Group Theory in Particle Physics

Joshua Albert

November 19, 2007

1 Group Theory

Group theory is a branch of mathematics which developed slowly over the years. Finding its origins in

algebraic equations, number theory, and geometry, this field would eventually be applied extensively to

physics in the 20th century.

1.1 What is a Group?

A group is a set of objects with an associated operation which satisfies certain properties. For a group G

which is a set of elements T, T ′, T ′′, ... with operator , we require:

1. Closure. ∀T, T ′ ∈ G : T T ′ ∈ G. This property of closure should be familiar to anyone with a linear

algebra background, as a vector space also fulfills this property, with addition as the operator.

2. Associativity. ∀T, T ′, T ′′ ∈ G : T (T ′ T ′′) = (T T ′) T ′′. This is just the associative property which

we often find in regular algebra. Note that this does not imply the commutative property.

3. Identity. ∃I ∈ G such that ∀T ∈ G : T I = I T = T . Thus, there exists an element I, called the

identity element in any group. This property is useful in defining the next:

4. Inverse. For each T ∈ G∃T−1 such that T T−1 = T−1 T = I. Any set with an operator satisfying

these four properties is a group.

For further examples, the operator may not be explicitly shown, but rather the implied multiplication

operation will represent whatever operation with respect to which the group is defined. The group operation

is often referred to as ”group multiplication”.

1

Of course, the way we defined a group is very abstract. We can use different operators and have different

types of objects as our group elements. Here are a few examples:

1. The set of integers under addition. The set of integers Z satisfies all the requirements of a group under

addition. Addition of two integers will never produce a non-integer, addition is associative, the integer

0 is the identity element, and every element has an inverse, specifically that element multiplied (in the

traditional sense) by -1.

2. The set of real numbers under multiplication. The set of real numbers R is a group under multiplication.

It is closed and associative, 1 is the identity and 1/x is the inverse of x. It should also be noted that

R is also a group under addition.

3. Parity. One can define a group consisting of two elements, one positive, one negative (or even and

odd). They can be represented at +1 and -1 and combine under the multiplication operation. The

parity group is an example of a simple group which ends up being used often in physics. Something

else worth noting is that while the previous two groups had infinite numbers of elements, this group

has a finite number of elements. The number of elements in a group is its ”order”.

4. Rotations in space. In the group of rotations in space, each element is a different rotation, which can be

specified by the three Euler angles. The operation for this group is the composition operation, that is,

performing one rotation, then the next. Obviously, two rotations put together are just another rotation

of the system. Associativity is maintained, the identity is the null rotation, and any rotation clearly

has an inverse to cancel it out. It should be noted that this group, unlike the other ones mentioned

so far, does not have commutivity as a property. That is, for two rotations R and R′, RR′ 6= R′R.

A group with non-commuting elements is called non-Abelian. Not surprisingly, a group with entirely

commuting elements, like the previous three examples, is called an Abelian group.

Thus, we have seen examples of groups with very different kinds of elements, and other groups can be

imagined. For example, a group of permutations on a Rubric’s cube, rearrangements of letters in a word,

or the set of unitary matrices. In its most abstract form, a finite group can be represented simply as a

multiplication table which indicates all the possible multiplications. Consider the group of the powers of

the imaginary constant i. A multiplication table for this group is shown in Table 1. While here the group

elements are labelled intuitively with values, an equivalent group can be represented using arbitrary labels,

such as A, B, C, D instead of i,−1,−i, 1. These tables are the most abstract way of representing a group,

but certainly not the most useful.

2

Table 1: Multiplication Table for Powers of i

i −1 −i 1i −1 −i 1 i−1 −i 1 i −1−i 1 i −1 −i1 i −1 −i 1

1.2 Representations

Clearly, groups range in type from very abstract, to very concrete. A good way of representing group

elements is needed to make things more managable. That is where matrix representations come in. Each

element in a group can be assigned a n × n matrix to represent it, provided it meets one condition. If each

element T ∈ G is represented by Γ(T ), we must have

Γ(TT ′) = Γ(T )Γ(T ′)

for each T, T ′ ∈ G. If this condition is met, then Γ is a d-dimensional representation of G. Note that matrix

multiplication is always used as the group multiplication operation.

