Zee A. Quantum Field Theory in a Nutshell

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522 | Closing Words to Part N

dynamics bends the causal structure of spacetime out of whack. The lack of strict locality

is not built into the laws of physics.

Of course, we also know how to imbue physics with non-locality right from the start.

We have Wilson’s lattice formulation of gauge theory, and more recently, Wen’s intriguing

lattice formulation of gravity.

When I showed the last three chapters to our friend SE, she mused, after some reflection,

“Now I see what theorists could always do when in doubt: enhance the symmetry and make

it local, complexify and bow to Cauchy, and take a square root when possible!”

I nodded, “These are the three ways of the warrior theorist: I call them the Einstein

way, the Heisenberg way, and the Dirac way. They were wildly successful in the past, and

perhaps they will work in the future as well.”

With this edition of my textbook, I can no doubt count on a new group of readers to

come up with fresh insights into field theory. As these new chapters suggest, there may

still be plenty of secret structures to uncover. And thus field theory marches on.

Finally, I reveal the origin of the quote at the start of the preface to the second edition. As a

kid, Feynman came across a calculus book

that proclaimed “What one fool can do, another

can.” He was thus inspired to master calculus. Now that you have mastered quantum field

theory, you can switch from the “understand” in the preface to the “do” in these closing

words.

Silvanus P. Thompson (1851–1916), Calculus Made Easy, 1910, updated by Martin Gardner, St. Martin’s Press,

(1998). I am kind of trying to do for quantum field theory what Thompson did for calculus.

Appendix A

Gaussian Integration and the Central

Identity of Quantum Field Theory

The basic Gaussian:



+∞

−∞

dxe

−

√

2π (1)

The scaled Gaussian:



+∞

−∞

dxe

−



2π



(2)

Moments:



+∞

−∞

dxe

−



2π



(2n − 1)(2n − 3)

...

1, n ≥ 1 (3)

Gaussian with source:



+∞

−∞

dxe

−

+Jx



2π



/2a

(4)



+∞

−∞

dxe

−

+iJx



2π



−J

/2a

(5)



+∞

−∞

dxe

iax

+iJx



2πi



−iJ

/2a

(6)



+∞

−∞



+∞

−∞

...



+∞

−∞

...

x+iJ



(2πi)

det[A]



−(i/2)J

−1

(7)



+∞

−∞



+∞

−∞

...



+∞

−∞

...

−

x+J



(2π)

det[A]



−1

(8)

In what follows, we omit an overall factor.

Central identity of quantum field theory:



Dϕe

−

ϕ−V(ϕ)+J

= e

−V(δ/δJ)

−1

(9)

524 | Appendix A. Gaussian Integration

A trivial variation:



Dϕe

−

ϕ+J

= e

−1

(10)

Variations:



Dϕe

(i/2)ϕ

ϕ+iJ

= e

−(i/2)J

−1

(11)



Dϕe



ϕ(x)Kϕ(x)+J(x)ϕ(x)]

= e



x[−

J(x)K

−1

J(x)]

(12)



Dϕe

−



ϕ(x)Kϕ(x)+J(x)ϕ(x)]

= e



J(x)K

−1

J(x)]

(13)

(where K or K

−1

or both may be nonlocal)

A specific example:



Dϕe



x[(λ/2)ϕ

+ϕ

ψψ]

= e



x[−(1/2λ)(

ψψ)

]

(14)

For K hermitean with ϕ complex:



Dϕ

†

Dϕe

−ϕ

†

ϕ+J

†

ϕ+ϕ

†

= e

†

−1

(15)

As noted earlier, various numerical factors have been swept under the integration measure. In applying these

formulas, be sure that these factors are not relevant for your purposes.

Appendix B A Brief Review of Group Theory

I give here a brief review of the group theory I will need in the text. I assume that you have been exposed to some

group theory, otherwise this instant review might not be intelligible. Most of the concepts are illustrated with

examples, and it goes without saying that you should work out all the examples and verify the assertions made

without proof.

