Version 1

Group Theory and the Quark Model

Milind V. Purohit (U. of South Carolina)

Abstract

Contents

1 Introduction 2

2 Symmetries and Conservation Laws 2

3 Introduction. Finite Groups 43.1 Subgroups, Cosets, Classes and Direct Products . . . . . . . . . . . . . . . . 43.2 Representations of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 The Symmetric Group Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3.1 S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3.2 S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4 The regular representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Reducible and Irreducible representations 84.1 Reduction of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Reduction of Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Introduction to Young Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Infinite groups and Lie groups 155.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 The Special Unitary Groups: SU(N) 166.1 Generators and algebra of SU(N) . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Roots and Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 Young tableaux in the context of SU(N) . . . . . . . . . . . . . . . . . . . . 20

6.4.1 Dimensionality of Representations . . . . . . . . . . . . . . . . . . . . 216.4.2 Reduction of direct products of representations . . . . . . . . . . . . 216.4.3 Basis Vectors of Representations . . . . . . . . . . . . . . . . . . . . . 22

7 The Quark Model 22

8 Conclusion and References 22

9 References 23

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1 Introduction

Our interest in group theory stems from its applications to particle physics, which are many.Fundamentally, when a group of transformation operators commutes with the Hamiltonian,the theory of the group they form can be brought to bear on the energy eigenstates. Grouptheory can then be used to classify these states, to specify their wavefunctions, to tell ushow to combine such states (generalization of the addition of angular momenta), to workout selection rules for matrix elements and so on.

Perhaps even more significant than this practical, calculational aspect is the central rolethat group theory plays in formulating the fundamental theories of electromagnetic, weak,strong and possible new interactions. This is the saga of gauge theories which has truly beenthe big leap in physics after relativity, quantum mechanics and field theory. For most of ourneeds the symmetry groups we shall require will be the infinite groups SU(N), which alsoserve as a model in other cases.

The groups SU(N) will be applied to angular momentum, isospin, and mainly to thequark model in this chapter. However, we will spend enough time on group theory just sothe reader is prepared to appreciate applications in the context of the standard model inparticular, and to theory-building in general.

Since symmetry operations typically satisfy the requirements of a “group” the vast math-ematical apparatus of group theory can be brought to bear on physical problems exhibitingsome kind of symmetry. For instance, we can use the properties of SU(2) to label states,to determine the multiplicity of “representations” (see below), to obtain wavefunctions andtheir symmetry properties etc. Lagrangians for theories are often built with a specific groupat their core describing the desired invariances of the physical theory under certain transfor-mations. We will study the bare minimum of group theory necessary for our purposes. Inparticular, we will focus on the finite group Sn and the infinite groups SU(N).

2 Symmetries and Conservation Laws

Symmetry considerations help us to think about and to classify properties of nature. Often,we have to construct laws of nature and an observed or expected symmetry guides us tothe correct form of the law. An example is finding the force on a charged particle due to acharged sphere. In solving such problems, we find that symmetries lead to conservation laws.Noether’s theorem (1917) shows how symmetries of nature lead to conservation laws andvice versa. For instance, translational symmetry leads to conservation of linear momentum,rotational symmetry leads to conservation of angular momentum, time evolution invarianceleads to conservation of energy and gauge invariance leads to conservation of various charges.Classically, of course, the Euler-Lagrange equations lead directly to conservation laws:

d

dt

(

∂L

∂q

)

=∂L

∂q= 0 if L does not depend on q

In quantum mechanics, we use the Schrodingerequation of motion

ih∂Ψ

∂t= HΨ

2

to show that if an operator A does not explicitly depend on time then it is a constant ofmotion if it commutes with the Hamiltonian operator H, i.e., that

d

dt〈A〉 =

i

h〈Ψ|[H,A]|Ψ〉

Even if we did know the detailed laws of nature, the underlying symmetries help us tothink about the resulting theory. We may get partial insight into problems without knowingthe full dynamics. Understanding atomic spectra is an example, as is the Wigner-Eckarttheorem. Recall that the Wigner-Eckart theorem allows us to factorize a matrix elementof a tensor operator into a Clebsch-Gordan coefficient and a reduced matrix element. Thereduced matrix element does not depend on the magnetic quantum numbers which specifythe orientation of a system in space, but only on the total angular momentum and other,non-geometric quantum numbers.

Classically, an example of symmetry operations is provided by rotations. If a classicalobject is located at some position ~x, then a rotation leaves the object in a new position ~x′

where~x′ = Rx

and R is the rotation matrix, an orthogonal 3 x 3 matrix. In quantum mechanics, allinformation about a quantum is incorporated in its state vector |Ψ〉. A rotation or othersymmetry operation is then carried out by operating on |Ψ〉 with an operator, let’s call itO, as shown:

|Ψ〉 → |Ψ′〉 = O|Ψ〉If we require that the scalar product of two state vectors remain invariant under these

operations it immediately follows that the operators are unitary, i.e.,

〈Ψ′|Φ′〉 = 〈Ψ|Φ〉implies that O†O = I where I is the identity operator. However, if we require only thatthe physically observable squares of matrix elements be invariant, i.e., only that

|〈Ψ′|Φ′〉|2 = |〈Ψ|Φ〉|2

then we find that the operator can also be antiunitary, in the sense that it is antilinear.Recall that an antilinear operator T has the property that

T a|Ψ〉 = a∗ T |Ψ〉

where a is any complex number.Further, Wigner showed that any such mapping |Ψ〉 → |Ψ′〉 which preserves proba-

bilities is rephase-invariant, i.e., that the phases of state vectors can be arbitrarily changedand retain the linear addition of states. Thus we shall mainly be concerned with unitary oranti-unitary operators, and most often with elements of the “Special Unitary” group wherethe overall phase is ignored.

