Quantum Field Theory for the Gifted Amateur

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Quantum Field Theory
for the Gifted Amateur
Tom Lancaster
Department of Physics, University of Durham

Stephen J. Blundell
Department of Physics, University of Oxford

3
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3

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Preface
BRICK: Well, they say nature hates a vacuum, Big Daddy.
BIG DADDY: That’s what they say, but sometimes I think
that a vacuum is a hell of a lot better than some of the stuff
that nature replaces it with.

Tennessee Williams (1911–1983) Cat on a Hot Tin Roof
Quantum field theory is arguably the most far-reaching and beautiful
physical theory ever constructed. It describes not only the quantum vacuum, but also the stuff that nature replaces it with. Aspects of quantum
field theory are also more stringently tested, as well as verified to greater
precision, than any other theory in physics. The subject nevertheless has
a reputation for difficulty which is perhaps well-deserved; its practitioners not only manipulate formidable equations but also depict physical
processes using a strange diagrammatic language consisting of bubbles,
wiggly lines, vertices, and other geometrical structures, each of which
has a well defined quantitative significance. Learning this mathematical
and geometrical language is an important initiation rite for any aspiring theoretical physicist, and a quantum field theory graduate course is
found in most universities, aided by a large number of weighty quantum
field theory textbooks. These books are written by professional quantum field theorists and are designed for those who aspire to join them
in that profession. Consequently they are frequently thorough, serious
minded and demand a high level of mathematical sophistication.
The motivation for our book is the idea that quantum field theory is
too important, too beautiful and too engaging to be restricted to the
professionals. Experimental physicists, or theoretical physicists in other
fields, would benefit greatly from knowing some quantum field theory,
both to understand research papers that use these ideas and also to
comprehend and appreciate the important insights that quantum field
theory has to offer. Quantum field theory has given us such a radically
different and revolutionary view of the physical world that we think
that more physicists should have the opportunity to engage with it.
The problem is that the existing texts require far too much in the way
of advanced mathematical facility and provide too little in the way of
physical motivation to assist those who want to learn quantum field
theory but not to be professional quantum field theorists. The gap
between an undergraduate course on quantum mechanics and a graduate
level quantum field theory textbook is a wide and deep chasm, and one
of the aims of this book is to provide a bridge to cross it. That being

said, we are not assuming the readers of this are simple-minded folk who

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vi Preface

1

After all, with the number of chapters
we ended up including, we could have
called it ‘Fifty shades of quantum field
theory’.

2

T.S. Eliot on Ulysses.

can be fobbed off with a trite analogy as a substitute for mathematical
argument. We aim to introduce all the maths but, by using numerous
worked examples and carefully worded motivations, to smooth the path
for understanding in a manner we have not found in the existing books.
We have chosen this book’s title with great care.1 Our imagined reader
is an amateur, wanting to learn quantum field theory without (at least
initially) joining the ranks of professional quantum field theorists; but
(s)he is gifted, possessing a curious and adaptable mind and willing to
embark on a significant intellectual challenge; (s)he has abundant curiosity about the physical world, a basic grounding in undergraduate
physics, and a desire to be told an entertaining and intellectually stimulating story, but will not feel patronized if a few mathematical niceties
are spelled out in detail. In fact, we suspect and hope that our book will
find wide readership amongst the graduate trainee quantum field theorists who will want to use it in conjunction with one of the traditional

texts (for learning most hard subjects, one usually needs at least two
books in order to get a more rounded picture).
One feature of our book is the large number of worked examples, which
are set in slightly smaller type. They are integral to the story, and flesh
out the details of calculations, but for the more casual reader the guts
of the argument of each chapter is played out in the main text. To
really get to grips with the subject, the many examples should provide
transparent demonstrations of the key ideas and understanding can be
confirmed by tackling the exercises at the end of each chapter. The
chapters are reasonably short, so that the development of ideas is kept
at a steady pace and each chapter ends with a summary of the key ideas
introduced.
Though the vacuum plays a big part in the story of quantum field theory, we have not been writing in one. In many ways the present volume
represents a compilation of some of the best ideas from the literature
and, as a result, we are indebted to these other books for providing
the raw material for many of our arguments. There is an extensive list
of further reading in Appendix A where we acknowledge our sources,
but we note here, in particular, the books by Zee and by Peskin and
Schroeder and their legendary antecedent: the lectures in quantum field
theory by Sidney Coleman. The latter are currently available online as
streamed videos and come highly recommended. Also deserving of special mention is the text by Weinberg which is ‘a book to which we are
all indebted, and from which none of us can escape.’2
It is a pleasure to acknowledge the help we have received from various sources in writing this book. Particular mention is due to Săonke
Adlung at Oxford University Press who has helped steer this project to
completion. No authors could wish for a more supportive editor and
we thank him, Jessica White and the OUP team, particularly Mike Nugent, our eagle-eyed copy editor. We are very grateful for the comments
and corrections we received from a number of friends and colleagues who
kindly gave up their time to read drafts of various chapters: Peter Byrne,
Claudio Castelnovo, John Chalker, Martin Galpin, Chris Maxwell, Tom