SO(N)

The special orthogonal group SO(N) consists of all N by N real matrices O that are orthogonal

O = 1 (1)

and have unit determinant

det O = 1 (2)

We denote the element in the ith row and j th column by O

. The group SO(N ) consists of rotations in N-

dimensional Euclidean space and its defining or fundamental representation is given by the N component vector

v ={v

, j = 1,...,N }, which transforms under the action of the group element O according to (as always, all

repeated indices are summed over)

→ v

i

= O

(3)

We define tensors as objects that transform as if they are equal to the product of vectors. For example, the tensor

ij k

transforms according to

ij k

→ T

ij k

= O

lmn

(4)

as if it is equal to the product v

. The emphasis is on the phrase “as if”: T

ij k

is not to be thought of as being

equal to v

It is important to develop some “feel” or intuition for groups and their representations. Some people find

it helpful to picture a certain number of objects being acted upon by the group and transformed into linear

combinations of each other. Thus, picture T

ij k

as N

objects being scrambled together.

Tensors furnish representations of the group. In our particular example, each group element is represented

by an N

by N

matrix acting on the N

objects T

ij k

. The number of objects in a tensor is called the dimension

of the representation.

It may well be that any given object in a representation does not transform, under all the elements of the group,

into a linear combination of all the other objects, but only into a subset of them. Let me illustrate with an example.

526 | Appendix B. Brief Review of Group Theory

Consider T

→ T

ij

= O

. Form the symmetric S

≡

+ T

) and antisymmetric combinations

≡

−T

). The symmetric combination S

transforms into O

, which is obviously symmetric.

Similarly, A

transforms into O

, which is obviously antisymmetric . In other words, the set of N

objects contained in T

split into two sets:

N(N +1) objects contained in S

and

N(N −1) objects contained

in A

. The S

’s transform among themselves and the A

’s transform among themselves.

The representation furnished by T

is said to be reducible: It breaks apart into two representations. Obviously,

representations that do not break apart are called irreducible.

We just exploited the obvious fact that the symmetry properties of a tensor under permutation of its indices is

not changed by the group transformation, namely that the indices on a tensor transform independently, as in (4).

The various possible symmetry properties may be classified with Young tableaux, which is useful in a general

treatment of group theory. Fortunately, in the field theory literature one rarely encounters a tensor with such

complex symmetry properties that one has to learn about Young tableaux.

Another way of saying this is that we can restrict our attention to tensors with definite symmetry properties

under permutation of their indices. In our specific example, we can always take T

to be either symmetric or

antisymmetric under the exchange of i and j .

We have yet to use the properties (1) and (2). Given a symmetric tensor T

consider the combination

T ≡δ

, known as the trace. Then T → δ

ij

= δ

= (O

)

= δ

= T , where

we used (1). In other words, T transforms into itself. We can subtract the trace from T

forming the traceless

tensor Q

≡ T

− (1/N)δ

T . The

N(N + 1) − 1 objects contained in Q

transform among themselves.

To summarize, given two vectors v and w, we can form a tensor, and decompose the tensor into a symmetric

traceless combination, a trace, and an antisymmetric tensor. This process is written as

N ⊗N = [

N(N + 1) − 1] ⊕ 1 ⊕

N(N − 1) (5)

In particular, for SO(3),3⊗3 = 5 ⊕ 1 ⊕ 3, a relation you should be familiar with from courses on mechanics

and electromagnetism.

There are two conventions for naming representations. We can simply give the dimension of the representa-

tion. (This can occasionally be ambiguous: Two distinct representations may happen to have the same dimension.)

