Contents
Collective Excitations: From Particles to Fields
Free Scalar Field Theory: Phonons ...................... 1
1.1.1 Classical Chain ............................. 2
1.1.2 Quantum Chain ............................. 8
tQuantum Electrodynamics .......................... 10
Problem Set ................................... 13
1.3.1 Questions on Collective Modes and Field Theories .......... 13
1.3.2 Answers ................................. 15
Second Quantisation
Notations and Definitions ........................... 19
Applications of Second Quantisation ..................... 25
2.2.1 Phonons ................................. 26
2.2.2 Interacting Electron Gas ........................ 26
2.2.3 Tight-Binding and the Mott-Hubbard Insulators ........... 28
2.2.4 tMott-Insulators and the Magnetic State ............... 32
2.2.5 Spin Waves ............................... 36
2.2.6 tHeisenberg Antiferromagnet ...................... 38
2.2.7 tDilute Bose Gas: Bogoluibov Theory ................. 40
2.2.8 tElectron-Phonon Interaction ..................... 43
Problem Set ................................... 46
2.3.1 Questions on the Second Quantisation ................ 46
2.3.2 Answers ................................. 52
Feynman Path Integral 59
3.1 The Path Integral - General Formalism .................... 59
3.1.1 Construction of the Path Integral ................... 60
3.1.2 Path Integral and Statistical Mechanics ................ 66
3.1.3 Path Integral and Classical Mechanics: Semiclassics ......... 69
3.1.4 Summary ................................ 72
3.2 Applications of the Feynman Path Integral .................. 73
3.2.1 Quantum Particle in a Well ...................... 74
3.2.2 Double Well Potential: Tunnelling and Instantons .......... 76
3.2.3 Unstable States and Bounces: False Vacuum ............. 83
3.2.4 tPath Integral for Spin: Topological Terms in Field Theory ..... 84
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3.2.5 ISecurity Derivatives and the Principles of Finance ......... 93
3.3 Appendix: Gaussian Integrals ......................... 96
3.3.1 One-dimensional Gaussian integrals .................. 96
3.3.2 Gaussian integration in more than one dimension .......... 97
3.3.3 Gaussian Functional Integration .................... 99
3.3.4 Grassmann Gaussian Integration ................... 100
3.4 Problem Set ................................... 103
3.4.1 Questions on the Feynman Path Integral ............... 103
3.4.2 Answers ................................. 110
Functional Field Theory 119
4.1 Construction of the Many-Body Path Integral ................ 120
4.1.1 Coherent States (Bosons) ....................... 120
4.1.2 Coherent States (Fermions) ...................... 123
4.2 Field Integral for Quantum Partition Function ................ 127
4.2.1 Partition Function of Non-Interacting Gas .............. 130
4.2.2 Connection to the Feynman Path Integral .............. 131
4.2.3 Partition Function of Harmonic Oscillator .............. 132
4.3 IWeakly Interacting Electron Gas ....................... 133
4.3.1 Field Theory of Partition Function .................. 135
4.3.2 Ground State Energy .......................... 140
4.4 Superconductivity ................................ 141
4.4.1 Mean-Field Theory of Superconductivity ............... 142
4.4.2 Superconductivity from the Path Integral ............... 145
4.4.3 Gap Equation .............................. 148
4.4.4 ISuperconductivity: Anderson-Higgs Mechanism ........... 150
4.4.5 Statistical Field Theory: Ferromagnetism Revisited ......... 151
4.5 INon-equilibrium Statistical Mechanics .................... 154
4.5.1 Formalism ................................ 156
4.5.2 Differences from quantum mechanics ................. 157
4.5.3 Relation to other formalisms ...................... 158
4.5.4 Branching and annihilating random walks .............. 159
4.6 Problem Set ................................... 161
4.6.1 Questions on the Functional Field Integral .............. 161
4.6.2 Answers ................................. 168
Relativistic Quantum Mechanics
Introduction ................................... 173
Klein-Gordon Equation ............................. 178
Dirac Equation ................................. 180
5.3.1 Density and Current .......................... 182
5.3.2 Relativistic Covariance ......................... 182
5.3.3 Angular Momentum and Spin ..................... 183
5.3.4 Parity .................................. 184
Free Particle Solution of the Dirac Equation ................. 185
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CONTENTS xi
5.4.1 Klein Paradox: Antiparticles ...................... 186
Canonical Quantisation of Relativistic Field ................. 190
5.5.1 Scalar Field: Klein-Gordon Equation Revisited ............ 190
5.5.2 tCharged Scalar Field ......................... 194
5.5.3 Dirac Field ............................... 195
Coupling to Electromagnetic Field ....................... 196
tFunctional Methods in Relativistic Theories ................. 197
tAttempt at a Synthesis ............................ 199
5.8.1 Fundamental Particles ......................... 199
5.8.2 Fundamental Interactions ....................... 201
5.8.3 Weinberg-Salam Electroweak Theory ................. 205
5.8.4 Experiment ............................... 207
5.8.5 Particle Decay .............................. 209
5.8.6 GUTS, Big Bang, and the Early Universe .............. 213
Appendix: Elements of Group Theory ..................... 215
Problem set ................................... 221
5.10.1 Questions on Relativistic Quantum Mechanics ............ 221
5.10.2 Answers ................................. 224
Tripos Questions 229
6.1 Questions .................................... 229
6.2 Answers ..................................... 237
Preface
The aim of this course is to provide a self-contained and coherent introduction to the
basic tools and concepts of quantum (and statistical) field theory including the method of
second quantisation, the Feynman and Coherent state path integral, as well as including
an introduction to relativistic quantum mechanics. The course is based fundamentally on
applications.
Importantly, this course is not intended to supplement theoretical courses offered in
part III mathematics. The lectures will differ substantially both in style as well as content.
Inevitably, and by design, there will however be substantial overlap with a variety of
different courses ranging from quantum field theory, to soft condensed matter, and from
solid state to particle physics.
Why study quantum field theory? The language of quantum field theory unites
all branches of physics: the fundamental equations of quantum field theory describe phase
transitions in the Early Universe equally well as those in magnetic insulators; dynamics
of quarks as well as fluctuations of cell membranes. Indeed within the same framework
one can describe equally well both classical and quantum systems.
Who should attend this course? Broadly speaking, a solid background in el-
ementary quantum, statistical, and particle mechanics, as well as elementary classical
electrodynamics will be assumed. But all the material should be accessible to those the-
oretically inclined. On the mathematical side, a knowledge of Stiirm-Liouville theory,
Fourier and complex analysis will be essential. As well as providing useful background
material for major and minor part III options, it is hoped that this course will prove to
be of general interest in its own right.
A course synopsis is outline below. Items indicated by a will be largely used as addi-
tional source material for problem sets, supervision, and useful background information.
The italicised items represent particular mathematical concepts which will be explored at
some length:
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