Jeevanjee N. An Introduction to Tensors and Group Theory for Physicists

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xiv Contents

4.4 From Lie Groups to Lie Algebras ..................111

4.5 Lie Algebras—Deﬁnition, Properties, and Examples ........115

4.6 The Lie Algebras of Classical and Quantum Physics ........121

4.7 Abstract Lie Algebras ........................127

4.8 HomomorphismandIsomorphismRevisited ............131

4.9 Problems...............................138

5 Basic Representation Theory ......................145

5.1 Representations:DeﬁnitionsandBasicExamples..........145

5.2 FurtherExamples ..........................150

5.3 Tensor Product Representations ...................159

5.4 Symmetric and Antisymmetric Tensor Product Representations . . 165

5.5 Equivalence of Representations ...................169

5.6 Direct Sums and Irreducibility . ...................177

5.7 More on Irreducibility ........................184

5.8 The Irreducible Representations of su(2), SU(2) and SO(3) ....188

5.9 RealRepresentationsandComplexiﬁcations ............193

5.10 The Irreducible Representations of sl(2, C)

, SL(2, C) and

SO(3, 1)

...............................196

5.11 Irreducibility and the Representations of O(3, 1) and Its Double

Covers ................................204

5.12Problems...............................208

6 The Wigner–Eckart Theorem and Other Applications ........213

6.1 Tensor Operators, Spherical Tensors and Representation Operators 213

6.2 Selection Rules and the Wigner–Eckart Theorem . .........217

6.3 Gamma Matrices and Dirac Bilinears ................222

6.4 Problems...............................225

Appendix Complexiﬁcations of Real Lie Algebras and the Tensor

Product Decomposition of sl(2,C

Representations .........227

A.1 Direct Sums and Complexiﬁcations of Lie Algebras ........227

A.2 Representations of Complexiﬁed Lie Algebras and the Tensor

Product Decomposition of sl(2, C)

Representations........229

References ...................................235

Index ......................................237

Notation

Some Mathematical Shorthand

R The set of real numbers

C The set of complex numbers

Z The set of positive and negative integers

∈ “is an element of”, “an element of”, i.e. 2 ∈R reads “2 is an

element of the real numbers”

/∈ “is not an element of”

∀ “for all”

⊂ “is a subset of”, “a subset of”

≡ Denotes a deﬁnition

f :A →B Denotes a map f that takes elements of the set A into

elements of the set B

f :a →b Indicates that the map f sends the element a to the element b

◦ Denotes a composition of maps, i.e. if f :A →B and

g :B →C, then g ◦f :A →C is given by

(g ◦f )(a) ≡g(f (a))

A ×B The set {(a,b)} of all ordered pairs where a ∈A, b ∈B.

Referred to as the cartesian product of sets A and B.

Extends in the obvious way to n-fold products A

×···×A

R ×···×R



 

n times

C ×···×C



 

n times

{A |Q} Denotes a set A subject to condition Q. For instance, the set

of all even integers can be written as {x ∈R |x/2 ∈Z}

 Denotes the end of a proof or example

xvi Notation

Dirac Dictionary

Standard Notation Dirac Notation

Vector ψ ∈H |ψ

Dual Vector L(ψ) ψ|

Inner Product (ψ, φ) ψ|φ

A(ψ), A ∈L(H)A|ψ

(ψ, Aφ) ψ|A|φ

⊗e



i,j

|ji|

⊗e

|i|j or |i, j

We summarize here all of the translations given in the text between quantum-mechanical Dirac

notation and standard mathematical notation.

Part I

Linear Algebra and Tensors

Chapter 1

A Quick Introduction to Tensors

The reason tensors are introduced in a somewhat ad-hoc manner in most physics

courses is twofold: ﬁrst, a detailed and proper understanding of tensors requires

mathematics that is slightly more abstract than the standard linear algebra and vec-

tor calculus that physics students use everyday. Second, students do not necessarily

need such an understanding to be able to manipulate tensors and solve problems

with them. The drawback, of course, is that many students feel uneasy whenever

tensors are discussed, and they ﬁnd that they can use tensors for computation but

do not have an intuitive feel for what they are doing. One of the primary aims of

this book is to alleviate those feelings. Doing that, however, requires a modest in-

vestment (about 30 pages) in some abstract linear algebra, so before diving into the

details we will begin with a rough overview of what a tensor is, which hopefully

will whet your appetite and tide you over until we can discuss tensors in full detail

in Chap. 3.

