A Unified Grand Tour of Theoretical Physics
Second Edition

A Unified Grand Tour of
Theoretical Physics
Second Edition
Ian D Lawrie
Reader in Theoretical Physics
The University of Leeds
Institute of Physics Publishing
Bristol and Philadelphia
c
 IOP Publishing Ltd 2002
All rights reserved. No part of this publication may be reproduced, stored
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agreement with the Committee of Vice-Chancellors and Principals.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN 0 7503 0604 1
Library of Congress Cataloging-in-Publication Data are available
First Edition published 1990
First Edition reprinted 1994, 1998
Commissioning Editor: James Revill
Production Editor: Simon Laurenson
Production Control: Sarah Plenty
Cover Design: Fr´ed´erique Swist
Marketing Executive: Laura Serratrice
Published by Institute of Physics Publishing, wholly owned by The Institute of

2.3.1 The affine connection 29
2.3.2 Geodesics 33
2.3.3 The Riemann curvature tensor 34
2.3.4 The metric 36
2.3.5 The metric connection 38
2.4 What is the Structure of Our Spacetime? 39
3 Classical Physics in Galilean and Minkowski Spacetimes 45
3.1 The Action Principle in Galilean Spacetime 46
3.2 Symmetries and Conservation Laws 50
3.3 The Hamiltonian 52
3.4 Poisson Brackets and Translation Operators 53
3.5 The Action Principle in Minkowski Spacetime 56
3.6 Classical Electrodynamics 61
3.7 Geometry in Classical Physics 64
3.7.1 More on tensors 65
3.7.2 Differential forms, dual tensors and Maxwell’s equations 67
vi
Contents
3.7.3 Configuration space and its relatives 73
3.7.4 The symplectic geometry of phase space 75
4 General Relativity and Gravitation 83
4.1 The Principle of Equivalence 83
4.2 Gravitational Forces 84
4.3 The Field Equations of General Relativity 87
4.4 The Gravitational Field of a Spherical Body 91
4.4.1 The Schwarzschild solution 91
4.4.2 Time near a massive body 93
4.4.3 Distances near a massive body 95
4.4.4 Particle trajectories near a massive body 96
4.5 Black and White Holes 97

7.5 Weyl and Majorana Spinors 159
7.6 Particles of Spin 1 and 2 163
7.6.1 Photons and massive spin-1 particles 163
7.6.2 Gravitons 166
7.7 Wave Equations in Curved Spacetime 168
Contents
vii
8 Forces, Connections and Gauge Fields 179
8.1 Electromagnetism 179
8.2 Non-Abelian Gauge Theories 185
8.3 Non-Abelian Theories and Electromagnetism 192
8.4 Relevance of Non-Abelian Theories to Physics 193
8.5 The Theory of Kaluza and Klein 194
9 Interacting Relativistic Field Theories 199
9.1 Asymptotic States and the Scattering Operator 200
9.2 Reduction Formulae 202
9.3 Path Integrals 205
9.3.1 Path integrals in non-relativistic quantum mechanics 205
9.3.2 Functional integrals in quantum field theory 208
9.4 Perturbation Theory 211
9.5 Quantization of Gauge Fields 214
9.6 Renormalization 218
9.7 Quantum Electrodynamics 224
9.7.1 The Coulomb potential 224
9.7.2 Vacuum polarization 227
9.7.3 The Lamb shift 229
9.7.4 The running coupling constant 229
9.7.5 Anomalous magnetic moments 231
10 Equilibrium Statistical Mechanics 235
10.1 Ergodic Theory and the Microcanonical Ensemble 236

12.7 Supersymmetry 328
12.7.1 The Wess–Zumino model 329
12.7.2 Superfields 330
12.7.3 Spontaneous supersymmetry breaking 332
12.7.4 The supersymmetry algebra 335
12.7.5 Supersymmetric gauge theories and supergravity 340
12.7.6 Some algebraic details 343
13 Solitons and So On 346
13.1 Domain Walls and Kinks 347
13.2 The Sine–Gordon Solitons 355
13.3 Vortices and Strings 359
13.4 Magnetic Monopoles 369
14 The Early Universe 379
14.1 The Robertson–Walker Metric 380
14.2 The Friedmann–Lemaˆıtre Models 385
14.3 Matter, Radiation and the Age of the Universe 390
14.4 The Fairly Early Universe 393
14.5 Nucleosynthesis 401
14.6 Recombination and the Horizon Problem 404
14.7 The Flatness Problem 405
14.8 The Very Early Universe 406
15 An Introduction to String Theory 425
15.1 The Relativistic Point Particle 427
15.2 The Free Classical String 431
15.2.1 The string action 431
15.2.2 Weyl invariance and gauge fixing 434
15.2.3 The Euclidean worldsheet and conformal invariance 437
15.2.4 Mode expansions 440
15.2.5 A useful transformation 445
15.3 Quantization of the Free Bosonic String 447

