examples of cyclic groups.

Definition: A subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ is a subset of the group elements which satisfies all the properties in the definition of a group under the inherited multiplication rule.

Example 3

Consider the set $\{0,1,2,3,\cdots ,N-1\}\,\!$ and identify $N\,\!$ and $0\,\!$. This is written as $0\equiv N\,\!$. The operation on this set will be addition. This is the group of integers modulo $N\,\!$ and is denoted $\mathbb {Z} _{N}\,\!$. To be concrete, let us consider the group $\mathbb {Z} _{3}\,\!$, consisting of $\{0,1,2;+\}\,\!$. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since $a+0=a\,\!$ for any integer $a\,\!$. Third, $1+2=3=0\,\!$ mod $3\,\!$. In other words, since $3\,\!$ and $0\,\!$ are equivalent, the sum of one and two is zero which is in the set. The order of the group is 3 (hence the subscript).

Example 1 Revisited

Recall Example 1 is a group with $\{+1,-1\}\,\!$ using multiplication. This is the simplest nontrivial cyclic group, since it is a cyclic group of order two. All elements of this group are obtained from powers of $-1\,\!$, namely $-1\,\!$ and $(-1)^{2}=1\,\!$. Notice that the generating element is special; one cannot just take any element of the group to be a generating element.

Example 4

We can represent the cyclic group of order $N\,\!$ in several ways. One is the set of elements $\{e^{2\pi in/(N-1)}\}\,\!$, $n=0,1,2,3,...,N-1\,\!$. Since this group can be seen as the consisting of the element $e^{2\pi i/(N-1)}\,\!$ and all its powers, then this is a cyclic group with generating element $e^{2\pi i/(N-1)}\,\!$.

Example 5

Include modular arithmetic under multiplication as a group.

Comparing Groups: Homomorphisms and Isomophisms

Let us consider two groups ${\mathcal {G}}_{1}\,\!$ and ${\mathcal {G}}_{2}\,\!$ with product rules symbolized by $\cdot \,\!$ and $\circ \,\!$ respectively. Let the elements of ${\mathcal {G}}_{1}\,\!$ be denoted $a_{1},a_{2},...\,\!$ and the elements of ${\mathcal {G}}_{2}\,\!$ be denoted $b_{1},b_{2},...\,\!$ When comparing two groups to see how similar they are, the relationship among the elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by $f:{\mathcal {G}}_{1}\rightarrow {\mathcal {G}}_{2}\,\!$, meaning $f(a_{1})\in {\mathcal {G}}_{2}\,\!$, then the two groups have the same (algebraic) structure if for all $a_{i},a_{j},a_{k}\in {\mathcal {G}}_{1}\,\!$,

	$a_{i}\cdot a_{j}=a_{k}\;\;\Rightarrow \;\;f(a_{i})\circ f(a_{j})=f(a_{k}).\,\!$	(D.3)

(Notice this can be true even if the map takes all of the elements $a_{i}\,\!$ to the identity.)

Definition: If the condition Eq.(D.3) is satisfied, the map is called a homomorpic map or a homomorphism.

Definition: If a homomorphism is one-to-one (each $a_{i}\,\!$ is mapped to one and only one $b_{j}\,\!$) and onto (each element in ${\mathcal {G}}_{2}\,\!$ has an element of ${\mathcal {G}}_{1}\,\!$ mapped to it), then the map is called an isomorphic map or an isomorphism.

These definitions are used repeatedly in the representation theory of groups.

Discussion

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so important to physics. Let us first discuss some of the important properties of the group multiplication table.

Group Multiplication Table

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that $ab=ba\,\!$ for abelian groups.) Another example is the presence of subgroups. This will be illustrated in this section.

Subgroups: Return to Example 2

In Example 2, Table D.1 immediately shows that the elements $\mathbb {I} ,R_{1},\,\!$ and $R_{2}\,\!$ form a subgroup since they are closed under multiplication. Another, somewhat less obvious, subgroup consists of the elements $\mathbb {I} \,\!$, and $r\,\!$. This is a convenient method for identifying subgroups, but is clearly limited to groups with a relatively small order.

The Rearrangement Theorem

Notice that each group element appears in each row and each column of Table D.1 once and only once. This is no coincidence, but is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element (due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two elements in a row that were the same, then $ac=ab\,\!$ for some $a,b,c\,\!$. But then $a^{-1}ac=a^{-1}ab\Rightarrow c=b\,\!$ which can't happen if all elements are distinct.)

Applications to Physics

These are too numerous to mention here. However, several comments come to mind. First, a general comment. If a system has a symmetry, and this can often be determined by inspection, then it has a constraint placed on it. This limits the acceptability of solutions - they must satisfy the symmetry requirement. Thus identifying symmetries is an excellent problem-solving technique. Choosing coordinates is an example of such symmetry identification.

Now, a group is a set of symmetries. To see this, suppose for example, that elements $A,B,C,D,...\,\!$ operate on an object in such a way that they do not change the object. Most often in physics the elements are matrices and the objects on which they act are vectors. If a vector, or set of vectors is unchanged by these operations, then the vectors have a symmetry described by the action of these operators. In Example 2 the vectors are the vertices of the triangle and the triangle is unchanged by their action. (Notice as an example of how a set of symmetries forms a group, that if the vector is $v\,\!$, then assuming $Av=v\,\!$, i.e. $A\,\!$ is a symmetry operation, and assuming $Bv=v\,\!$, then $ABv=v\,\!$, thus the set is close under multiplication.) One way to think of this is quite literal. If a symmetry operation is applied to the equilateral triangle and the triangle is still an equilateral triangle and the vertices are indistinguishable, then the operation did not change anything discernable.