Many older books deﬁne a tensor as a collection of objects which carry indices

and which ‘transform’ in a particular way speciﬁed by those indices. Unfortunately,

this deﬁnition usually does not yield much insight into what a tensor is. One of the

main purposes of the present text is to promulgate the more modern deﬁnition of

a tensor, which is equivalent to the old one but is more conceptual and is in fact

already standard in the mathematics literature. This deﬁnition takes a tensor to be a

function which eats a certain number of vectors (known as the rank r of the tensor)

and produces a number. The distinguishing characteristic of a tensor is a special

property called multilinearity, which means that it must be linear in each of its r

arguments (recall that linearity for a function with a single argument just means

that T(v+cw) =T(v)+cT (w) for all vectors v and w and numbers c). As we

will explain in a moment, this multilinearity enables us to express the value of the

function on an arbitrary set of r vectors in terms of the values of the function on

r basis vectors like

y, and

z. These values of the function on basis vectors are

nothing but the familiar components of the tensor, which in older treatments are

usually introduced ﬁrst as part of the deﬁnition of the tensor.

To make this concrete, consider a rank 2 tensor T , whose job it is to eat two

vectors v and w and produce a number which we will denote as T(v,w). For such

a tensor, multilinearity means

N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists,

DOI 10.1007/978-0-8176-4715-5_1, © Springer Science+Business Media, LLC 2011

4 1 A Quick Introduction to Tensors

T(v

+cv

,w)=T(v

,w)+cT (v

,w) (1.1)

T(v,w

+cw

) =T(v,w

) +cT (v, w

) (1.2)

for any number c and all vectors v and w. This means that if we have a coordinate

basis for our vector space, say

y and

z, then T is determined entirely by its values

on the basis vectors, as follows: ﬁrst, expand v and w in the coordinate basis as

v =v

x +v

y +v

w =w

x +w

y +w

Then by (1.1) and (1.2)wehave

T(v,w)=T(v

x +v

y +v

z,w

x +w

y +w

x,w

x +w

y +w

z) +v

y,w

x +w

y +w

z,w

x +w

y +w

x) +v

y) +v

z) +v

y) +v

z) +v

x) +v

z).

If we then abbreviate our notation as

≡T(

(1.3)

and so on, we have

T(v,w)=v

(1.4)

which may look familiar from discussion of tensors in the physics literature. In that

literature, the above equation is often part of the deﬁnition of a second rank tensor;

here, though, we see that its form is really just a consequence of multilinearity.

Another advantage of our approach is that the components {T

,...} of

T have a meaning beyond that of just being coefﬁcients that appear in expressions

like (1.4); from (1.3), we see that components are the values of the tensor when

evaluated on a given set of basis vectors. This fact is crucial in getting a feel for

tensors and what they mean.

Another nice feature of our deﬁnition of a tensor is that it allows us to derive

the tensor transformation laws which historically were taken as the deﬁnition of a

tensor. Say we switch to a new set of basis vectors {



} which are related to

the old basis vectors by



x +A



y +A



x +A



y +A



z (1.5)



x +A



y +A



1 A Quick Introduction to Tensors 5

This does not affect the action of T , since T exists independently of any basis,but

if we would like to compute the value of T(v,w) in terms of the new components



) and (w



) of v and w, then we will need to know what the new

components {T



,...} look like. Computing T



, for instance, gives







=T(A



x +A



y +A



z,A



x +A



y +A



. (1.6)

You probably recognize this as an instance of the standard tensor transformation

law, which used to be taken as the deﬁnition of a tensor. Here, the transformation

law is another consequence of multilinearity! In Chap. 3 we will introduce more

convenient notation, which will allow us to use the Einstein summation convention,

so that we can write the general form of (1.6)as



, (1.7)

a form which may be more familiar to some readers.

One common source of confusion is that in physics textbooks, tensors (usually

of the second rank) are often represented as matrices, so then the student wonders

what the difference is between a matrix and a tensor. Above, we have deﬁned a

tensor as a multilinear function on a vector space. What does that have to do with

a matrix? Well, if we choose a basis {

z}, we can then write the corresponding

components {T

,...} in the form of a matrix [T ] as

[T ]≡

⎛

⎝

⎞

⎠

. (1.8)

Equation (1.4) can then be written compactly as

T(v,w)=(v

)

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

(1.9)

where the usual matrix multiplication is implied. Thus, once a basis is chosen, the

action of T can be neatly expressed using the corresponding matrix [T ]. It is crucial

to keep in mind, though, that this association between a tensor and a matrix depends

entirely on a choice of basis, and that [T ] is useful mainly as a computational tool,

not a conceptual handle. T is best thought of abstractly as a multilinear function,

and [T ] as its representation in a particular coordinate system.

One possible objection to our approach is that matrices and tensors are often

thought of as linear operators which take vectors into vectors, as opposed to objects

which eat vectors and spit out numbers. It turns out, though, that for a second rank

6 1 A Quick Introduction to Tensors

tensor these two notions are equivalent. Say we have a linear operator R; then we

can turn R into a second rank tensor T

(v, w) ≡v ·Rw (1.10)

where · denotes the usual dot product of vectors. You can roughly interpret this

tensor as taking in v and w and giving back the “component of Rw lying along v”.

You can also easily check that T

is multilinear, i.e. it satisﬁes (1.1) and (1.2). If we

compute the components of T

we ﬁnd that, for instance,

)

(

x ·R

x ·(R

x +R

y +R

so the components of the tensor T

are the same as the components of the linear

operator R! In components, the action of T

then looks like

(v, w) =(v

)

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

(1.11)

which is identical to (1.9). This makes it obvious how to turn a linear operator R into

a second rank tensor—just sandwich the component matrix in between two vectors!