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3 Introduction. Finite Groups

A group is defined as a set of quantities (or operations) a, b, c, ... for which “multiplication”is defined and which have the following four properties:

• Closure under multiplication: the multiplication operation ab must result in anothergroup element c, i.e., c must be a member of the group. This means that the “multi-plication table” consists entirely of group elements.

• Associativity under group multiplication: As in the usual sense, i.e., the product a(bc)must equal the product (ab)c.

• Identity: A unique element I exists with the property

Ia = aI = a

for any element a in the group.

• Inverse: Every element a has an inverse a−1 which is also an element of the group suchthat

aa−1 = a−1a = I

For instance, the natural numbers 1, 2, 3, . . . do not form a group under addition becausethere is no identity element (zero is not a natural number) and there is no inverse. However,when we consider the set of integers (which include zero and negative numbers) they do forma group under addition. Another really simple example of a group is provided by the set oftwo operations “Identity” (or no-op) and “Mirror Reflection”.

If the group has a finite number of elements it is called a “finite” group, otherwise an“infinite” group. The number of elements of a finite group is called the “order” of the group.

The laws of commutativity may or may not hold, i.e., it may or may not be true thatgroup multiplication is commutative. If the commutation relation holds, the group is calledAbelian. For instance, the group R(2) which describes rotations in two-dimensional space isAbelian (convince yourself of this). On the other hand, R(3), the group describing rotationsin three-dimensional space, is non-Abelian.

Sometimes a group is extended to form an algebra, which means that the elements forma linear vector space. Thus, we can also add elements to each other, not just multiply them.The addition operation leads to a zero for the algebra, but the zero of the algebra does nothave an inverse.

3.1 Subgroups, Cosets, Classes and Direct Products

It may happen that a subset of the elements of a group G themselves form a group. In thatcase the subset is called a subgroup. If g is an element of a group and S is a subset of thegroup of order hS, then the set obtained by multiplying each element of S by g on the leftis called the left coset gS of the group. Similarly, we can define the right coset Sg. Clearly,the cosets have the same number of elements hS as the subgroup itself.

Note that

4

• (i) Two cosets of a subgroup are either identical or have no element in common. Wecan see this as follows. Let the subgroup S have elements si. If two elements of leftcosets are equal e.g.,

g1s1 = g2s2

Then we see that g2 = g1s1s−12 = g1s3 where s3 is an element of the subgroup S. Thus,

the left coset g2S = g1s3S = g1S because of the closure property of the subgroup S.

• (ii) Each element g of the group G appears in at least one coset. This is because e.g.,the left coset gs−1

i contains g and because gs−1i is a group element.

• (iii) Therefore, the whole group splits up into cosets of S, each of order hS. Therefore,hS is a factor of h.

Two elements g1 and g2 of G are said to be equivalent if there exists some group elementg such that

g−1g1g = g2

A subset of elements in which each element is equivalent to another is called a class. Classestypically contain elements which are similar to one another in some physical way: thus, anelement in a class can be obtained from another by first executing a third group element,followed by the other element in the class, followed by the inverse of the first operation. Anexample is given in the discussion of the group S3 described below, for instance.

The identity element is in a class by itself. If a subgroup S has the property g−1Sg = Sfor every element g in G, then S is called an invariant subgroup. Invariant subgroups arethus made up of a whole class or whole classes.

If there is a mapping from a group G1 onto a group G2 which preserves group multiplica-tion, then the two groups are called homomorphic. If the correspondence between the groupelements is one-to-one then the groups are called isomorphic.

The direct product of two groups G1 and G2 exists if each element of G1 commutes withevery element of G2. These elements act on different spaces. Thus, the elements of the directproduct group G can be written as g1g2 or as an ordered pair (g1, g2). Group multiplicationis then defined by

g′′ = gg′ = (g1, g2)(g′1, g

′2) = (g1g

′1, g2g

′2)

3.2 Representations of Groups

In general, elements of a group may be represented by linear operators acting on a linearvector space (a homomorphism), but almost always we will be interested only in matrixrepresentations. The elements of a group can be represented by square matrices, whichact on the vectors in the vector space, which in turn are linear combinations of the “basisvectors”. A representaion D of a groupG, is such that there is a matrixD(g) which representseach element g and group multiplication of elements is preserved by multiplication of thecorresponding matrices. In other words, for any two elements a and b of the group if c is theelement which satisfies the condition

ab = c

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the corresponding matrices M(a), M(b) and M(c) of the representation must satisfy thesame condition in matrix form:

M(a)M(b) = M(c)

If the matrices are n×n in size then we call it an n-dimensional representation. Of course, wecan always represent all the elements of a group by a unit matrix in one or more dimensions;such a representation would be called a trivial representation. Mostly, we will be interestedin a faithful representation in which the matrices corresponding to different group elementsare themselves distinct.