It turns out that group theory has been applied with great success to many areas of quantum physics -- solid-state physics including crystallography, nuclear physics, atomic physics, molecular physics, and particle physics. It has also been applied in classical physics and relativity.

Some groups of infinite order such as Lie groups, were originally studied in large part in order to understand the symmetries of differential equations. This is the set of groups which is discussed next.

A Little Representation Theory

A group is specified by a set of elements, its product rule, and the relations among the elements of the group under the product rule. So the group multiplication table is how one identifies a group or shows that two groups are isomorphic. If they are isomorphic, then they are said to be the same group, even though they are represented in different ways. For our purposes, one of the most important theorems in group theory is that a set of group can always be represented by a set of matrices. This, along with ordinary matrix multiplication for the product rule provides a representation of the group. This is true for groups which have a finite order as well as infinite order (discussed later). In physics, it is most common to only discuss matrix representations of the groups.

Let us consider an example of the representation of the group from Example 2. This is a group of operations which will take any permutation of the vertices to any other permutation. This is also the set of permutations of three objects. This group is often denoted $S_{3}\,\!$. The set of matrices representing the rotations, reflection, and rotations combined with reflection provides one way of representing this group. Another way to represent this group is to use $3\times 3\,\!$ matrices rather than the $2\times 2\,\!$ matrices given in the example. Let us consider the following set of matrices.

	${\begin{aligned}\mathbb {I} _{3}=\left({\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right),\;\;\;&P_{2}=\left({\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{array}}\right),\\P_{4}=\left({\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}}\right),\;\;\;&P_{1}=\left({\begin{array}{ccc}1&0&0\\0&0&1\\0&1&0\end{array}}\right),\\P_{3}=\left({\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}}\right),\;\;\;&P_{5}=\left({\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1\end{array}}\right).\end{aligned}}\,\!$	(D.4)

Clearly when these matrices act on a column vector labelling the vertices,

	$\left({\begin{array}{c}1\\2\\3\end{array}}\right).\,\!$	(D.5)

the result is one of the permutations of three objects. These orientations correspond to the same action as the $2\times 2\,\!$ matrices given in the example above. Therefore, these two sets of matrices represent the same group, $S_{3}\,\!$. These representations are clearly different. In fact the dimension of the matrices representing the group is different for the two different representations. There are other representations that can be immediately constructed. Consider a set of matrices like the following:

	${\begin{aligned}\mathbb {I} _{5}=\left({\begin{array}{cc}\mathbb {I} _{3}&0\\0&\mathbb {I} _{2}\end{array}}\right),\;\;\;&g_{2}=\left({\begin{array}{cc}P_{2}&0\\0&R_{1}\end{array}}\right),\\g_{4}=\left({\begin{array}{cc}P_{4}&0\\0&R_{2}\end{array}}\right),\;\;\;&g_{1}=\left({\begin{array}{cc}P_{1}&0\\0&r\end{array}}\right),\\g_{3}=\left({\begin{array}{cc}P_{3}&0\\0&r_{1}\end{array}}\right),\;\;\;&g_{5}=\left({\begin{array}{cc}P_{5}&0\\0&r_{2}\end{array}}\right).\end{aligned}}\,\!$	(D.6)

This set of matrices is said to be block-diagonal since it only has non-zero elements in blocks along the diagonal. The $0\,\!$ represents a block of zeroes which is either $3\times 2\,\!$ (upper right) or $2\times 3\,\!$ (upper left). This set of matrices clearly satisfies the same relations as the sets given above ($\{\mathbb {I} _{3},P_{1},P_{2},P_{3},P_{4},P_{5}\}\,\!$ and $\{\mathbb {I} _{2},R_{1},R_{2},r,r_{1},r_{2}\}\,\!$) since the matrices multiply in blocks. Therefore this is another representation of the group $S_{3}\,\!$.

Infinite Order Groups: Lie Groups

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called Lie groups. More succinctly, a Lie group is a group which is also a differentiable manifold (see Analysis, Manifolds, and Physics, 1991).

Example 6

The Lie group most often used as the introductory example is the group consisting of the set $e^{i\theta }\,\!$, for all $\theta \,\!$. This group has an infinite number of elements (an infinite order), and has one parameter, $\theta \,\!$. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices

	$\left({\begin{array}{cc}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{array}}\right).\,\!$	(D.7)

Example 7

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, $2\times 2\,\!$ matrices, which satisify,

	$U^{\dagger }U=\mathbb {I} =UU^{\dagger }.\,\!$	(C.2)

This group is called $U(2)\,\!$. This is called the set of unitary $2\times 2\,\!$ matrices. (Hence the $U\,\!$.) Notice that the determinant of this set is $e^{i\alpha }\,\!$, where $\alpha \,\!$ is a real number, since

	$1=\det(\mathbb {I} )=\det(UU^{\dagger })=\det(U)\det(U^{\dagger })=\det(U)(\det(U))^{*}.\,\!$	(D.8)

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted $SU(2)\,\!$, and is called the special unitary group. (Hence the S.)

Example 8

One can immediately generalize the unitary and special unitary groups to $N\times N\,\!$ matrices. These are denoted $U(N)\,\!$ and $SU(N)\,\!$ respectively.

Example 9

The complex General Linear group is the set of invertible $N\times N\,\!$ matrices with complex entries. It is denoted $GL(N,\mathbb {C} )\,\!$.

More Representation Theory

There is a theorem which states that any group is isomorphic to a subgroup of the general linear group. This is a very important theorem since it means that we can always use matrices to represent a group. This is what is most often done in physics, groups are almost always represented by matrices. Therefore, here and throughout these notes, representations which are discussed will be matrix representations unless otherwise stated.