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Preface vii

McLeish, Johannes Mă
oller, Paul Tulip and Rob Williams. They deserve
much credit for saving us from various embarrassing errors, but any that
remain are due to us; those that we find post-publication will be posted
on the book’s website:
/>For various bits of helpful information, we thank Hideo Aoki, Nikitas Gidopoulos, Paul Goddard and John Singleton. Our thanks are also due to
various graduate students at Durham and Oxford who have unwittingly
served as guinea pigs as we tried out various ways of presenting this
material in graduate lectures. Finally we thank Cally and Katherine for
their love and support.
TL & SJB
Durham & Oxford
January 2, 2014

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Contents
0 Overture
0.1 What is quantum field theory?
0.2 What is a field?
0.3 Who is this book for?

0.4 Special relativity
0.5 Fourier transforms
0.6 Electromagnetism

1
1
2
2
3
6
7

I

9

The Universe as a set of harmonic oscillators

1 Lagrangians
1.1 Fermat’s principle
1.2 Newton’s laws
1.3 Functionals
1.4 Lagrangians and least action
1.5 Why does it work?
Exercises

10
10
10
11

14
16
17

2 Simple harmonic oscillators
2.1 Introduction
2.2 Mass on a spring
2.3 A trivial generalization
2.4 Phonons
Exercises

19
19
19
23
25
27

3 Occupation number representation
3.1 A particle in a box
3.2 Changing the notation
3.3 Replace state labels with operators
3.4 Indistinguishability and symmetry
3.5 The continuum limit
Exercises

28
28
29
31

31
35
36

4 Making second quantization work
4.1 Field operators
4.2 How to second quantize an operator
4.3 The kinetic energy and the tight-binding Hamiltonian
4.4 Two particles

37
37
39
43
44

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x Contents

4.5 The Hubbard model
Exercises

II

Writing down Lagrangians

46
48


49

5 Continuous systems
5.1 Lagrangians and Hamiltonians
5.2 A charged particle in an electromagnetic field
5.3 Classical fields
5.4 Lagrangian and Hamiltonian density
Exercises

50
50
52
54
55
58

6 A first stab at relativistic quantum mechanics
6.1 The Klein–Gordon equation
6.2 Probability currents and densities
6.3 Feynman’s interpretation of the negative energy states
6.4 No conclusions
Exercises

59
59
61
61
63
63


7 Examples of Lagrangians, or how to write down a theory
7.1 A massless scalar field
7.2 A massive scalar field
7.3 An external source
7.4 The φ4 theory
7.5 Two scalar fields
7.6 The complex scalar field
Exercises

64
64
65
66
67
67
68
69

III

71

The need for quantum fields

8 The passage of time
8.1 Schră
odingers picture and the time-evolution operator
8.2 The Heisenberg picture
8.3 The death of single-particle quantum mechanics

8.4 Old quantum theory is dead; long live fields!
Exercises

72
72
74
75
76
78

9 Quantum mechanical transformations
9.1 Translations in spacetime
9.2 Rotations
9.3 Representations of transformations
9.4 Transformations of quantum fields
9.5 Lorentz transformations
Exercises

79
79
82
83
85
86
88

10 Symmetry
10.1 Invariance and conservation

90

90

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Contents xi

10.2 Noether’s theorem
10.3 Spacetime translation
10.4 Other symmetries
Exercises

92
94
96
97

11 Canonical quantization of fields
11.1 The canonical quantization machine
11.2 Normalizing factors
11.3 What becomes of the Hamiltonian?
11.4 Normal ordering
11.5 The meaning of the mode expansion
Exercises

98
98
101
102
104

106
108

12 Examples of canonical quantization
12.1 Complex scalar field theory
12.2 Noether’s current for complex scalar field theory
12.3 Complex scalar field theory in the non-relativistic limit
Exercises

109
109
111
112
116

13 Fields with many components and
massive electromagnetism
13.1 Internal symmetries
13.2 Massive electromagnetism
13.3 Polarizations and projections
Exercises

117
117
120
123
125

14 Gauge fields and gauge theory
14.1 What is a gauge field?

14.2 Electromagnetism is the simplest gauge theory
14.3 Canonical quantization of the electromagnetic field
Exercises

126
126
129
131
134

15 Discrete transformations
135
15.1 Charge conjugation
135
15.2 Parity
136
15.3 Time reversal
137
15.4 Combinations of discrete and continuous transformations 139
Exercises
142

IV

Propagators and perturbations

16 Propagators and Green’s functions
16.1 What is a Green’s function?
16.2 Propagators in quantum mechanics
16.3 Turning it around: quantum mechanics from the

propagator and a first look at perturbation theory
16.4 The many faces of the propagator
Exercises

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143
144
144
146
149
151
152


xii Contents

17 Propagators and fields
17.1 The field propagator in outline
17.2 The Feynman propagator
17.3 Finding the free propagator for scalar field theory
17.4 Yukawa’s force-carrying particles
17.5 Anatomy of the propagator
Exercises

154
155
156
158
159

162
163

18 The S-matrix
18.1 The S-matrix: a hero for our times
18.2 Some new machinery: the interaction representation
18.3 The interaction picture applied to scattering
18.4 Perturbation expansion of the S-matrix
18.5 Wick’s theorem
Exercises

165
166
167
168
169
171
174

19 Expanding the S-matrix: Feynman diagrams
19.1 Meet some interactions
19.2 The example of φ4 theory
19.3 Anatomy of a diagram
19.4 Symmetry factors
19.5 Calculations in p-space
19.6 A first look at scattering
Exercises

175
176

177
181
182
183
186
187

20 Scattering theory
20.1 Another theory: Yukawa’s ψ † ψφ interactions
20.2 Scattering in the ψ † ψφ theory
20.3 The transition matrix and the invariant amplitude
20.4 The scattering cross-section
Exercises

188
188
190
192
193
194

V

195

Interlude: wisdom from statistical physics

21 Statistical physics: a crash course
21.1 Statistical mechanics in a nutshell
21.2 Sources in statistical physics

21.3 A look ahead
Exercises

196
196
197
198
199

22 The generating functional for fields
22.1 How to find Green’s functions
22.2 Linking things up with the Gell-Mann–Low theorem
22.3 How to calculate Green’s functions with diagrams
22.4 More facts about diagrams
Exercises

201
201
203
204
206
208

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Contents xiii

VI


Path integrals

209

23 Path integrals: I said to him, ‘You’re crazy’
23.1 How to do quantum mechanics using path integrals
23.2 The Gaussian integral
23.3 The propagator for the simple harmonic oscillator
Exercises

210
210
213
217
220

24 Field integrals
24.1 The functional integral for fields
24.2 Which field integrals should you do?
24.3 The generating functional for scalar fields
Exercises

221
221
222
223
226

25 Statistical field theory
25.1 Wick rotation and Euclidean space

25.2 The partition function
25.3 Perturbation theory and Feynman rules
Exercises

228
229
231
233
236

26 Broken symmetry
26.1 Landau theory
26.2 Breaking symmetry with a Lagrangian
26.3 Breaking a continuous symmetry: Goldstone modes
26.4 Breaking a symmetry in a gauge theory
26.5 Order in reduced dimensions
Exercises

237
237
239
240
242
244
245

27 Coherent states
27.1 Coherent states of the harmonic oscillator
27.2 What do coherent states look like?
27.3 Number, phase and the phase operator

27.4 Examples of coherent states
Exercises

247
247
249
250
252
253

28 Grassmann numbers: coherent states
and the path integral for fermions
28.1 Grassmann numbers
28.2 Coherent states for fermions
28.3 The path integral for fermions
Exercises

255
255
257
257
258

VII

259

Topological ideas

29 Topological objects

29.1 What is topology?
29.2 Kinks
29.3 Vortices
Exercises

260
260
262
264
266

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xiv Contents

30 Topological field theory
30.1 Fractional statistics `a la Wilczek:
the strange case of anyons
30.2 Chern–Simons theory
30.3 Fractional statistics from Chern–Simons theory
Exercises

267

VIII

273

Renormalization: taming the infinite


267
269
271
272

31 Renormalization, quasiparticles and the Fermi surface 274
31.1 Recap: interacting and non-interacting theories
274
31.2 Quasiparticles
276
31.3 The propagator for a dressed particle
277
31.4 Elementary quasiparticles in a metal
279
31.5 The Landau Fermi liquid
280
Exercises
284
32 Renormalization: the problem and its solution
32.1 The problem is divergences
32.2 The solution is counterterms
32.3 How to tame an integral
32.4 What counterterms mean
32.5 Making renormalization even simpler
32.6 Which theories are renormalizable?
Exercises

285
285

287
288
290
292
293
294

33 Renormalization in action:
propagators and Feynman diagrams
295
33.1 How interactions change the propagator in perturbation
theory
295
33.2 The role of counterterms: renormalization conditions
297
33.3 The vertex function
298
Exercises
300
34 The renormalization group
34.1 The problem
34.2 Flows in parameter space
34.3 The renormalization group method
34.4 Application 1: asymptotic freedom
34.5 Application 2: Anderson localization
34.6 Application 3: the Kosterlitz–Thouless transition
Exercises

302
302

304
305
307
308
309
312

35 Ferromagnetism: a renormalization group tutorial
35.1 Background: critical phenomena and scaling
35.2 The ferromagnetic transition and critical phenomena
Exercises

313
313
315
320

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Contents xv

IX

Putting a spin on QFT

321

36 The Dirac equation
322

36.1 The Dirac equation
322
36.2 Massless particles: left- and right-handed wave functions 323
36.3 Dirac and Weyl spinors
327
36.4 Basis states for superpositions
330
36.5 The non-relativistic limit of the Dirac equation
332
Exercises
334
37 How to transform a spinor
37.1 Spinors aren’t vectors
37.2 Rotating spinors
37.3 Boosting spinors
37.4 Why are there four components in the Dirac equation?
Exercises

336
336
337
337
339
340

38 The quantum Dirac field
38.1 Canonical quantization and Noether current
38.2 The fermion propagator
38.3 Feynman rules and scattering
38.4 Local symmetry and a gauge theory for fermions

Exercises

341
341
343
345
346
347

39 A rough guide to quantum electrodynamics
39.1 Quantum light and the photon propagator
39.2 Feynman rules and a first QED process
39.3 Gauge invariance in QED
Exercises

348
348
349
351
353

40 QED scattering: three famous cross-sections
40.1 Example 1: Rutherford scattering
40.2 Example 2: Spin sums and the Mott formula
40.3 Example 3: Compton scattering
40.4 Crossing symmetry
Exercises

355
355

356
357
358
359

41 The renormalization of QED and two great results
360
41.1 Renormalizing the photon propagator: dielectric vacuum 361
41.2 The renormalization group and the electric charge
364
41.3 Vertex corrections and the electron g-factor
365
Exercises
368

X Some applications from the world
of condensed matter

369

42 Superfluids
42.1 Bogoliubov’s hunting license

370
370

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xvi Contents


42.2 Bogoliubov’s transformation
42.3 Superfluids and fields
42.4 The current in a superfluid
Exercises

372
374
377
379

43 The many-body problem and the metal
43.1 Mean-field theory
43.2 The Hartree–Fock ground state energy of a metal
43.3 Excitations in the mean-field approximation
43.4 Electrons and holes
43.5 Finding the excitations with propagators
43.6 Ground states and excitations
43.7 The random phase approximation
Exercises

380
380
383
386
388
389
390
393
398


44 Superconductors
44.1 A model of a superconductor
44.2 The ground state is made of Cooper pairs
44.3 Ground state energy
44.4 The quasiparticles are bogolons
44.5 Broken symmetry
44.6 Field theory of a charged superfluid
Exercises

400
400
402
403
405
406
407
409

45 The fractional quantum Hall fluid
45.1 Magnetic translations
45.2 Landau Levels
45.3 The integer quantum Hall effect
45.4 The fractional quantum Hall effect
Exercises

411
411
413
415

417
421

XI Some applications from the world
of particle physics

423

46 Non-abelian gauge theory
46.1 Abelian gauge theory revisited
46.2 Yang–Mills theory
46.3 Interactions and dynamics of Wµ
46.4 Breaking symmetry with a non-abelian gauge theory
Exercises

424
424
425
428
430
432

47 The Weinberg–Salam model
47.1 The symmetries of Nature before symmetry breaking
47.2 Introducing the Higgs field
47.3 Symmetry breaking the Higgs field
47.4 The origin of electron mass
47.5 The photon and the gauge bosons
Exercises


433
434
437
438
439
440
443

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Contents xvii

48 Majorana fermions
48.1 The Majorana solution
48.2 Field operators
48.3 Majorana mass and charge
Exercises

444
444
446
447
450

49 Magnetic monopoles
49.1 Dirac’s monopole and the Dirac string
49.2 The ’t Hooft–Polyakov monopole
Exercises


451
451
453
456

50 Instantons, tunnelling and the end of the world
50.1 Instantons in quantum particle mechanics
50.2 A particle in a potential well
50.3 A particle in a double well
50.4 The fate of the false vacuum
Exercises

457
458
459
460
463
466

A Further reading

467

B Useful complex analysis
B.1 What is an analytic function?
B.2 What is a pole?
B.3 How to find a residue
B.4 Three rules of contour integrals
B.5 What is a branch cut?
B.6 The principal value of an integral


473
473
474
474
475
477
478

Index

480

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0

Overture
To begin at the beginning
Dylan Thomas (1914–1953)

0.1 What is quantum field theory?
1
0.2 What is a field?

Beginnings are always troublesome

George Eliot (1819–1880)

0.1

What is quantum field theory?

Every particle and every wave in the Universe is simply an excitation
of a quantum field that is defined over all space and time.
That remarkable assertion is at the heart of quantum field theory. It
means that any attempt to understand the fundamental physical laws
governing elementary particles has to first grapple with the fundamentals
of quantum field theory. It also means that any description of complicated interacting systems, such as are encountered in the many-body
problem and in condensed matter physics, will involve quantum field
theory to properly describe the interactions. It may even mean, though
at the time of writing no-one knows if this is true, that a full theory
of quantum gravity will be some kind of quantum upgrade of general
relativity (which is a classical field theory). In any case, quantum field
theory is the best theory currently available to describe the world around
us and, in a particular incarnation known as quantum electrodynamics
(QED), is the most accurately tested physical theory. For example, the
magnetic dipole moment of the electron has been tested to ten significant
figures.
The ideas making up quantum field theory have profound consequences. They explain why all electrons are identical (the same argument works for all photons, all quarks, etc.) because each electron is
an excitation of the same electron quantum field and therefore it’s not
surprising that they all have the same properties. Quantum field theory
also constrains the symmetry of the representations of the permutation
symmetry group of any class of identical particles so that some classes
obey Fermi–Dirac statistics and others Bose–Einstein statistics. Interactions in quantum field theory involve products of operators which are
found to create and annihilate particles and so interactions correspond
to processes in which particles are created or annihilated; hence there


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2

0.3 Who is this book for?

2

0.4 Special relativity

3

0.5 Fourier transforms

6

0.6 Electromagnetism

7


2 Overture

is also the possibility of creating and destroying virtual particles which
mediate forces.

0.2

xµ


φ(xµ)
Fig. 1 A field is some kind of machine that takes a position in spacetime, given by the coordinates xµ , and
outputs an object representing the amplitude of something at that point in
spacetime. Here the output is the
scalar φ(xµ ) but it could be, for example, a vector, a complex number, a
spinor or a tensor.

What is a field?

This is all very well, but what is a field? We will think of a field as some
kind of machine that takes a position in spacetime and outputs an object representing the amplitude of something at that point in spacetime
(Fig. 1). The amplitude could be a scalar, a vector, a complex number,
a spinor or a tensor. This concept of a field, an unseen entity which
pervades space and time, can be traced back to the study of gravity due
to Kepler and ultimately Newton, though neither used the term and
the idea of action-at-a-distance between two gravitationally attracting
bodies seemed successful but nevertheless utterly mysterious. Euler’s
fluid dynamics got closer to the matter by considering what we would
now think of as a velocity field which modelled the movement of fluid at
every point in space and hence its capacity to do work on a test particle imagined at some particular location. Faraday, despite (or perhaps
because of) an absence of mathematical schooling, grasped intuitively
the idea of an electric or magnetic field that permeates all space and
time, and although he first considered this a convenient mental picture
he began to become increasingly convinced that his lines of force had an
independent physical existence. Maxwell codified Faraday’s idea and the
electromagnetic field, together with all the paraphernalia of field theory,
was born.
Thus in classical physics we understand that gravity is a field, electromagnetism is a field, and each can be described by a set of equations
which governs their behaviour. The field can oscillate in space and time

and thus wave-like excitations of the field can be found (electromagnetic
waves are well-known, but gravity waves are still to be observed). The
advent of quantum mechanics removed the distinction between what had
been thought of as wave-like objects and particle-like objects. Therefore
even matter itself is an excitation of a quantum field and quantum fields
become the fundamental objects which describe reality.

0.3

Who is this book for?

Quantum field theory is undoubtedly important, but it is also notoriously difficult. Forbidding-looking integrals and a plethora of funny
squiggly Feynman diagrams are enough to strike fear in many a heart
and stomach. The situation is not helped by the fact that the many excellent existing books are written by exceedingly clever practioners who
structure their explanations with the aspiring professional in mind. This
book is designed to be different. It is written by experimental physicists
and aimed at the interested amateur. Quantum field theory is too interesting and too important to be reserved for professional theorists.
However, though our imagined reader is not necessarily an aspiring pro-

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0.4

Special relativity 3

fessional (though we hope quite a few will be) we will assume that (s)he
is enthusiastic and has some familiarity with non-relativistic quantum
mechanics, special relativity and Fourier transforms at an undergraduate physics level. In the remainder of this chapter we will review a few
basic concepts that will serve to establish some conventions of notation.


0.4

Special relativity

Quantum fields are defined over space and time and so we need a proper
description of spacetime, and so we will need to use Einstein’s special
theory of relativity which asserts that the speed c of light is the same
in every inertial frame. This theory implies that the coordinates of an
event in a frame S and a frame S¯ (moving relative to frame S at speed
v along the x-axis) are related by the Lorentz transformation
vx
t¯ = γ t − 2 ,
c
x
¯ = γ(x − vt),
y¯ = y,
z¯ = z,

(1)

where γ = (1 − β 2 )−1/2 and β = v/c. Because the speed of light sets
the scale for all speeds, we will choose units such that c = 1. For similar
reasons1 we will also set = 1.
A good physical theory is said to be covariant if it transforms sensibly under coordinate transformations.2 In particular, we require that
quantities should be Lorentz covariant if they are to transform appropriately under the elements of the Lorentz group (which include the
Lorentz transformations of special relativity, such as eqn 1). This will
require us to write our theory in terms of certain well-defined mathematical objects, such as scalars, vectors and tensors.3
• Scalars: A scalar is a number, and will take the same value in
every inertial frame. It is thus said to be Lorentz invariant.

Examples of scalars include the electric charge and rest mass of a
particle.
• Vectors: A vector can be thought of as an arrow. In a particular
basis it can be described by a set of components. If the basis
is rotated, then the components will change, but the length of
the arrow will be unchanged (the length of a vector is a scalar).
In spacetime, vectors have four components and are called fourvectors. A four-vector is an object which has a single time-like
component and three space-like components. Three-vectors will be
displayed in bold italics, such as x or p for position and momentum
respectively. The components of three-vectors are listed with a
Roman index taken from the middle of the alphabet: e.g. xi , with
i = 1, 2, 3. Four-vectors are made from a time-like part and a
space-like part and are displayed in italic script, so position in

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1

In some of the early chapters we will
include the factors of and c so that
the reader can make better contact with
what they already know, and will give
notice when we are going to remove
them.
2

A good counterexample is the twocomponent ‘shopping vector’ that contains the price of fish and the price of
bread in each component. If you approach the supermarket checkout with
the trolley at 45◦ to the vertical, you
will soon discover that the prices of

your shopping will not transform appropriately.
3

We will postpone discussion of tensors
until the end of this section.


4 Overture

spacetime is written x where x = (t, x). Components for fourvectors will be given a Greek index, so for example xµ where µ =
0, 1, 2, 3. We say that the zeroth component, x0 , is time-like.

Example 0.1
4

These are written with c = 1.

Some other examples of four-vectors4 are:
ã the energy-momentum four-vector p = (E, p),

5

Though strictly à refers only to the
µth component of the four-vector operator ∂, rather than to the whole thing,
we will sometimes write a subscript (or
superscript) in expressions like this to
indicate whether coordinates are listed
with the indices lowered or with them
raised.


6

7

In full, this equation would be
ô
X x
à
a
à =
a .
x


These unfortunate terms are due
to the English mathematician J. J.
Sylvester (1814–1897). Both types of
vectors transform covariantly, in the
sense of ‘properly’, and we wish to retain this sense of the word ‘covariant’
rather than using it to simply label one
type of object that transforms properly.
Thus we will usually specify whether
the indices on a particular object are
‘upstairs’ (like aµ ) or ‘downstairs’ (like
∂µ φ) and their transformation properties can then be deduced accordingly.

• the current density four-vector j = (ρ, j),

• the vector potential four-vector A = (V, A).


The four-dimensional derivative operator ∂µ is also a combination of a time-like part
and a space-like part, and is defined5 by
„
«
« „
∂
∂
∂ ∂ ∂ ∂
∂µ ≡
=
.
(2)
,
∇
=
,
,
,
∂xµ
∂t
∂t ∂x ∂y ∂z
Note the lower index written in ∂µ , contrasting with the upper index on four-vectors
like xµ , which means that the four-dimensional derivative is ‘naturally lowered’. This
is significant, as we will now describe.

A general coordinate transformation from one inertial frame to another
maps {xµ } → {¯
xµ }, and the vector aµ transforms as
a
¯µ =


∂x
¯µ
∂xν

aν .

(3)

Here we have used the Einstein summation convention, by which
twice repeated indices are assumed to be summed.6 Certain other
vectors will transform differently. For example, the gradient vector
∂µ φ ≡ ∂φ/∂xµ transforms as
∂φ
=
∂x
¯µ

∂xν
∂x
¯µ

∂φ
.
∂xν

(4)

The jargon is that aµ transforms like a contravariant vector and
∂φ/∂xµ ≡ ∂µ φ transforms like a covariant vector,7 though we will

avoid these terms and just note that aµ has its indices ‘upstairs’ and
∂µ φ has them ‘downstairs’ and they will then transform accordingly.
The Lorentz transformation (eqn 1) can be rewritten in matrix form
as

 


t¯
γ
−βγ 0 0
t
 x
 


γ
0 0 
 ¯  =  −βγ
 x ,
(5)
 y¯   0


0
1 0
y 
z¯
0
0

0 1
z
or for short as

x
¯µ = Λµν xν ,
Λµν

µ

(6)

ν

where
≡ (∂ x
¯ /∂x ) is the Lorentz transformation matrix. In the
same way, the energy-momentum four-vector transforms as
p¯µ = Λµν pν .

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(7)


0.4

A downstairs vector aµ transforms as
a
¯µ = Λµν aν ,


(8)

¯µ ) is the inverse of the Lorentz transformation
where Λµν ≡ (∂xν /∂ x
µ 8
matrix Λ ν .
The Lorentz transformation changes components but leaves the length
of the four-vector x unchanged. This length is given by the square root9
of
|x|2 = x · x = (x0 )2 − (x1 )2 − (x2 )2 − (x3 )2 .
(10)
10

In general, the four-vector inner product

is

Special relativity 5

8
Note that in the equation x
¯µ
we have
0
γ
βγ 0 0
B βγ
γ
0 0

à = B
@ 0
0
1 0
0
0
0 1

In fact

à à =



x
à
x

ô

x
x
à

ô

= à xν
1

C

C.
A

(9)

= δνρ ,

where the Kronecker delta δij is defined by

1 i=j
δij =
0 i = j.

0 0

a · b = a b − a · b,

(11)

a · b = gµν aµ bν ,

(12)

The δij symbol is named after Leopold
Kronecker (1823–1891).

(13)

9
Of course |x| can be negative in special

relativity, so it is better to deal with
|x|2 , the square of the length.

which we can write
where the metric tensor gµν is

1
 0

gµν = 
0
0

given by
0
−1
0
0


0
0
0
0 
.
−1 0 
0 −1

10


Upstairs and downstairs vectors are related by the metric tensor via
aµ = gµν aν ,

(14)

so that we can lower or raise an index by inserting the metric tensor.
The form of the metric tensor in eqn 13 allows us to write
a0 = a0

ai = −ai ,

Note that other conventions are possible and some books write a · b =
−a0 b0 + a · b and define their metric
tensor differently. This is an entirely legitimate alternative lifestyle choice, but
it’s best to stick to one convention in a
single book.

(15)
Exercise: You can check that

and hence
µ ν

µ

µ

a · b = gµν a b = aµ b = a bµ .

Note also that a · b = g


µν

aµ bν and gµν = g

µν

(16)

.

ρ
gµν g νρ = δµ

and also that
Λµν = gµκ Λκρ g ρν .

Example 0.2
(i) An example of an inner product is
p · p = pµ pµ = (E, p) · (E, p) = E 2 − p2 = m2 ,

(17)

where m is the rest mass of the particle.
ν

∂x
ν
µ = 4
(ii) The combination ∂µ xν = ∂x

µ = δµ and hence the inner product ∂µ x
(remember the Einstein summation convention).

(iii) The d’Alembertian operator ∂ 2 is given by a product of two derivative operators (and is the four-dimensional generalization of the Laplacian operator).
It is written as
∂2
∂2
∂2
∂2
∂ 2 = ∂ µ ∂µ =
−
−
−
(18)
∂t2
∂x2
∂y 2
∂z 2
∂2
=
− ∇2 .
(19)
∂t2

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Named in honour of the French mathematician Jean le Rond d’Alembert
(1717–1783).
In some texts the
d’Alembertian is written as ✷ and in

some as ✷2 . Because of this confusion,
we will avoid the ✷ symbol altogether.


6 Overture
i···k
To complete this discussion, we can also define a general tensor Tℓ···n
with an arbitrary set of upstairs and downstairs indices. This transforms
as
′
′
′
∂x
¯k ∂xℓ
∂xn i···k
∂x
¯i
···k′
·
·
·
T
.
(20)
T¯ℓi′ ···n
′ =
′ ···
i
k
ℓ

∂x
∂x ∂ x
¯
∂x
¯n′ ℓ···n

Example 0.3
11

It has two indices (hence second
rank) and one is upstairs, one is downstairs (hence mixed).

12

This is named after Italian physicist Tullio Levi-Civita (1873–1941).
Useful relationships with the LeviCivita symbol include results for threedimensional vectors:
(b × c)i = εijk bj ck ,

(i) The Kronecker delta δji is a ‘mixed tensor of second rank’,11 and one can check
that it transforms correctly as follows:
∂x
¯i ∂xk
∂x
¯i ∂xℓ ℓ
δ =
= δji .
(21)
δ¯ij =
∂xk ∂ x
¯j k

∂xk ∂ x
¯j
ij
Note that whenever δij or δ are written, they are not tensors and are simply a
shorthand for the scalar 1, in the case when i = j, or 0 when i = j.
(ii) The antisymmetric symbol or Levi-Civita symbol12 εijkℓ is defined in four
dimensions by (i) all even permutations ijkℓ of 0123 (such as ijkℓ = 2301) have
εijkℓ = 1; (ii) all odd permutations ijkℓ of 0123 (such as ijkℓ = 0213) have εijkℓ =
−1; (iii) all other terms are zero (e.g. ε0012 = 0). The Levi-Civita symbol can be
defined in other dimensions.13 We will not treat this symbol as a tensor, so the
version with downstairs indices εijk is identical to εijk .

a · (b × c) = εijk ai bj ck ,

and matrix algebra:

det A = εi1 i2 ···in A1i1 A2i2 · · · Anin ,
where A is a n × n matrix with components Aij .
13

The version in two dimensions is
rather simple:
ε01 = −ε10 = 1,
ε00 = ε11 = 0.
In three dimensions, the nonzero components are:
ε012 = ε201 = ε120 = 1,
021

ε


210

=ε

102

=ε

= −1.

Jean Baptiste Joseph Fourier (1768–
1830)

0.5

Fourier transforms

We will constantly be needing to swap between representations of an
object in spacetime and in the corresponding frequency variables, that
is spatial and temporal frequency. Spatial frequency k and temporal
frequency ω also form a four-vector (ω, k) and using E = ω and p =
k we see that this is the energy-momentum four-vector (E, p). (In
fact, with our convention
= 1 the two objects are identical!) To
swap between representations, we define the four-dimensional Fourier
transform f˜(k) of a function f (x) of spacetime x as
f˜(k) =

d4 x eik·x f (x),


(22)

where four-dimensional integration is defined by
d4 x =

dx0 dx1 dx2 dx3 .

(23)

d4 k −ik·x ˜
e
f (k),
(2π)4

(24)

The inverse transform is
f (x) =

and contains four factors of 2π that are needed for each of the four
integrations. Another way of writing eqn 22 is
14

Getting these right is actually important: if you have (2π)4 on the top of an
equation and not the bottom, your answer will be out by a factor of well over
two million.

f˜(ω, k) =

d3 xdt ei(ωt−k·x) f (t, x).


(25)

In the spirit of this definition, we will try to formulate our equations so
that every factor of dk comes with a (2π), hopefully eliminating one of
the major causes of insanity in the subject, the annoying factors14 of 2π.

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0.6

Electromagnetism 7

Example 0.4
The Dirac delta function δ(x) is a function localized at the origin and which
has integral unity. It is the perfect model of a localized particle. The integral of a
d-dimensional Dirac delta function δ (d) (x) is given by
Z
dd x δ (d) (x) = 1.
(26)
It is defined by

Z

dd x f (x)δ (d) (x) = f (0).

(27)

Consequently, its Fourier transform is given by

Z
δ˜(d) (k) =
dd x eik·x δ (d) (x) = 1.

(28)

Hence, the inverse Fourier transform in four-dimensions is
Z
d4 k −ik·x
e
= δ (4) (x).
(2π)4

0.6

(29)

Electromagnetism

In SI units Maxwell’s equations in free space can be written:
∇ · E = ǫρ0 ,
∇ · B = 0,

∇ × E = − ∂B
∂t ,
∇ × B = µ0 J +

1 ∂E
c2 ∂t .


James Clerk Maxwell (1831–1879)

(30)

In this book we will choose15 the Heaviside–Lorentz16 system of units
(also known as the ‘rationalized Gaussian CGS’ system) which can be
obtained from SI by setting ǫ0 = µ0 = 1. Thus the electrostatic potential
V (x) = q/4πǫ0 |x| of SI becomes
V (x) =

q
,
4π|x|

(31)

in Heaviside–Lorentz units, and Maxwell’s equations can be written
∇ · E = ρ,
∇ · B = 0,

∇ × E = − 1c ∂B
∂t ,
∇ × B = 1c (J + ∂E
∂t ).

(32)

Using our other choice of c = = 1 obviously removes the factors of
c from these equations. In addition, the fine structure constant α =
1

e2 /4π c ≈ 137
simplifies to
α=

e2
.
4π

(33)

Note that we will give electromagnetic charge q in units of the electron
charge e by writing q = Q|e|. The charge on the electron corresponds
to Q = −1.

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15

Although SI units are preferable for
many applications in physics, the desire
to make our (admittedly often complicated) equations as simple as possible
motivates a different choice of units for
the discussion of electromagnetism in
quantum field theory. Almost all books
on quantum field theory use Heaviside–
Lorentz units, though the famous textbooks on electrodynamics by Landau
and Lifshitz and by Jackson do not.
16

These units are named after the English electrical engineer O. Heaviside

(1850–1925) and the Dutch physicist
H. A. Lorentz (1853–1928).