Now that we can look at groups in a more concrete way, this is a good time to discuss some group

properties. First of all, using matrices to represent group elements has some clear advantages, since matrices,

like non-Abelian group elements, do not necessarily commute. Matrices also automatically have the property

of being associative, and the identity matrix is already defined. The existence of an inverse group element

simply requires that all our group elements be represented by square non-singular matrices.

Equivalent representations are representations which are related by a similarity tranfsormation. For Ω a

fixed operator (a constant matrix), and T ∈ G,

Γ′(T ) = ΩΓ(T )Ω−1

and Γ′(T ) and Γ(T ) are equivalent representations. It is easily demonstrated that the new representation is

still valid:

Γ′(T1)Γ′(T2) = ΩΓ(T1)Ω

−1ΩΓ(T2)Ω−1

= ΩΓ(T1)Γ(T2)Ω−1

= ΩΓ(T1T2)Ω−1

= Γ′(T1T2)

3

By using similarity transformations, we can change one representation into a potentially more useful

one. An important concept in representations is the idea of reducibility. If, by similarity transformation,

a group representation can be converted to block diagonal form, it is a reducible representation. If such a

transformation is not possible, it is an irreducible representation.

This brings up one more group concept– the direct product of groups. The direct product is not exclusive

the realm of representations; on the contrary, it is a property of the groups themselves. Essentially, two groups

are combined in a direct product (⊗) to produce elements which have a component from each group. These

elements operate on each other as follows: For Ti ∈ group G, and Sj ∈ group H:

Uij = Ti ⊗ Sj ∈ G ⊗H

Uij Ukl = (Ti Tj) ⊗ (Sk Sl)

The representation for a direct product group is a block diagonal matrix, with each block as one of

the irreducible representations. Since different blocks in block diagonal matrices do not interact through

multiplication, and multiply normally with themselves, this is a good representation.

1.3 Some Examples of Useful Groups

Most of the useful finite groups have already been described in section 1.1. In fact, the main useful finite

group to be described in this report is the parity group. This is related to the parity conservation which is

so useful in calculating possible transitions and so on in nuclear physics. Parity is the Z2 group.

The groups which are most important for our purposes are Lie groups. A Lie group is defined as ”a group

in which the elements are labelled by a set of continuous parameters with a multiplication law that depends

smoothly on the parameters.” [2] These Lie groups can describe many continuous symmetries, and, once Lie

algebras are developed, can describe some discrete phenomena too.

The first group of note is the set of all rotations in 3-dimensional space. This group is called O(3). It is

represented by the group of all 3-dimensional real orthogonal matrices. Because the matrix entries should

be real, this group is sometimes referred to by O(3, R). This group describes both proper and improper

rotations, so not every element of the group is connected smoothly to every other element. Any improper

rotation can be represented as a product of a proper rotation and an inversion of spatial coordinates.

Of course, if all proper rotations form a group of their own, then this group can also be identified. This

group is called SO(3), or SO(3, R). This is the set of all 3 × 3 orthogonal matrices with determinant 1 (the

4

special orthogonal group). Both the orthogonal and special orthogonal groups can be generalized to a higher

(or lower) number of dimensions. For example, SO(4, R) is the set of all 4 × 4 orthogonal matrices with

determinant 1. In general, groups SO(n, R) and O(n, R) can be constructed for any n ≥ 0, although for the

lowest values of n these are trivial.

Another similar set of groups are the unitary (U(n)) and special unitary (SU(n)) groups. These consist

of all unitary n × n matrices and all unitary n × n matrices with determinant 1, respectively. There is no

restriction that the entries be real. It is not immediately obvious that these have a physical signficance.