Index 555

Preface to the Second Edition
In preparing this revised edition of the Tour, I have corrected several errors and
misprints for which I would like to take this opportunity of apologizing to readers
of the first edition.
By now, supersymmetry and string theory have become so prominent in
the theoretical physics literature (despite the more or less total absence of any
experimental evidence of their relevance to the real world!) as to be obligatory in
a book with this title. Accordingly, I have added introductory accounts of these
topics in §12.7 and chapter 15. A comprehensive treatment of either topic (were
I competent to write it) would require a book in itself, but I hope that the short
accounts I have given will serve to make the extensive technical literature a little
more accessible. I confess that I am no expert on string theory; Chris Hull and
Jim Gates have given me advice which is perhaps enough to ensure that what I
say is not grossly misleading, and I thank them for it.
Other new material in this edition includes a section on the applications of
differential geometry to Newtonian mechanics and classical electromagnetism
(§3.7) and a chapter on magnetic monopoles and other topological defects
(chapter 13). I have also expanded my discussions of quantum fields in curved
spacetimes (§7.7), grand unified theories (§12.6) and inflationary cosmology
(§14.8) and attempted to improve and update my presentation of various other
matters in minor ways.
I would like to thank IoP Publishing for giving me the opportunity of revising
and extending the Tour. I am grateful to Jim Revill for his continual friendship and
encouragement, and to Simon Laurenson for his unfailing patience and courtesy
in dealing with the technicalities of bringing the final product into being.
Ian D Lawrie
October 2001
xi

will be found suitable as a basis for such courses, and have tried to arrange the
material so that lecturers may select topics from it according to their own tastes.
Postgraduate students will no doubt find, as I have done, the need to acquire
some familiarity with a wide range of material which is treated adequately only
in rather forbidding technical treatises. They, I hope, will find here a palatable
xiii
xiv
Preface to the First Edition
introduction to much of what they need and, indeed, a sufficient coverage of those
topics which are peripheral to their chosen speciality.
Third, I have tried to provide for professional scientists and engineers who
are not theoretical physicists. They, I conceive, may find themselves unsatisfied
by semi-popular accounts of advances in the subject but without time for a
full-scale assault on the technical literature. For them, this book may perhaps
constitute a useful half-way house.
Responsibility for what appears herein is, of course, my own, but I should
like to acknowledge the assistance I have received along the way. Much of what
I understand of statistical mechanics was imparted some time ago by Michael
Fisher. Others who have benefitted from his wisdom may recognize his influence
in what I have to say, but he naturally bears no responsibility for anything I
failed to understand properly. During 1986–7, I spent a sabbatical year at the
University of British Columbia, where I had my first opportunity to teach a
substantial graduate course on quantum field theory. The discipline of preparing
the lectures and the perceptive response of the students who took the course did
much to sharpen the somewhat less advanced presentation offered here. Euan
Squires was instrumental in securing a contract for the book to be written. I have
greatly appreciated his enthusiastic support during the writing and his comments
on the first draft of the manuscript. I am also grateful to Gary Gibbons, who
read the chapters on relativity and gravitation and saved me from a number of
faux pas. Professor Jim Gates reviewed the entire manuscript, and I have greatly

A
↔
∂
µ
B antisymmetric derivative (= A∂
µ
B −(∂
µ
A)B) 142
|(|) ket (bra) vector 112
A
T
transpose of a matrix A
ˆ
A operator in the Hilbert space of state vectors 114
(in later chapters, the circumflex is omitted)
ˆ
A
†
adjoint (or Hermitian conjugate) operator 115
∗
T dual tensor 70
/
a contraction with Dirac matrices (= γ
µ
a
µ
) 152
{A, B}
P

worldsheet metric of a relativistic string 432

µ
νσ
affine connection coefficients 31, 39
d exterior derivative 70
dx
a
basis one-form 66
δ
ij
, δ
ij
, δ
i
j
Kronecker delta symbol 518
xv
xvi
Glossary of Mathematical Symbols
δ(x − y) Dirac delta function 518
e fundamental charge 184
e
µ
a
(x ) vierbein 170
F
µν
field strength tensor 62, 187
g