Alternatively, we can specify the symmetry properties of the tensor furnishing the representation. For instance,

the representation furnished by a totally antisymmetric tensor of n indices is often denoted by [n] and the rep-

resentation furnished by a totally symmetric traceless tensor of n indices by {n}. Obviously, [1] ={1}. In this

notation, the decomposition in (5) can be written as {1}⊗{1}={2}⊕{0}⊕[2]. For the group SO(3), with its

long standing in physics, the confusion over names is almost worse than in reading Russian novels: For instance,

{1} is also known as p and {2} as d.

We have yet to use (2). Using the antisymmetric symbol ε

123. . .N

, we write (2) as

...i

...O

= 1 (6)

or equivalently

...i

...O

= ε

...j

(7)

By multiplying (7) by O

repeatedly, we can obviously generate more identities. Instead of drowning in a sea of

indices, let me explain this point by specializing to say N =3. Thus, multiplying (7) by (O

)

, we obtain

= ε

)

Speaking loosely, we can think of moving some of the O’s on the left hand side of (7) to the right hand side,

where they become O

’s.

Using these identities, you can easily show that [n] is equivalent to [N − n], that is, these two representations

transform in the same way. For example, as is well known, in SO(3) the antisymmetric 2-index tensor is equivalent

to the vector. (The cross product of two vectors is a vector.)

Any orthogonal matrix can be written as O = e

. The conditions (1) and (2) imply that A is real and

antisymmetric, so that A may be expressed as a linear combination of N(N − 1)/2 antisymmetric matrices

denoted by iJ

: O = e

iθ

(with repeated indices summed over). We have defined J

as imaginary and

antisymmetric and hence hermitean. Since the commutator [J

, J

] is antihermitean, it can be written as a

linear combination of the iJ’s.

Ironically, some students are confused at this point because of their familiarity with SO(3), which has special

properties that do not generalize to SO(N ).

In speaking about rotations in 3-dimensional space we can specify a rotation as either around say the third

axis, with the corresponding generator J

, or as in the (1-2)-plane, with the corresponding generator J

=−J

Appendix B. Brief Review of Group Theory | 527

In higher dimensions, for example 10-dimensional space, we can speak of a rotation in the (6-7)-plane, with the

corresponding generator J

=−J

, but it is nonsense to speak of a rotation around the fifth axis. Thus, to

generalize to higher dimensions we should write the standard commutation relation [J

, J

] = iJ

for SO(3) as

, J

] = iJ

, which can be generalized immediately to

, J

] =i(δ

− δ

+ δ

− δ

) (8)

The right hand side reflects the antisymmetric character of J

=−J

. A potential confusion some students

may have about the notation: J

denotes a matrix generating rotation in the (i-j )-plane, a matrix with element

)

in the k-th row and l-th column. The indices i , j , k, and l all run from 1 to N , but in (J

)

the set {ij }

and the set {kl}should be distinguished conceptually: The former labels the generator and the latter are matricial

indices when the generator is regarded as a matrix. As an exercise, write down (J

)

explicitly and obtain (8) by

direct computation.

In studying group theory, as I have already remarked, one source of confusion comes from the fact that

some of the smaller groups, which we tend to encounter first in our studies, have special properties that do not

generalize. The special property of SO(3) we just noted is due to the fact that the antisymmetric symbol ε

ij k

carries

three indices and thus J

may be written as J

≡

ij k

.ForSO(4) the antisymmetric symbol ε

ij kl

carries four

indices and we can form the combinations

ij kl

). Define J

≡

±J

), J

≡

±J

), and

≡

± J

). By explicit computation, show that [J

, J

] = iε

ij k

,[J

−

, J

−

] = iε

ij k

−

, and [J

, J

−

] = 0.

This proves the well-known theorem that SO(4) is locally isomorphic to SO(3) ⊗ SO(3).

I assume that you know that SO(3) is locally isomorphic to SU(2). If you don’t, I give a brief review below.

With a few i’s included here and there, these two results prove the statement that the Lorentz group SO(3, 1)

is locally isomorphic to SU(2) ⊗ SU(2), which we proved explicitly in chapter II.3. The Lorentz group can be

thought of as an “analytic continuation” of the rotation group SO(4). See below for a more precise statement.

One highly non-obvious result of group theory is that SO(N) contains representations other than vector and

tensor. I develop the relevant group theory for the spinor representations in chapter VII.7.

SU(N)

We next turn to the special unitary group SU(N) consisting of all N by N matrices U that are unitary

†

U = 1 (9)

and have unit determinant

det U = 1 (10)

The story of SU (N ) has more or less the same plot as the story of SO(N) with the crucial difference that the

tensors of the unitary groups can carry both upper and lower indices. We denote the element in the ith row and

jth column by U

; the wisdom of this notation will soon become apparent.

The defining or fundamental representation of SU (N ) consists of N objects ϕ

, j =1,...,N , that transform

under the action of the group element U according to

→ ϕ

i

= U

(11)

Taking the complex conjugate of (11) we have

∗i

→ (U

)

∗

∗j

= (U

†

)

∗j

(12)

We invite ourselves to define an object we write as ϕ

that transforms in the same way as ϕ

∗i

; thus

→ ϕ



= (U

†

)

(13)

Note that we did not say that ϕ

is equal to ϕ

∗i

; we merely said that ϕ

and ϕ

∗i

transform in the same way.

As before, we can have tensors. The tensor ϕ

, for example, transforms as if it is equal to the product ϕ

→ ϕ

ij

= U

†

)

(14)

528 | Appendix B. Brief Review of Group Theory

Again, we emphasize that we did not say that ϕ

is equal to ϕ

. (In some books ϕ

is called a covariant vector

and ϕ

a contravariant vector. A tensor ϕ

......

...

with m upper indices and n lower indices is defined to transform as

if it is equal to the product of m covariant vectors and n contravariant vectors.)

The possibility of complex conjugation in SU (N ) leads naturally to having indices “upstairs” and “downstairs.”

Note that (9) can be written out explicitly as (U

†

)

= δ

and thus the Kronecker delta in SU (N ) carries one

upper and one lower index. It is important when taking traces that we set an upper index equal to a lower index

and sum over them: for example, we can consider δ

≡ ϕ

, which transforms as

→ U

†

)

= U

(15)

where we have used (9). In other words, ϕ

, the trace of ϕ

, denote N objects that transform into linear

combinations of each other in the same way as ϕ

. Thus, given a tensor, we can always subtract out its trace.

As in the discussion for SO(N), tensors furnish representations of the group. The discussion proceeds as

before. The symmetry properties of a tensor under permutation of its indices are not changed by the group

transformation.

Another way of saying this is that given a tensor we can always take it to have definite symmetry properties

under permutation of its upper indices and under permutation of its lower indices. In our specific example, we

can always take ϕ

to be either symmetric or antisymmetric under the exchange of i and j and to be traceless.

Thus, the symmetric traceless tensor ϕ

furnishes a representation with dimension

(N + 1) − N and the

antisymmetric traceless tensor ϕ

a representation with dimension

(N −1) −N .

Thus, in summary, the irreducible representations of SU (N ) are realized by traceless tensors with definite

symmetry properties under permutation of indices. For example, in SU(5), some commonly encountered

representations are ϕ

, ϕ

(antisymmetric), ϕ

(symmetric), ϕ

, ϕ

(antisymmetric in the upper indices and

traceless) with dimensions 5, 10, 15, 24, and 45, respectively. Convince yourself that for SU (N) the dimensions

of the representations defined by these tensors are N , N(N − 1)/2, N(N +1)/2, N

−1, and

(N −1) −N,

respectively.

The representation defined by the traceless tensor ϕ

is known as the adjoint representation. By definition,

it transforms according to ϕ

→ ϕ

i

= U

†

)

= U

†

)

. We are thus invited to regard ϕ

as a matrix

transforming according to

ϕ → ϕ



= UϕU

†

(16)

Note that if ϕ is hermitean it stays hermitean, and thus we can take ϕ to be a hermitean traceless matrix. (If ϕ

is antihermitean we can always multiply it by i .) Another way of saying this is that given a hermitean traceless

matrix X, UXU

†

is also hermitean and traceless if U is an element of SU (N).

As in the SO(N) story, representations of SU (N ) have many names. For example, we can refer to the

representation furnished by a tensor with m upper and n lower indices as (m, n). Alternatively, we can refer

to them by their dimensions, with an asterisk to distinguish representations with mostly lower indices from the

representations with mostly upper indices. For example, an alias for (1, 0) is N and for (0, 1) is N

∗

. A square

bracket is used to indicate that the indices are antisymmetric and a curly bracket indicate that the indices are

symmetric. Thus, the 10 of SU(5) is also known as [2, 0] = [2], where as indicated the 0 (no lower index) is

suppressed. Similarly, 10

∗

is also known as [0, 2] = [2]

∗

The condition (10) can be written as either

...i

...U

= 1 (17)

...i

...U

= 1 (18)

Thus, we have two antisymmetric symbols ε

...i

and ε

...i

that we can use to raise and lower indices. Again,

we can immediately generalize (17) to

...i

...U

= ε

...j

and multiplying this identity by (U

†

)

and summing over j

we obtain

...p

...U

N−1

= ε

...j

†

)

Appendix B. Brief Review of Group Theory | 529

Clearly, by repeating this process, we can peel off the U ’s on the left hand side and put them back as U

†

’s on the

right hand side. We can play a similar game with (18).

To avoid drowning in a sea of indices, let me show you how to raise and lower indices in a specific example

rather than in general. Consider the tensor ϕ

in SU(4). We expect that the tensor ϕ

kpq

≡ϕ

ijpq

will transform

as a tensor with three lower indices. Indeed,

kpq

≡ ϕ

ijpq

→ ε

ijpq

†

)

= ε

lmst

†

)

†

)

†

)

= (U

†

)

†

)

†

)

nst

As in SO(N ) we can look at the generators of SU (N) by noting that any unitary matrix can be written as

U = e

, with H hermitean and traceless as required by (9) and (10). There are (N

− 1) linearly independent

N by N hermitean traceless matrices T

(a = 1,2,...,N

− 1). Any N by N hermitean traceless matrix can be

written as a linear combination of the T

’s and thus we can write U =e

iθ

, where θ

are real numbers and the

index a is summed over.

Since the commutator [T

, T

] is antihermitean and traceless, it can also be written as a linear combination

of the T

’s:

, T

] =if

abc

(19)

(with the index c summed over.) The commutation relations (19) define the Lie algebra of SU (N ), and f

abc

are

known as the structure constants. For SU(2) the structure constants f

abc

are simply given by the antisymmetric

symbol ε

abc

Sometimes students are confused by how the generators act. Consider an infinitesimal transformation

U  1 + iθ

. On the defining representation, ϕ

→ U

 ϕ

+ iθ

)

. Thus, the ath generator acting

on the defining representation gives T

ϕ. Now consider the adjoint representation (16)

ϕ → ϕ



 (1 + iθ

)ϕ(1 + iθ

)

†

 ϕ + iθ

ϕ − ϕiθ

= ϕ + iθ

, ϕ] (20)

In other words, the ath generator acting on the adjoint representation gives [T

, ϕ]. Perhaps some students are

confused by the fact that ϕ is used as a generic symbol to denote different objects.

Since the adjoint representation ϕ is hermitean and traceless it can also be written as a linear combination of

the generators, thus ϕ =ϕ

. Using (19) we can thus also write (20) as ϕ

→ϕ

c

ϕ

−f

abc

. In particular

for SU(2), the three objects ϕ

transform as a 3-vector. (Note the notation: ϕ

is not to be confused with ϕ

:in

SU(2) the index a = 1, 2, 3 while i = 1, 2.)