However, the (special) unitary groups can have close ties to the (special) orthogonal groups. For example,

consider U(1), which is the group of complex numbers with unit modulus. This group is actually isomorphic

to SO(2, R) [1], and each describe rotations in a plane. To say that two groups are isomorphic means that

there is a one-to-one mapping from one group to the other satisfying

Φ(T1)Φ(T2) = Φ(T1T2)

for Φ(T ) the isomorphic mapping of group elements Ti. Similarly, a homomorphic (like isomorphic, but not

one-to-one) mapping exists between SU(2) and SO(3).

Other useful groups are best introduced by their physical manifestations. The Lorentz group is the group

of all 3-space rotations and Lorentz boosts. The Lorentz group defines a set of transformations under which

physical laws must be invariant (lest they violate special relativity) but which leaves one point invariant

(specifically, the origin). This is a 6-dimensional Lie group, and the definition of the dimensionality of a Lie

group will be described in section 2.2. When combined with the 4-dimensional Lie group of translations in

space and time (x,y,z,t), the result is the Poincare group, the full symmetry group of relativistic field theory

[5].

2 Lie Groups and Symmetries

Lie groups are a very special class of continuous groups. In this report only classical Lie groups will be

considered; this restriction allows us to examine some fascinating properties of Lie groups. Through these

groups we will find connections to particle physics.

5

2.1 Lie Algebras

As described before, a Lie Group has elements which are defined by continuous parameters. It is possible to

construct an element of a Lie group by considering how far away from the identity element the given element

is. This is somewhat confusing, but it can be made fairly clear with an example.

Consider the Lie group of translations in one dimension [5](this is easily generalized to more dimensions).

The translation operator operates on a position defined state so that:

T (x)|x0〉 = |x + x0〉 (1)

We can define a generator of translation P (the symbol is not random) such that an infinitesimal displacement

is given by:

T (dx) ≡ I − idxP (2)

Our group properties let us write:

T (x + dx) = T (x) + dxdT (x)

dx(3)

and

T (x + dx) = T (dx)T (x) (4)

Then, substituting 2 into 4 and equating to 3 we get

dT (x)

dx= −iPT (x) (5)

which solves to:

T (x) = e−iPx (6)

We see that the translation operator can be written as an exponentiation of infinitesimal translations. In

quantum physics, the generator of translations, P, is actually the momentum operator. In general, any simple

Lie group can be defined as an exponentiation of infinitesimal generators, the generators being known as Lie

algebras.

Lie algebras have their own defining properties, and much can be learned about a Lie group by studying

the algebras which generate it. Lie algebras and their representations have equivalence relations just like the

groups. The dimension of a Lie group is given by the number of independent Lie algebras which generate it.

6

Thus, the group of all translations in 3-dimensional space is 3-dimensional, as it is generated by the three

translation operators for the three cartesian directions.

2.2 Continuous Symmetries

The translation operator brings up a symmetry immediately: translation symmetry. All physical laws are

equivalent if the coordinate axes are shifted by some spatial translation. This is represented by the Lie

group of translation operators. It is trivial to check that this group obeys all four rules for being a group.

Moreover, because this is a continuous symmetry, Noether’s theorem requires that there be a conservation

law associated with it. In this case, it is conservation of linear momentum.

This pattern of the generators being represented by the operators for a conserved quantity can be found

in rotational symmetry as well. The angular momentum operators Lx, Ly and Lz are the Lie algebras for the

rotation group. The spin one angular momentum operators (for SO(3)) have the same commutation structure

as the SU(2) generators, which are the Pauli matrices. While the Pauli matrices always exponentiate to

SU(2), constructing SO(3) can be a little bit tricky [1], and will not be treated here.

Finally, the time translation operator is the Hamiltonian, and this leads to the connection between time

invariance and conservation of energy. Now the Poincare group’s definition as being a 10-dimensional Lie

group is clear. Three algebras correspond to the three generators of rotations, and three more correspond

to the generators of boosts. These six are the algebras of the Lorentz group. Then, the three translation

operators and the time translation operator fill out the remaining four Lie algebras.

3 Particles and Forces

Lie groups and Lie algebras have enough depth of complexity to study in great detail, far beyond the scope

of this report. The concepts of classes, weights, roots, and so on can be useful in this analysis, but are also,

in general, beyond the scope of this report. We will, however, examine SU(3) in detail, as this group plays a

very important role in particle physics. We will find one inexact symmetry in SU(3) (the up-down-strange

quarks yielding the Eightfold Way), and one exact symmetry (the QCD representation of gluons).

7

3.1 Taming the Particle Zoo (Approximate SU(3))

The generators of SU(3) can be written in different bases, but the standard one is the Gell-Mann λ matrices.

λ1 =

0 1 0

1 0 0

0 0 0

λ2 =

0 −i 0

i 0 0

0 0 0

λ3 =

1 0 0

0 −1 0

0 0 0

λ4 =

0 0 1

0 0 0

1 0 0

λ5 =

0 0 −i

0 0 0

i 0 0

λ6 =

0 0 0

0 0 1

0 1 0

λ7 =

0 0 0

0 0 −i

0 i 0

λ8 = 1√3

1 0 0

0 1 0

0 0 −2

The generators we want are actually given by Ti = λi/2. It should be noted that there are only eight

generators, despite the appearance that there would be nine degrees of freedom (from nine matrix entries).

However, a property of Lie algebras is that they must be traceless, and this effectively eliminates one degree

of freedom, thus yielding only eight generators. We note that T1, T2 and T3 form a subalgebra of SU(2).

This generates the isospin subgroup.

Taking T3 and T8, the two diagonal elements, we get what is called a Cartan subalgebra [2]. As diagonal

matrices, they will naturally have the three cartesian unit vectors as eigenvectors. In the |H1, H2〉 basis, for

H1 = T3 and H2 = T8, we find:

1

0

0

corresponds to |12,

1

2√

3〉

0

1

0

corresponds to | − 1

2,

1

2√

3〉

8

0

0

1

corresponds to |0,− 1√3〉

Plotting H2 vs. H1 gives three points in an equilateral triangle. These points correspond to the up, down,

and strange quark. The H1 number corresponds to the z-component of isospin, I3, and H2 =√

3Y/2 for Y

the hypercharge, defined by the sum of the baryon number and strageness (strangely, a strange quark has

strangeness -1). Now, from here, we can easily begin producing combinations of these quarks, and plotting

them on Y vs. I3. For the mesons, see figure 1.

TTTTT

TT

TTT

r

r

r

r

r rr

r

K0(ds) K+(us)

K−(su) K0(sd)

π+(ud)π−(du)π0

η

s = 1

s = 0

s = −1

q = −1 q = 0

q = 1

Figure 1: The meson octet (used LATEX code from [6]). π0 is given by√

1

2(uu − dd)) and

ηis given by (√

1

6(uu + dd − 2ss)).

Other, similar pictures can be produced for baryons as well. Because baryons have three quarks, the

group involved is actually 3⊗ 3⊗ 3 = 10⊕ 8⊕ 8⊕ 1 giving us the 27 elements we should expect from all the

permutations of the three quark flavors in three type positions [4]. However, this representation of SU(3)

is reduced into four irreducible representations, each of which is a group by itself. Just like the mesons fit

into the octet, there is a baryonic decouplet, two octets, and a singlet. A nice feature of these organizations,

commonly referred to as the ”eightfold-way”, is that the particles can be placed into these arrangements

based on their physical properties alone. While organization by hypercharge and isospin is exact, there is

also an approximate pattern in the masses. Because the up and down quarks are so similar in mass, and

because the strange quark is also a light quark, the rows of particles in these arrangements tend to have

similar masses. The baryon decouplet is shown in figure 2.