Two representations are equivalent if their matrices are related by an equivalence trans-formation:

M ′(a) = U−1M(a)U

That is, a single unitary matrix U must simultaneously transform all matrices in one rep-resentation into the corresponding matrices in the other. We will be interested in unitarymatrices U which transform the basis states upon which M(a) perform symmetry operationsinto a new set. The equivalence transformation is analogous to the transformation of opera-tors and one can easily verify that the multiplication table for an equivalent representationis the same as the original.

All the matrices representing elements belonging to a single class have the same “char-acter” (group-theory speak for trace) in any given representation. In fact, we often think ofelements belonging to a class as being of a given “type”, for instance elements could form aclass of rotations, or a class of reflections. The fact that all elements in a class have the samecharacter makes it easy to create “character tables” and to reduce a representation into asum of irreducible representations (the Clebsch-Gordan decomposition).

3.3 The Symmetric Group Sn

This is the group of permutations of n distinct objects. Clearly, Sn has n! elements. You maywant to prove that all elements of Sn which are related to each other by a cyclic permutationform a class. The group Sn is centrally important in quantum physics. Symmetry underpermutations is, of course, crucial in systems with identical particles. The symmetric groupsare also important because they are connected with the groups U(N) and SU(N) and alsobecause all finite groups are isomorphic either to Sn or to one of its subgroups (see also thediscussion of regular representations later).

3.3.1 S2

For example, the objects 1 and 2 can be arranged as 12 or 21. We say that the symmetricgroup S2 has the elements e (identity element) and P12 where P12 exchanges the positions of1 and 2. Thus, a wavefunction Ψ(1, 2) (which may or may not be expressed as a product oftwo one-particle functions Ψ(1, 2) = Ψa(1)Ψb(2)) is transformed by P12 as follows:

P12Ψ(1, 2) = Ψ(2, 1)

Using P12 we can create the symmetrizer S12 and anti-symmetrizer A12 as follows:

S12 = (e+ P12) and A12 = (e− P12)

6

[Note: Not all functions can be anti-symmetrized. For instance, A12Ψa(1)Ψa(2) = 0. Also,note that after symmetrizing or antisymmetrizing you may need to normalize the wavefunc-tions.]

3.3.2 S3

If we have three objects, for instance “1”, “2” and “3”, then the element [1 2 3] means leavethe objects in their place, but [2 1 3] stands for interchanging “1” and “2” while leaving “3”in place. If we have 3 objects, say “u”, “d” and “s”, then

[123]uds = uds

[213]uds = dus

[132][213]uds = dsu

But[231]uds = dsu

therefore, [2 3 1] = [1 3 2] × [2 1 3].There are a total of 6 elements of the group S3: e = [123], σ1 = [132], σ3 = [213],

C+ = [312], C− = [231], σ2 = [321].Note that we may use a different notation. In our notation [2 3 1] means that element

“2” is put into the first position etc. One can imagine writing the reverse. For instance, wemay choose to write this as (3 1 2) meaning that element “1” is put into location “3”, “2”goes to location “1” etc. in which case again uds → dsu. Yet another notation, perhapseven more explicit, for [2 3 1] or (3 1 2) is

(

1 2 33 1 2

)

In this notation, which is very explicit, the order of the columns does not matter.Regardless of notation, it should be clear that S3 has order 6 and the identity element is e.

A pictorial way to view S3 is shown in Fig. 1. Here we can view S3 as the group of symmetryoperations on an equilateral triangle: we label the vertices “1”, “2” and “3” and considerthe elements to be no change (identity), reflections σ around the medians (3 operations) androtations C by 120◦ and 240◦. These operations all leave the triangle invariant.

Note that here there are 3 classes: the identity element, the three reflections, and the tworotations. The reflections, for example, are equivalent to each other and can be obtained fromeach other by preceding the operation with a rotation and following the reflection with theinverse rotation. The elements that result from group multiplcation of two group elementsare shown in the multiplication table below.

Since we discussed symmetry properties in the context of S2, it worth noting that thesymmetrizer and anti-symmetrizer for S3 can be written as

S123 = 1 + P12 + P23 + P31 + P12P13 + P13P12

A123 = 1 − P12 − P23 − P31 + P12P13 + P13P12

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1

2

3e

3

1

2+C

2

3

1-C

1

3

21σ

3

2

12σ

2

1

33σ

Figure 1: Elements of S3 as operations on an equilateral triangle. The identity is the elementcalled e and shows the triangle undisturbed. The C± operations represent rotations of thetriangle, while the σ operations involve reflections around the three medians of the triangle.

3.4 The regular representation

To motivate the connection between finite groups and the symmetric group in particular,we should recognize a special representation called the regular representation, where we usethe group elements themselves as the basis. That is, the basis vectors are n-dimensionalwhere n is the order of the group, and the basis vector representing the ith element is zeroexcept for the ith component which is equal to 1. The matrices representing each elementare simply permutation matrices, which reproduce the group multiplication table. We seehere a connection between an arbitrary group and the symmetric group.

4 Reducible and Irreducible representations

If there is an equivalence transformation which simultaneously turns all the matrices in agiven representation into submatrices along the diagonal (block diagonal form) it is calleda reducible representation. If no such transformation can be found we have an irreducible

representation. (The block diagonal form is one like the matrix below in which only thesubmatrices along the diagonal contain non-zero elements).

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Table 1: Group Multiplication table for S3

e σ1 σ2 σ3 C+ C−

e e σ1 σ2 σ3 C+ C−

σ1 σ1 e C+ C− σ2 σ3

σ2 σ2 C− e C+ σ3 σ1

σ3 σ3 C+ C− e σ1 σ2

C+ C+ σ3 σ1 σ2 C− eC− C− σ2 σ3 σ1 e C+

......

......

. . . . . . . . . . . . . . . . . . . . . . . ....

......

......

.... . . . . . . . . . . . . . . . . . . . . . . .

......

We will mostly be interested in unitary representations, i.e., representations by unitarymatrices. Since all irreducible representations can be shown to be equivalent to unitaryirreducible representations, in all that follows we will implictly assume that we are dealingwith unitary irreducible representations.

The number of distinct (not equivalent) irreducible representations of a group are limitedby the following relation between their dimensionalities di and the order n of the group:

∑

i

d2i = n

How do we know whether a given representation is reducible or not? It turns out to besufficient to find the sum of squares of the characters. Denoting the character of element gk inthe ith representation as χ(i)(gk), the sum over character squares determines the reducibilityof a representation as follows:

∑

k

|χ(i)(gk)|2 = n (Irreducible Representation)

∑

k

|χ(i)(gk)|2 > n (Reducible Representation)

4.1 Reduction of Representations

A powerful theorem called the orthogonality theorem helps in the process of reducing ordecomposing a reducible matrix representation into irredicuble ones along the diagonal. The

9

theorem states that given two representations Γ(i)(gk) and Γ(j)(gk) of group elements gk, thefollowing relation holds:

∑

k

Γ(i)∗αβ (gk)Γ

(j)γδ (gk) =

n

di

δijδαβδγδ

where α, β, γ, and δ are matrix indices, n is the order of the group, and di is the dimension-ality of the ith representation.

This orthogonality theorem implies that the n-component vector of matrix elementsin a specified row and column of a given representation is orthogonal to all other suchvectors! An interesting corollary follows. Since every group has a trivial one-dimensionalirreducible representation where every element is represented by unity, we see that in allother representations all the matrices must add up to zero.

There are many other important relations that follow from this theorem, but here wewill mention only one: the character orthogonality theorem, which follows from the grouporthogonality theorem and which states that

∑

k

χ(i)∗(gk)χ(j)(gk) = nδij

This theorem can be used to reduce a representation into irreducible representations, sincethe character of a group element in any representation is simply the sum of the charactersof all contained irreducible representations and characters do not change under similaritytransformations. If the ith irreducible representation is contained ai times, i.e., if

Γ = a1Γ(1) ⊕ a2Γ

(2) ⊕ . . .⊕ aNΓ(N)

then the characters of elements gk in representation Γ will be given by

χ(gk) =∑

i

aiχ(i)(gk).

Together with the character orthogonality theorem this relation immediately tells us that

ai = 〈χ(i)∗(gk)χ(gk)〉

where 〈〉 denotes an average over all group elements, and we have decomposed the represen-tation Γ into irreducible representations.

4.2 Reduction of Sn

We can now try to construct matrix representations of Sn. Let us start with S2. The simplestbasis states we can imagine are

(

10

)

,(

01

)

The group elements are the identity e and the permutation P12 which transforms each stateinto the other:

e =(

1 00 1

)

, P12 =(

0 11 0

)

10

Now we try to “reduce” these two matrices directly, i.e., by finding a suitable basis. Wesee that the similarity transformation using the matrix

S =1√2

(

1 11 −1

)

will transform the basis vectors to

s =1√2

(

11

)

and a =1√2

(

1−1

)

These are eigenvectors of P12 with eigenvalues ±1. Under the permutation P12 the newbasis vectors thus transform into themselves (up to a phase factor) and we have “reduced”the representation.

Indeed, in the new representation the elements e and P12 are transformed into blockdiagonal form:

g → SgST

transforms e and P12 into

e′ =(

1 00 1

)

, P ′12 =

(

1 00 −1

)

We see that the new matrices are “reducible” into two 1x1 matrices.Indeed, if we have any arbitrary two-particle state ψ(12) then we can define two basis

functions as eitherψ(12) and ψ(21)

or as a symmetric and antisymmetric combination

S12 = 1 + P12 and A12 = 1 − P12

The latter pair form the basis states of the symmetric and antisymmetric representations ofthe S2 group.

What if we tried this for S3? Now, of course, there are 3! = 6 basis states:

ψ(123), ψ(132), ψ(213), ψ(312), ψ(231), ψ(321)

The totally symmetric and antisymmetric combinations

ψS = ψ(123) + ψ(231) + ψ(312) + ψ(132) + ψ(321) + ψ(213)

andψA = ψ(123) + ψ(231) + ψ(312) − ψ(132) − ψ(321) − ψ(213)

form the basis states of one-dimensional representations of S3. Two other representations ofS3 exist, both of which are two-dimensional. We could form six column vectors, each with 6rows, with a single 1 and five 0’s to represent the six different arrangements of three objects.Then we would have to create matrix representations of the 6 group operations and reducethe space to subspaces where the group operations do not mix the subspaces. This workhas been done by mathematicians and here we will quote the results. It turns out that thedimensionality, number and basis states of such reduced subspaces can be neatly summarizedby the use of Young tableaux.

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4.3 Introduction to Young Tableaux

In order to enumerate irreducible representations and their dimensionalities in general it isbetter to turn our attention to the use of Young tableaux, a powerful tool for the study ofgroups.

Young tableaux consist of rows of boxes stacked on top of each other. To be precise, thestacks of empty boxes are sometimes called Young “frames” to distinguish them from filledboxes which are called tableaux, but we will just use the term “tableaux” for both. Theproperties of groups are translated into rules for constructing the tableaux. For instance,one rule states that all rows must be left-justified, i.e., all rows are aligned at their left edge.

An example of an allowed tableau is

We may place numbers into the boxes, for instance to denote the number of the “object”(or particle) in each “state” (box). Thus, we can write

1 23

In the case of the symmetric group, we must enforce three rules for Young tableaux:

• The number of boxes in any row must not exceed the number in the row above.

• The numbers in the boxes may not decrease as we go across a row.

• The numbers in the boxes must increase as we go down a column.

Examples of legitimate tableaux include:

1 12

and 1 32

However, the following are NOT allowed:

2 13

and 1 21

We are now ready to see how to get the dimensionalities of irreducible representations.For Sn, create all possible tableaux that satisfy the rules given above with the numbers “1”through “n” entered into the tableau. Each configuration represents one or more irreduciblerepresentations. The dimensionality of each irreducible representation is given by the numberof ways in which the numbers can legitimately be entered in the boxes for the correspondingconfiguration. Thus, in the case of S2 we have two possible tableaux:

and

12

For each of these, there is only one possible way in which we may enter the numbers “1”and “2”:

1 2 and 12

Thus, S2 has only two irreducible representations each of dimensionality 1, as expected fromour discussion of S2 above.

13

A more complicated example is provided by enumerating the irreducible representationsof S4:

1 2 3 4

1 2 34

1 2 43

1 3 42

1 23 4

1 32 4

1 234

1 324

1 423

1234

It turns out that in Sn there are exactly as many (inequivalent) irreducible representationsas the dimensionality of a tableau. Thus, there are exactly 2 irreducible representations inS4 of the kind

Thus we reduce the 24 element group S4 as follows:

4! = 1 ⊕ 32 ⊕ 22 ⊕ 32 ⊕ 1 = 24

The representations with the number of rows and columns interchanged are called “con-jugate” representations:

and are conjugate

and are conjugate

is self-conjugate

In general, the rows represent symmetry and columns represent antisymmetry. For in-stance, the tableau

1 2

represents a symmetrized version of a two-particle state while

12

14

represents the anti-symmetrized version. Thus, one sees that the rule for not repeating anumber as one goes down a column makes sense: antisymmetrizing a state which is alreadysymmetric will give zero. Of course, tableaux such as

are of mixed symmetry: symmetric across the top row, but antisymmetric with respect toexchange of any two particles in a column.

5 Infinite groups and Lie groups

5.1 Introduction

As mentioned earlier, in quantum mechanics almost all transformation operators are unitary.Of course, these operators operate on states defined in some basis, and two operations (suchas two rotations) are equivalent to a single, combined operation. Although infinite in number,the unitary operations typically are quantified using a finite number of parameters such asangles, and together they form a group (satisfy closure and the other group properties).Therefore, we can benefit from the theory of infinite groups, and specifically the theory ofunitary groups.

A Lie group is an infinite group of elements which depend in a continuous, differentiable

manner on some parameters. The groups of rotations in two and three dimensions, R(2)and R(3), are examples of Lie groups: their elements are continuous, differentiable matrixfunctions of the angles of rotation. A continuous group that has a finite number, n, ofparameters is called a n-parameter finite continuous group. Thus, we may write the elementof a Lie group with parameters θ1, θ2 . . . θn as A(θ1, θ2 . . . θn) or A(~θ).

For instance, the group of all n× n unitary matrices known as U(n) can be “generated”

from parameters ~θ and “generators” ~λ (which are Hermitian matrices, constant for eachrepresentation):

U(~θ) ≡ exp(−i~θ · ~λ)

If, in addition, we do not care about an overall phase and are interested in only those unitarymatrices which have determinant +1, then U(n) becomes the “special unitary” group SU(n)and the generators must be Hermitian as well as traceless.

In the late 19th century Sophus Lie showed that essentially we can understand an entirecontinuous group just by studying the properties of elements lying close to the identity i.e.,those generated by a infinitesimal changes to the identity element using the generators. Anexample of such a continuous group is SO(n), the group of “special orthogonal” matriceswhich describes rotations in n-dimensions. In an odd number of dimensions this excludesthe parity transformation xi → −xi.