This whole process is also reversible: we can turn a second rank tensor T into a

linear operator R

by deﬁning the components of the vector R

(v) as



(v)



≡T(v,

x), (1.12)

and similarly for the other components. One can check that these processes are

inverses of each other, so this sets up a one-to-one correspondence between linear

operators and second rank tensors and we can thus regard them as equivalent. Since

the matrix form of both is identical, one often does not bother trying to clarify

exactly which one is at hand, and often times it is just a matter of interpretation.

How does all this work in a physical context? One nice example is the rank 2

moment-of-inertia tensor I, familiar from rigid body dynamics in classical mechan-

ics. In most textbooks, this tensor usually arises when one considers the relationship

between the angular velocity vector ω and the angular momentum L or kinetic en-

ergy KE of a rigid body. If one chooses a basis {

z} and then expresses, say, KE

in terms of ω, one gets an expression like (1.4) with v =w =ω and T

.From

this expression, most textbooks then ﬁgure out how the I

must transform under a

change of coordinates, and this behavior under coordinate change is then what iden-

tiﬁes the I

as a tensor. In this text, though, we will take a different point of view.

We will deﬁne I to be the function which, for a given rigid body, assigns to a state

of rotation (described by the angular momentum vector ω) twice the corresponding

kinetic energy KE. One can then calculate KE in terms of ω, which yields an ex-

pression of the form (1.4); this then shows that I is a tensor in the old sense of the

1 A Quick Introduction to Tensors 7

term, which also means that it is a tensor in the modern sense of being a multilinear

function. We can then think of I as a second rank tensor deﬁned by

KE =

I(ω, ω). (1.13)

From our point of view, we see that the components of I have a physical meaning;

for instance, I

= I(

x) is just twice the kinetic energy of the rigid body when

ω =

x. When one actually needs to compute the kinetic energy for a rigid body,

one usually picks a convenient coordinate system, computes the components of I

using the standard formulas,

and then uses the convenient matrix representation to

compute

KE =

(ω

,ω

)

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎝

⎞

⎠

, (1.14)

which is the familiar expression often found in mechanics texts.

As mentioned above, one can also interpret I as a linear operator which takes

vectors into vectors. If we let I act on the angular velocity vector, we get

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎝

⎞

⎠

(1.15)

which you probably recognize as the coordinate expression for the angular momen-

tum L. Thus, I can be interpreted either as the rank 2 tensor which eats two copies

of the angular velocity vector ω and produces the kinetic energy, or as the linear

operator which takes ω into L. The two deﬁnitions are equivalent.

Many other tensors which we will consider in this text are also nicely understood

as multilinear functions which happen to have convenient component and matrix

representations. These include the electromagnetic ﬁeld tensor of electrodynam-

ics, as well as the metric tensors of Newtonian and relativistic mechanics. As we

progress we will see how we can use our new point of view to get a handle on these

usually somewhat enigmatic objects.

See Example 3.14 or any standard textbook such as Goldstein [6].

Chapter 2

Vector Spaces

Since tensors are a special class of functions deﬁned on vector spaces, we must

have a good foundation in linear algebra before discussing them. In particular, you

will need a little bit more linear algebra than is covered in most sophomore or ju-

nior level linear algebra/ODE courses. This chapter starts with the familiar material

about vectors, bases, linear operators etc. but eventually moves on to slightly more

sophisticated topics that are essential for understanding tensors in physics. As we

lay this groundwork, hopefully you will ﬁnd that our slightly more abstract view-

point also clariﬁes the nature of many of the objects in physics you have already

encountered.

2.1 Deﬁnition and Examples

We begin with the deﬁnition of an abstract vector space. We are taught as under-

graduates to think of vectors as arrows with a head and a tail, or as ordered triples

of real numbers, but physics, and especially quantum mechanics, requires a more

abstract notion of vectors. Before reading the deﬁnition of an abstract vector space,

keep in mind that the deﬁnition is supposed to distill all the essential features of

vectors as we know them (like addition and scalar multiplication) while detaching

the notion of a vector space from speciﬁc constructs, like ordered n-tuples of real or

complex numbers (denoted as R

and C

, respectively). The mathematical utility of

this is that much of what we know about vector spaces depends only on the essential

properties of addition and scalar multiplication, not on other properties particular to

or C

. If we work in the abstract framework and then come across other math-

ematical objects that do not look like R

or C

but that are abstract vector spaces,

then most everything we know about R

and C

will apply to these spaces as well.

Physics also forces us to use the abstract deﬁnition since many quantum-mechanical

vector spaces are inﬁnite-dimensional and cannot be viewed as C

or R

for any n.

An added dividend of the abstract approach is that we will learn to think about vector

spaces independently of any basis, which will prove very useful.

N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists,