Quantum Field Theory for the Gifted Amateur

504 Pages·2014·15.335 MB·English

by  Tom Lancaster, Stephen J. Blundell

Quantum Field Theory for the Gifted Amateur
Quantum Field Theory
for the Gifted Amateur
Tom Lancaster
Department of Physics, University of Durham
Stephen J. Blundell
Department of Physics, University of Oxford
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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
physicaltheoryeverconstructed. Itdescribesnotonlythequantumvac-
uum,butalsothestuffthatnaturereplacesitwith. Aspectsofquantum
fieldtheoryarealsomorestringentlytested,aswellasverifiedtogreater
precision,thananyothertheoryinphysics. Thesubjectneverthelesshas
a reputation for difficulty which is perhaps well-deserved; its practition-
ers 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
hasawelldefinedquantitative significance. Learningthismathematical
and geometrical language is an important initiation rite for any aspir-
ing theoretical physicist, and a quantum field theory graduate course is
foundinmostuniversities, aidedbyalargenumberofweightyquantum
field theory textbooks. These books are written by professional quan-
tum 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
betweenanundergraduatecourseonquantummechanicsandagraduate
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,wearenotassumingthereadersofthisaresimple-mindedfolkwho
vi Preface
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.
1Afterall,withthenumberofchapters Wehavechosenthisbook’stitlewithgreatcare.1Ourimaginedreader
we ended up including, we could have is an amateur, wanting to learn quantum field theory without (at least
calledit‘Fiftyshadesofquantumfield
initially) joining the ranks of professional quantum field theorists; but
theory’.
(s)he is gifted, possessing a curious and adaptable mind and willing to
embark on a significant intellectual challenge; (s)he has abundant cu-
riosity about the physical world, a basic grounding in undergraduate
physics, and a desire to be told an entertaining and intellectually stim-
ulating story, but will not feel patronized if a few mathematical niceties
arespelledoutindetail. Infact,wesuspectandhopethatourbookwill
find wide readership amongst the graduate trainee quantum field theo-
rists 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).
Onefeatureofourbookisthelargenumberofworkedexamples,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
atasteadypaceandeachchapterendswithasummaryofthekeyideas
introduced.
Thoughthevacuumplaysabigpartinthestoryofquantumfieldthe-
ory, 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
Schroederandtheirlegendaryantecedent: thelecturesinquantumfield
theory by Sidney Coleman. The latter are currently available online as
streamed videos and come highly recommended. Also deserving of spe-
cial mention is the text by Weinberg which is ‘a book to which we are
2T.S.EliotonUlysses. all indebted, and from which none of us can escape.’2
It is a pleasure to acknowledge the help we have received from var-
ious sources in writing this book. Particular mention is due to S¨onke
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 Nu-
gent, oureagle-eyedcopyeditor. Weareverygratefulforthecomments
andcorrectionswereceivedfromanumberoffriendsandcolleagueswho
kindlygaveuptheirtimetoreaddraftsofvariouschapters: PeterByrne,
Claudio Castelnovo, John Chalker, Martin Galpin, Chris Maxwell, Tom
Preface vii
McLeish, Johannes M¨oller, Paul Tulip and Rob Williams. They deserve
muchcreditforsavingusfromvariousembarrassingerrors,butanythat
remain are due to us; those that we find post-publication will be posted
on the book’s website:
http://www.dur.ac.uk/physics/qftgabook
Forvariousbitsofhelpfulinformation,wethankHideoAoki,NikitasGi-
dopoulos,PaulGoddardandJohnSingleton. Ourthanksarealsodueto
various graduate students at Durham and Oxford who have unwittingly
served as guinea pigs as we tried out various ways of presenting this
materialingraduatelectures. FinallywethankCallyandKatherinefor
their love and support.
TL & SJB
Durham & Oxford
January 2, 2014

Contents
0 Overture 1
0.1 What is quantum field theory? 1
0.2 What is a field? 2
0.3 Who is this book for? 2
0.4 Special relativity 3
0.5 Fourier transforms 6
0.6 Electromagnetism 7
I The Universe as a set of harmonic oscillators 9
1 Lagrangians 10
1.1 Fermat’s principle 10
1.2 Newton’s laws 10
1.3 Functionals 11
1.4 Lagrangians and least action 14
1.5 Why does it work? 16
Exercises 17
2 Simple harmonic oscillators 19
2.1 Introduction 19
2.2 Mass on a spring 19
2.3 A trivial generalization 23
2.4 Phonons 25
Exercises 27
3 Occupation number representation 28
3.1 A particle in a box 28
3.2 Changing the notation 29
3.3 Replace state labels with operators 31
3.4 Indistinguishability and symmetry 31
3.5 The continuum limit 35
Exercises 36
4 Making second quantization work 37
4.1 Field operators 37
4.2 How to second quantize an operator 39
4.3 The kinetic energy and the tight-binding Hamiltonian 43
4.4 Two particles 44
x Contents
4.5 The Hubbard model 46
Exercises 48
II Writing down Lagrangians 49
5 Continuous systems 50
5.1 Lagrangians and Hamiltonians 50
5.2 A charged particle in an electromagnetic field 52
5.3 Classical fields 54
5.4 Lagrangian and Hamiltonian density 55
Exercises 58
6 A first stab at relativistic quantum mechanics 59
6.1 The Klein–Gordon equation 59
6.2 Probability currents and densities 61
6.3 Feynman’s interpretation of the negative energy states 61
6.4 No conclusions 63
Exercises 63
7 ExamplesofLagrangians, orhowtowritedownatheory 64
7.1 A massless scalar field 64
7.2 A massive scalar field 65
7.3 An external source 66
7.4 The φ4 theory 67
7.5 Two scalar fields 67
7.6 The complex scalar field 68
Exercises 69
III The need for quantum fields 71
8 The passage of time 72
8.1 Schr¨odinger’s picture and the time-evolution operator 72
8.2 The Heisenberg picture 74
8.3 The death of single-particle quantum mechanics 75
8.4 Old quantum theory is dead; long live fields! 76
Exercises 78
9 Quantum mechanical transformations 79
9.1 Translations in spacetime 79
9.2 Rotations 82
9.3 Representations of transformations 83
9.4 Transformations of quantum fields 85
9.5 Lorentz transformations 86
Exercises 88
10 Symmetry 90
10.1 Invariance and conservation 90