L
n
Virasoro generators 442–3
 cosmological constant 88

µ

µ
coordinate transformation matrix 26
configuration space 73
the real line 16
R
µ
νστ
Riemann curvature tensor 35
R
µν
Ricci tensor 36
R Ricci curvature scalar 39
ρ phase-space probability density 54, 237
ˆρ density operator 251
S action 47
σ
i
Pauli matrices 147
T
µν
stress tensor 60, 89
T
ab

a
bν
spin connection 171
 symplectic 2-form 76
(t) cosmological density ratio 387
Chapter 1
Introduction: The Ways of Nature
In the eighteenth century, it became fashionable for wealthy young Englishmen
to undertake the Grand Tour, an excursion which may have lasted several years,
their principal destinations being Paris and the great cultural centres of Italy—
Rome, Venice, Florence and Naples. For many, no doubt, the joys of traveling
and occasional revelry were a sufficient inducement. For others, the opportunity
to observe at first hand the social, literary and artistic achievements of other
nations represented the completion of their liberal education. For a few, perhaps,
it was the starting point of an independent intellectual career. It is in somewhat
the same spirit that I wish to offer readers of this book a guided grand tour
of theoretical physics. The members of my party need be neither wealthy (my
publisher permitting), young, English nor male. I am, however, going to assume
that they have a sound knowledge of basic physics, such as a student in his or her
final year of undergraduate study ought to possess.
Our itinerary cannot, of course, include everything that is important in
theoretical physics. Our principal destinations are those central ideas which form
the foundations of our understanding of how the world works—our knowledge,
as it now stands, of the ways of nature. In outline, the topics I plan to explore
are: the theories of relativity, which concern themselves with the geometrical
structure of space and time and from which emerge an account of gravitational
phenomena; quantum mechanics and quantum field theory, which describe the
constitution of matter at the most microscopic level that is currently accessible to
experiments; and statistical mechanics, which, up to a point, allows us to deduce
from this microscopic constitution the properties of the macroscopic systems of

or reinterpreted as a constituent part of some more comprehensive theory. Every
time this happens, we improve our understanding of what the world is really like:
we gain a clearer picture of the ways of nature.
The way in which such transformations in our understanding come about
is not necessarily apparent at the point where a detailed theoretical prediction is
confronted with an experimental datum. Take, for example, the transformation
of classical Newtonian mechanics into quantum mechanics. We have discovered,
amongst other things, that electrons can be diffracted by crystals: a phenomenon
for which quantum mechanics can account but classical mechanics cannot.
Therefore, it is often said, classical mechanics must be wrong, or at least no
more than an approximation to quantum mechanics with a restricted range of
usefulness. It is indeed true that, under appropriate circumstances, the predictions
of classical mechanics can be regarded as a good approximation to those of
quantum mechanics, but that is the less interesting part of the truth. There is,
as we shall see, a level of description (which is not especially esoteric) at which
classical and quantum mechanics are virtually identical, apart from a change of
interpretation, and it is the reinterpretation that is vital and profound. It is, I
maintain, at such a level of description that an understanding of the ways of nature
is to be sought, and it is that level of description that is emphasized in this book.
It would, of course, be absurd to lay claim to any understanding of the
ways of nature if our theories could not be tested in detail against experimental
observations. Unfortunately, the task of deriving from our fundamental theories
precise predictions that can be subjected to stringent experimental tests is often a
long and highly technical one. This task, like the devising of the experiments
Introduction: The Ways of Nature
3
themselves, is essential and intellectually challenging but, for want of the
necessary space, I shall not often describe in detail how it can be accomplished.
I do not think that this requires any apology. The basic conceptual understanding
I hope to provide can, on first acquaintance, be obscured by the technical details

these should serve as previews of the more detailed accounts that follow and
enable readers to preserve a sense of direction and purpose while the mathematical
formalism is developed. Ideally, readers should already be acquainted with special
relativity, the wave-mechanical version of quantum mechanics and their simpler
applications. Readers who are thus equipped may prefer to skip these introductory
sections or to regard them and the more elementary exercises as a short revision
course.
In the main, my treatment of mathematical formalism is intended to be
complete and explicit. Where I have omitted the algebraic details needed to derive
an equation, readers should be able to supply them, and should usually not be
4
Introduction: The Ways of Nature
satisfied until they have done so. In some cases, the exercises offer guidance.
The exercises should, indeed, be regarded as an integral part of the tour; some
of them introduce important ideas that are not dealt with fully in the main text.
Occasionally, it is necessary for me merely to quote the result of a calculation that
is too lengthy or technical to be reproduced in detail, and I shall indicate when
this is so.
There is one other aspect of theoretical physics that I should like readers to
be aware of. It has become apparent that there are many similarities, some of
them physical and others mathematical, between areas of physics which, on the
face of it, appear to be quite separate. In the course of this book, I emphasize
two of these unifying themes particularly. One is that the geometrical ideas we
need to describe the structure of space and time also lie at the root of the gauge
theories of fundamental forces, described in chapters 8 and 12, of which the most
familiar is electromagnetism. Indeed, once we realize the importance of these
ideas, the existence of both gravitational and other forces is seen to be almost
inevitable, even if we had not already been aware of them. The other is a basic
mathematical similarity between quantum field theory and statistical mechanics
which, as I discuss in chapter 10, can appear in several different guises. This is not

or where it provides a useful historical perspective, but I have by no means listed
every paper in these categories. I have certainly not attempted to refer explicitly to
the work of every scientist who has made important contributions to the subjects
I discuss. To do so would require a book in itself.
It is time for our tour to begin.
Chapter 2
Geometry
Our tour of theoretical physics begins with geometry, and there are two reasons
for this. One is that the framework of space and time provides, as it were, the
stage upon which physical events are played out, and it will be helpful to gain a
clear idea of what this stage looks like before introducing the cast. As a matter of
fact, the geometry of space and time itself plays an active role in those physical
processes that involve gravitation (and perhaps, according to some speculative
theories, in other processes as well). Thus, our study of geometry will culminate,
in chapter 4, in the account of gravity offered by Einstein’s general theory of
relativity. The other reason for beginning with geometry is that the mathematical
notions we develop will reappear in later contexts.
To a large extent, the special and general theories of relativity are ‘negative’
theories. By this I mean that they consist more in relaxing incorrect, though
plausible, assumptions that we are inclined to make about the nature of space
and time than in introducing new ones. I propose to explain how this works in
the following way. We shall start by introducing a prototype version of space
and time, called a ‘differentiable manifold’, which possesses a bare minimum of
geometrical properties—for example, the notion of length is not yet meaningful.
(Actually, it may be necessary to abandon even these minimal properties if, for
example, we want a geometry that is fully compatible with quantum theory and
I shall touch briefly on this in chapter 15.) In order to arrive at a structure
that more closely resembles space and time as we know them, we then have to
endow the manifold with additional properties, known as an ‘affine connection’
and a ‘metric’. Two points then emerge: first, the common-sense notions of

The change of motion is proportional to the motive force impressed; and is
made in the direction of the right line in which that force is impressed.
(Newton 1686)
So, from these definitions alone, we have no way of deciding whether some
observed acceleration of a body relative to a given frame should be attributed, on
the one hand, to the action of a force or, on the other hand, to an acceleration of
the frame of reference. Eddington has made this point by a facetious re-rendering
of the first law:
Every body tends to move in the track in which it actually does move, except
insofar as it is compelled by material impacts to follow some other track than
that in which it would otherwise move.
(Eddington 1929)
The extra assumption we need, of course, is that forces can arise only from the
influence of one body on another. An inertial frame is one relative to which any
body sufficiently well isolated from all other matter for these influences to be
negligible does not accelerate. In practice, needless to say, this isolation cannot
be achieved. The successful application of Newtonian mechanics depends on our
being able systematically to identify, and take proper account of, all those forces
8
Geometry
Figure 2.1. Two systems of Cartesian coordinates in relative motion.
that cannot be eliminated. To proceed, we must take it as established that, in
principle, frames of reference can be constructed, relative to which any isolated
body will, as a matter of fact, always refuse to accelerate. These frames we call
inertial.
Obviously, any two inertial frames must either be relatively at rest or have a
uniform relative velocity. Consider, then, two inertial frames, S and S

(standing
for Systems of coordinates) with Cartesian axes so arranged that the x and x

= yz

= zt

= t. (2.1)
Since the path of a moving particle is just a sequence of events, we easily find that
its velocity relative to S, in vector notation u = dx/dt, is related to its velocity
u

= dx

/dt

relative to S

by u

= u − v, with v = (v, 0, 0), and that its
acceleration is the same in both frames, a

= a.
Despite its intuitive plausibility, the common-sense view turns out to be
mistaken in several respects. The special theory of relativity hinges on the fact
that the relation u

= u −v is not true. That is to say, this relation disagrees with
experimental evidence, although discrepancies are detectable only when speeds
are involved whose magnitudes are an appreciable fraction of a fundamental
speed c, whose value is approximately 2.998 × 10
8

travels at the same speed, c, relative to the apparatus used to observe it.
In his paper of 1905, Einstein makes the fundamental assumption (though
he expresses things a little differently) that light travels with exactly the same
speed, c, relative to any inertial frame. Since this is clearly incompatible with
the Galilean transformation law given in (2.1), he takes the remarkable step of
modifying this law to read
x

=
x − vt
(1 −v
2
/c
2
)
1/2
y

= y
z

= zt

=
t − vx/c
2
(1 −v
2
/c
2

. Using the transformation (2.2), we
easily find that its equation at time t

relative to S

is x
2
+ y
2
+ z
2
= c
2
t
2
.
Many of the elementary consequences of special relativity follow directly
from the Lorentz transformation, and we shall meet some of them in later
chapters. What particularly concerns us at present—and what makes Einstein’s
interpretation of the transformation equations so remarkable—is the change that