A continuous group is called compact if the parameters are all bounded and if their domainof variation is closed, i.e., convergent sequences converge to a point within the domain. Wewill not concern ourselves with further qualifications, such as “simple” Lie groups and “semi-simple” Lie groups; suffice it to say that U(1) is not simple while the SU(n) are simple Liegroups.

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6 The Special Unitary Groups: SU(N)

In quantum mechanics, all transformations that we study are unitary, because they preservethe norms of state vectors:

If |α〉 → |α′〉 = U ′|α〉then to preserve the norm we require that

〈α′|α′〉 = 〈α|U ′†U ′|α〉 = 〈α|α〉 = 1

This implies that U ′ should be a unitary matrix. Now, all unitary matrices U ′ can be writtenas exp(iH) where H is a Hermitian matrix. [This is quite easy to prove - try it as an exercise.]

Let us consider the case of 2×2 unitary matrices. Each matrix has four complex ele-ments, but the condition of unitarity reduces the number of real parameters back to four.Similarly, the Hermitian matrices in the exponential have four real parameters. A general2×2 Hermitian matrix can be written as

H = θ0I + ~θ · ~σ

where I is the identity matrix, ~σ is the triplet of Pauli spin matrices, and ~θ is a vector ofparameters. Since I commutes with all 2×2 matrices, we can write

U ′ = exp(θ0I) exp(~θ · ~σ)

The first term simply gives an overall phase to the states, which is typically not interesting.Hence, we focus on the remaining unitary matrix

U = exp(~θ · ~σ)

Consider next the group of all unitary matrices which have determinant +1. Recallthat unitary matrices interest us because they leave scalar products invariant. All unitarymatrices of a given order which have determinant +1 form a Lie group called SU(N). Thesecan be obtained continuously from the identity matrix, while matrices with determinant -1cannot. Thus SU(N) will turn out to be very useful in describing transformations (symmetryoperations) in particle physics. Interest in SU(N) stems from the fact that we remove anoverall phase factor from unitary matrices to form SU(N) matrices (in quantum mechanicsthe overall phase is often irrelevant). (The eigenvalues of a unitary matrix must all havemodulus unity and thus their product, which is the matrix determinant, must also havemodulus unity.)

6.1 Generators and algebra of SU(N)

We define the generators of SU(N) by

Gi ≡ i limdθi→0

(

U(0, . . . , dθi, . . . , 0) − I

dθi

)

≡ i∂U

∂θi

∣

∣

∣

∣

~θ=0

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where the identity element I is the element for which ~θ = 0. Thus, each element U(~θ) =

exp(−i~θ · ~G).Clearly, SU(N) has N 2 − 1 parameters because the matrices must be unitary (N 2 condi-

tions) and have determinant +1 (one condition). Writing the elements of matrices infinites-imally different from the identity matrix as

Uij = δij + duij

we see, using the properties of unitarity and +1 determinant, that

duik = −du∗ki

and∑

i

duii = 0

Thus we see that the generators of SU(N) are Hermitian and traceless. Clearly, again thereare N2 − 1 independent Hermitian and traceless generators. How many of these commute?If we were to simultaneously diagonalize the maximum number of them, we would have atmost N − 1 such generators because of the tracelessness condition.

Using the fact that the Gi are Hermitian, we see that their commutators must be anti-Hermitian. Since the Gi span the space of traceless Hermitian matrices, we must be able towrite down each commutator as the following linear combination:

[Gi, Gj] = igijkGk

where the gijk are real coefficients. These coefficients are called the structure constants andserve to define the Lie group in a representation of any dimensionality. The above relationis called the Lie algebra of the generators - the generators form a Lie algebra while theoriginal group matrices form the Lie group. Taking the Hermitian adjoint of this equationimmediately leads to the fact that the gijk are antisymmetric with respect to interchange ofthe first two indices. In fact, it can be shown that they are antisymmetric with respect tointerchange of any two indices. An important point to keep in mind about Lie Algebras isthis: the Lie algebra of a group is very important from a physics point of view, since thecommutator encapsulates the physics being described by the group theory. An example ofa Lie algebra is provided by the generators of the group SU(2) - the components of angularmomentum. These are known to satisfy the relation

[Ji, Jj] = ihεijkJk

and therefore the εijk are the structure constants of SU(2) (in representations of all dimen-sionalities).

Using the Jacobi identity for matrices A, B and C

[[A,B], C] + [[B,C], A] + [[C,A], B] = 0

it is possible to show that the structure constants themselves satisfy the Lie algebra. Inother words a representation of the group (called the adjoint representation) can be created

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by taking the generators to be the structure constants gijk written as the matrices (gk)ij.Clearly, these generators are n× n matrices where n = N 2 − 1. Finally, one can show thatif Gi satisfy the commutator algebra, so do the −G†

i . This new representation may or maynot be equivalent to the original.

Conventionally, the following prescription for writing down the generators of SU(N) (upto a constant) is often used. The commuting, diagonal, generators are written as

1 0 . . . 00 −1 . . . 0...

.... . . 0

0 0 . . . 0

,

1 0 0 . . . 00 1 0 . . . 00 0 −2 . . . 0...

......

. . . 00 0 0 . . . 0

,

1 0 0 0 . . . 00 1 0 0 . . . 00 0 1 0 . . . 00 0 0 −3 . . . 0...

......

.... . . 0

0 0 0 0 . . . 0

etc.

The matrices with offdiagonal terms can be written as the combinations (T ji + T ij) andi(T ji − T ij) where T ij has a 1 in the i, j location (i 6= j) and zeros elsewhere. Convinceyourself that you can generate the Pauli spin matrices in this fashion. The result is that wecan write the generators of the fundamental representation of SU(2) as

G1 =1

2

(

0 11 0

)

, G2 =1

2

(

0 −ii 0

)

, G3 =1

2

(

1 00 −1

)

Similarly, the generators of the fundamental representation of SU(3) are eight in numberand are given by the Gell-Mann matrices

λ1 =1

2

0 1 01 0 00 0 0

, λ2 =1

2

0 −i 0i 0 00 0 0

, λ3 =1

2

1 0 00 −1 00 0 0

,

λ4 =1

2

0 0 10 0 01 0 0

, λ5 =1

2

0 0 −i0 0 0i 0 0

, λ6 =1

2

0 0 00 0 10 1 0

,

λ7 =1

2

0 0 00 0 −i0 i 0

, λ8 =1

2√

3

1 0 00 1 00 0 −2

We see from the form of the SU(3) generators that the first seven include the Paulispin matrices as 2×2 components, and thus include SU(2) transformations. SU(3) for flavorcontains the three SU(2) transformations u ↔ d, d ↔ s, and s ↔ u, where the necessarydiagonal SU(2) generator is a linear combination of the SU(3) diagonal generators λ3 andλ8. These two diagonal generators are used to label states within a multiplet and can beidentified, for flavor SU(3), with I3 and 1 + 3S.

6.2 Roots and Rank

For any Lie algebra we can attempt to find the maximum number of commuting generatorsGD. These are generators which either commute with all the other generators or can be

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made to do so by taking appropriate linear combinations of the other generators. Thus, theysolve the eigenvalue equation

[GD, GX ] = ρGX

where GX is a generator or a linear combination of generators. The eigenvalues ρ for a givendiagonal generator are called “roots”. The maximum number of diagonal generators for agroup is called the “rank” (r) of the group. For instance, in SU(2) there is only one diagonalgenerator: Jz. The roots are 0,±h:

[Jz, Jz] = 0, [Jz, J±] = ±hJ±

A “root diagram” in SU(2) would simply be a straight line with the roots indicated bydots. There are some important properties of the rank of a group worth mentioning. Webegin by defining a Casimir operator. A Casimir operator is a non-linear combination ofgenerators which commutes with all the generators of a group. Thus, for instance, J 2 is theCasimir operator for the SU(2) of angular momentum: it commutes with each component

of ~J . It turns out that the number of Casimir operators is equal to the rank of a group.The eigenvalues of a Casimir operator are the same for all members of a “representation” or“multiplet”. Thus, both the spin up and spin down state of an electron are characterized bys = 1/2 while each state is characterized by a different value of m. In general, we can labelmultiplets by the eigenvalue of the Casimir operators and their members by the eigenvaluesof the diagonal generators, called the weights (or weight vectors, in general). SU(3) thushas two Casimir operators: one which is the sum of the squares of the eight generators, andanother cubic in the generators.

There are other interesting properties of the rank r of a group. For instance, in the caseof SU(N) where r = N − 1, there are r fundamental representations of the group. A funda-mental representation of SU(N) is a representation with dimensionality N . Thus, in SU(3)of color or flavor, the quarks and anti-quarks both have their own, distinct, fundamentalrepresentations.

A more important property of the rank r lies in the connection to physics. There areonly a finite number of Lie algebras with a given rank - and these were all classified byCartan. Thus, if a physical problem appears to involve multiplets characterized by a certainnumber r of observables, we know immediately to look at only those Lie algebras of rank ralready classified by Cartan. Further, if we are talking about quantum-mechanical unitarytransformations, we almost certainly want to investigate SU(r+1).

6.3 Clebsch-Gordan coefficients

Clebsch-Gordan coefficients are used in the addition of angular momenta and, in general, inthe decomposition of the direct product of two representations of a group into irreduciblerepresentations. To see what this means, we first return to the familiar problem of additionof angular momenta.

Given two systems of angular momenta j1 and j2 and quantum numbers m1 and m2,the complete set of states which describe the combined object are most easily written as|j1m1〉|j2m2〉. These states span a complete set in the so-called direct product representation.We simply took the direct product of single-particle states; in other words if we were talking

19

of wavefunctions we simply multiply the (2j1 + 1) eigenfunctions of one particle with the(2j2 + 1) eigenfunctions of the other to get a total of (2j1 + 1)(2j2 + 1) direct productwavefunctions. The combined system can have angular momenta j which vary from |j1 − j2|to ji + j2 and for each of which m varies, as usual, from −j to +j. It is easy to show thatthere are still (2j1 + 1)(2j2 + 1) states labeled with the quantum numbers j and m. (Trythis as an exercise!). How do we transform between these two bases?