The SU(3) symmetry exhibited in generating the Eightfold Way is a result of the approximate symmetry

9

TTTTTTTTTTTTT

r

r

r

r

r rr

r

r r

∆−(ddd) ∆0(ddu) ∆+(duu) ∆++(uuu)

Ξ∗−(dss) Ξ∗0(uss)

Σ∗+(uus)Σ∗−(dds) Σ∗0(uds)

Ω−(sss)

s = 0

s = −1

s = −2

s = −3

q = −1

q = 0

q = 1

q = 2

Figure 2: The baryon decouplet (used LATEX code from [6]). Especially notable is the Ω−, which waspredicted using the Eightfold Way by Gell-Mann before it was discovered [3].

between the three light quarks, up, down and strange. Isospin is closer to an exact symmetry because the

up and down quarks have nearly the same mass. This model is not very useful when expanded to SU(4)

with the charm quark, because its mass is much greater than that of the strange quark, and much much

greater than that of the up or down quark. Thus, trying to unite all six quarks under a SU(6) symmetry

can be misleading, as this is not an exact symmetry. However, the Eightfold Way can provide a convenient

and somewhat intuitive organization method for hadronic particles. However, there is nothing unique about

this representation, it is merely a consequence of the quarks which make up the baryons and mesons.

3.2 Constructing The Force Carriers: SU(3) and Beyond

While SU(3) is an approximate flavor symmetry due to quark mass differences, it is an exact color symmetry!

The symmetry SU(3) is in fact the basis for QCD, and its 8-dimensional generator basis (the Gell-Mann λ

matrices) is reflected in the octet of gluons. While all combinations of red, green and blue with their anti-

colors should give nine states, one of them is a color singlet, and would have some non-gluon like properties

such as a long range, and thus it is believed that this does not exist (at least as a gluon) [3]. The color octet

10

is:

|1〉 = (rb + br)/√

2 |2〉 = −i(rb − br)/√

2

|3〉 = (rr − bb)/√

2 |4〉 = (rg + gr)/√

2

|5〉 = −i(rg − gr)/√

2 |6〉 = (bg + gb)/√

2

|7〉 = −i(bg − gb)/√

2 |8〉 = (rr + bb − 2gg)/√

6

and the color singlet is:

|9〉 = (rr + bb + gg)/√

3

Gluons are the carriers of the strong force, and the strong force is described by SU(3) symmetry. Of course,

attempts have been made to associate the other forces with groups. There has actually been considerable

success here, as the weak and electromagnetic forces are actually united in this way. The electroweak

interaction is based on a combined Lie algebra of U(1) ⊕ SU(2). Salam, Weinberg and Glashow were

awarded the 1979 Nobel Prize in Physics for their contributions to unifying electromagnetism and the weak

interaction.

Attempts to unify the strong force, and even gravity with the electroweak into one single interaction are

underway. It is major goal of theoretical physics to show that all the forces are in fact manifestations of a

single phenomenon. Group theory, with its organization of like elements into recognizable patterns, may be

an important part of achieving this goal.

References

[1] J.F. Cornwell. Group Theory in Physics: An Introduction. Academic Press, Cambridge, UK, 1997.

[2] Howard Georgi. Lie Algebras in Particle Physics. The Benjamin/Cummings Publishing Company, Mas-

sachusetts, USA, 1982.

[3] David Griffiths. Introduction to Elementary Particles. John Wiley and Sons, Inc., New York, USA, 1987.

[4] Francis Halzen and Alan D. Martin. Quarks and Leptons: An Introductory Course in Modern Particle

Physics. John Wiley and Sons, New York, USA, 1984.

[5] Wu-Ki Tung. Group Theory in Physics. World Scientific, Philadelphia, USA, 1985.

11

[6] Eightfold way (physics).

http://en.wikipedia.org/wiki/Eightfold_way_(physics)

.

12