This last remark essentially amounts to a proof that SU(2) is locally isomorphic to SO(3). I will now give a

somewhat more formal proof. Any 2 by 2 hermitean traceless matrix X can be written as a linear combination of

the three Pauli matrices X =x

σ with three real coefficients (x

, x

), which we regard as the components of a

3-vector x. For any element U of SU(2), X



≡U

†

XU is hermitean and traceless, so that we can write X



=x



σ .

Note that we have implicitly used the first defining property of an SU(2) matrix (9). By explicit computation,

we find detX =−x

. Invoking the second defining property of an SU(2) matrix (10), we obtain detX



= detX

and thus x

2

=x

. The 3-vector x is rotated into the 3-vector x



. Thus we can associate a rotation with any given

U . Since U and −U are associated with the same rotation, this gives a double covering of SO(3) by SU(2).A

physicist would just say that when a spin

particle is rotated through 2π, its wave function changes sign. The

map clearly preserves group multiplication: if two elements U

and U

of SU(2) are mapped to the rotations R

and R

respectively, then the element U

is mapped to the rotation R

. Alternatively, noting that trX

=x

and trX

2

= trX

, we obtain the same conclusion.

Once again, the two special unitary groups that most students learn first, namely SU(2) and SU(3), have

special properties that do not generalize to SU(N), just as SO(3) has special properties that do not generalize to

SO(N ), possibly leading to confusion.

For SU(2), because the antisymmetric symbol ε

and ε

carry two indices, it suffices to consider only tensors

with upper indices, all symmetrized: We can raise all lower indices of any tensor by contracting with ε

repeatedly.

After this is done, we can remove any pair of indices in which the tensor is antisymmetric by contracting with

In particular, ϕ

= ε

, which can be stated equivalently in terms of a special property of the Pauli matrices

∗

=−σ

(21)

530 | Appendix B. Brief Review of Group Theory

so that



θ σ

)

∗

= e



θ σ

(22)

For SU(2) (11) becomes

→ ϕ

i

= (e



θ σ

)

Complex conjugating, we obtain

∗i

→ [(e



θ σ

)

]

∗

∗j

= [(−iσ



θ σ

(iσ

)]

∗j

and so

iσ

∗

→ e



θ σ

(iσ

∗

)

We learn that iσ

∗

transform in the same way as ϕ. Recall that we define ϕ

to transform in the same way as ϕ

∗i

Thus, ε

transforms in the same way as ϕ

. In the jargon, SU(2) is said to have only real and pseudoreal

representations, but not complex representations. A pseudoreal representation is equivalent to its complex

conjugate upon a similarity transformation. Recall that (21) figures into our discussion of charge conjugation in

chapter II.1 and of the Higgs doublet in chapter VII.2.

For SU(3) it suffices to consider only tensors with all their upper indices symmetrized and all their lower

indices symmetrized. Thus, the representations of SU(3) are uniquely labeled by two integers (m, n), where m

and n denote the number of upper and lower indices. The reason is that the antisymmetric symbols ε

ij k

and

ij k

carry three indices. We can always trade a pair of lower indices in which the tensor is antisymmetric for one

upper index, and similarly for upper indices.

You can see easily that these special properties do not generalize beyond SU(2) and SU(3).

Multiplying representations together

In a course on quantum mechanics you learn how to combine angular momentum. We have already encountered

this concept in (5), which when specialized to SO(3), tells us that 3 ⊗ 3 = 5 ⊕ 1 ⊕ 3, as we noted. This is

sometimes described by saying that when we combine two angular momentum L = 1 states we obtain L = 0, 1, 2.

Students are justifiably confused when this procedure is also known as addition of angular momentum.