Inserting the completeness relation

I =∑

m1,m2

|m1m2〉〈m1m2| (1)

we can write the expansion

|jm〉 =∑

m1,m2

|m1m2〉〈m1m2|jm〉 (2)

where the scalar products 〈m1m2|jm〉 describe the expansion of |jm〉 in terms of the directproduct states |m1m2〉 and are called the Clebsch-Gordan coefficients.

Similarly, one can define Clebsch-Gordan coefficients for the decomposition of the directproduct of two SU(3) representations into irreducible representations. These and otherClebsch-Gordan coefficients are tabulated extensively in the literature.

6.4 Young tableaux in the context of SU(N)

Young tableaux for SU(N) are similar to those for Sn with only a few important differences.First, we now use the boxes to denote the particles (the first box represents the first particle,the second box the second particle and so on) and the states of particle are written into theboxes. Thus, the triplet and singlet configurations of a spin-1/2 particle may be written as

Triplet: + + − + and − −

Singlet: −+

Note that − + denotes the symmetric state (| + −〉 + | − +〉)/√

2. On the other hand −+

denotes the antisymmetric state (| + −〉 − | − +〉)/√

2.There are two new rules that are used in constructing Young tableaux for SU(N), the

first of which is:

• SU(N) tableaux can have a maximum of N columns. The reason for this is thatwe put the states of the fundamental representation into each box and any columnwith more than N boxes will necessarily repeat a state. It will then be impossibleto antisymmetrize with respect to interchange of those two boxes, as required by theantisymmetry of columns.

The rules for Young tableaux stated earlier in section 4.3 continue to apply.

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6.4.1 Dimensionality of Representations

What is the dimensionality of the representation pictured by a Young tableau? If we startwith the top row and write the number of boxes in each row as λi, we can denote a tableauby the set of N numbers (λ1, λ2, . . . λN ). Thus the tableau

is represented by ~λ = (4, 3, 3, 1). Alternatively we can denote the tableau by the set ofN − 1 numbers (p1, p2, . . . pN−1) which are the differences between the λi: p1 = λ1 − λ2, . . .pi = λi − λi+1 etc. In our case here ~p = (1, 0, 2).

It can be shown that the dimensionality d of SU(2) tableaux is given by

d2 = (p1 + 1)

Similarly, the dimensionalities of representations of SU(3), SU(4) etc. are given by

d3 =1

2!(p1 + 1)(p2 + 1)(p1 + p2 + 2)

d4 =1

2!3!(p1 + 1)(p1 + p2 + 2)(p1 + p2 + p3 + 3)(p2 + 1)(p2 + p3 + 2)(p3 + 1)

etc. These formulas give us the dimensionality of irreducible representations; we will needthem in what follows.

6.4.2 Reduction of direct products of representations

We will be interested in the direct product of representations and the dimensionality ofirreducible representations formed after the direct product is reduced. For instance, whenwe combine the fundamental representations of quarks and anti-quarks to form mesons, whatkinds of multiplets (irreducible representations) should we expect the mesons to fall into?For this task, we will need a final rule, the second rule specific to SU(N) Young tableaux.This rule applies to the reduction of the direct product of two irreducible representations.

• When combining two irreducible representations, put a row number in each row for thesecond representation. For the resulting representations, put the boxes of the secondrepresentation to the right of or below the boxes of the first representation. Draw apath starting from the rightmost box of the top row, going to the leftmost box andcontinuing in a similar fashion with the next row. At every point in this path thenumber of boxes encountered with the number m must be greater than or equal tothe number of boxes encountered with the number m + 1. Also, continue to applythe earlier rules regarding numbers in tableaux’ boxes to these numbers, i.e., theymust increase going down in columns and they may not decrease from left to right ina row. The distinct tableaux thus formed are the irreducible representations in thedecomposition.

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Thus, we can reduce the direct product of the two fundamental representations of SU(3)as follows:

⊗ 1 = 1 ⊕1

6.4.3 Basis Vectors of Representations

We expect 3 × 3 states to give us 6 symmetric and 3 antisymmetric states. Think of twoquarks with flavors u, d and s combining to give symmetric versions of uu, dd, ss, ud, dsand su, while yielding antisymmetric versions of ud, ds and su.

7 The Quark Model

8 Conclusion and References

In conclusion, group theory is a vast subject and we have examined only a very few featuresof greatest relevance to us in our study of quantum mechanics and particle physics. Grouptheory can be extensively employed in understanding various aspects of physics. The math-ematics itself is quite fascinating and books on group theory tend to be rather enjoyable.There is an element of magic to it: we gain understanding in leaps by discovering hiddenconnections and find regularities we didn’t know even existed! For a physicist the trick is toconstantly ask oneself how the mathematics describes the physics.

Students of particle physics will find the books by Jones, Lichtenberg and Georgi to beparticularly good references.

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9 References

References

[1] Group Theory by M. Hammermesh, Addison-Wesley, (1962).

[2] “Lie Algebras in Particle Physics” (Frontiers in Physics) by H. Georgi, Perseus Press(1999).

[3] “Unitary Symmetry and Elementary Particles” by D. B. Lichtenberg, Academic Press(1970).

[4] “Groups, representations, and physics” by H. F. Jones, A. Hilger, (1990).

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