Given two tensors ϕ and η of SU (N ), with m upper and n lower indices and with m



upper and n



lower indices,

respectively, we can consider a tensor T with (m + m



) upper and (n + n



) lower indices that transforms in the

same way as the product ϕη. We can then reduce T by the various operations described above. This operation of

multiplying two representations together is of course of fundamental importance in physics. In quantum field

theory, for example, we multiply fields together to construct the Lagrangian.

As an example, multiply 5

∗

and 10 in SU(5). To reduce T

= ϕ

we separate out the trace ϕ

(which

transforms as a 5) after which there is nothing more we can do. Thus,

∗

⊗ 10 = 5 ⊕ 45 (23)

As another example, consider 10 ⊗ 10: ϕ

. It is easiest to write η

equivalently as a tensor with three lower

indices ε

mnhkl

. The product 10 ⊗ 10 then carries two upper and three lower indices and we will write it as T

mnh

Taking traces, we separate out T

mij

, which we recognize as 5

∗

, and the traceless part of T

mnj

, which we recognize

as 45

∗

(see above), thus obtaining:

10 ⊗ 10 = 5

∗

⊕ 45

∗

⊕ 50

∗

(24)

As exercises you can work out

5 ⊗5 = 10 ⊕ 15 (25)

and

5 ⊗5

∗

= 1 ⊕ 24 (26)

You should recognize the 24 as the adjoint.

Appendix B. Brief Review of Group Theory | 531

In physics we are often called upon to multiply a tensor by itself. Statistics then plays a role. For instance,

SU(5) grand unification contains a scalar field ϕ

transforming as 5. Because of Bose statistics, the product ϕ

contains only the 15.

Restriction to subgroup

To explain the next group theoretic concept, let me take a physical example. The SU (3) of Gell-Mann and Ne’eman

transforms the three quarks u, d , and s into linear combinations of each other. It contains as a subgroup the

isospin SU(2) of Heisenberg, which transforms u and d, but leaves s alone. In other words, upon restriction to

the subgroup SU(2) the irreducible representation 3 of SU(3) decomposes as

3 →2 ⊕ 1 (27)

Consider an irreducible representation with dimension d of some group G. When we restrict our attention

to a subgroup H , the set of d objects will in general decompose into n subsets, containing d

, d

,...,d

objects,

such that the objects of each subset only transform among themselves under the action of H . This makes obvious

sense since there are fewer transformations in H than in G.

The decomposition of the fundamental or defining representation specifies how the subgroup H is embedded

in G. Since all representations may be built up as products of the fundamental representation, once we know

how the fundamental representation decomposes, we know how all representations decompose. For example,

in SU(3)

3 ⊗3

∗

= 8 ⊕ 1 (28)

while in SU(2)

(2 ⊕ 1) ⊗ (2 ⊕ 1) = (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1 (29)

Comparing (28) and (29) we learn that

8 → 3 ⊕ 1 ⊕ 2 ⊕ 2. (30)

Alternatively, we can simply look at the tensors involved. Consider ϕ

of SU(3) where the index i takes on the

value 1, 2, 3. Let the index μ takes on the value 1, 2. Obviously, ϕ

={ϕ

, ϕ

} corresponds to an explicit display

of (27). Then ϕ

={¯ϕ

, ϕ

}, where the bar on ¯ϕ

is to remind us that it is traceless. This corresponds

precisely to (30).

Actually, SU(3) also contains the larger subgroup SU(2) ⊗ U(1), where the U(1) is generated by the traceless

hermitean matrix

⎛

⎜

⎝

−100

0 −10

002

⎞

⎟

⎠

We can then write (27) as 3 → (2, −1) ⊕ (1, 2), where the notation is almost self-explanatory. Thus, (2, −1)

denotes a 2 under SU(2) with “charge” −1 under U(1).

In the text, we will decompose various representations of SU(5) and SO(10). Everything we do there will

simply be somewhat more elaborate versions of what we did here.