Quantum field theory is arguably the most far-reaching and beautiful physical theory ever constructed, with aspects more stringently tested and verified to greater precision than any other theory in physics. Unfortunately, the subject has gained a notorious reputation for difficulty, with forbidding looking mathematics and a peculiar diagrammatic language described in an array of unforgiving, weighty textbooks aimed firmly at aspiring professionals. However, quantum field theory is too important, too beautiful, and too engaging to be restricted to the professionals. This book on quantum field theory is designed to be different. It is written by experimental physicists and aims to provide the interested amateur with a bridge from undergraduate physics to quantum field theory. The imagined reader is a gifted amateur, possessing a curious and adaptable mind, looking to be told an entertaining and intellectually stimulating story, but who will not feel patronised if a few mathematical niceties are spelled out in detail. Using numerous worked examples, diagrams, and careful physically motivated explanations, this book will smooth the path towards understanding the radically different and revolutionary view of the physical world that quantum field theory provides, and which all physicists should have the opportunity to experience.

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

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 Examples of Lagrangians, or how to write down a theory 64

7.1 A massless scalar field 64

7.2 A massive scalar field 65

7.3 An external source 66

7.4 The φ 4 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 Schrödinger’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

10.2 Noether’s theorem 92

10.3 Spacetime translation 94

10.4 Other symmetries 96

Exercises 97

11 Canonical quantization of fields 98

11.1 The canonical quantization machine 98

11.2 Normalizing factors 101

11.3 What becomes of the Hamiltonian? 102

11.4 Normal ordering 104

11.5 The meaning of the mode expansion 106

Exercises 108

12 Examples of canonical quantization 109

12.1 Complex scalar field theory 109

12.2 Noether’s current for complex scalar field theory 111

12.3 Complex scalar field theory in the non-relativistic limit 112

Exercises 116

13 Fields with many components and massive electromagnetism 117

13.1 Internal symmetries 117

13.2 Massive electromagnetism 120

13.3 Polarizations and projections 123

Exercises 125

14 Gauge fields and gauge theory 126

14.1 What is a gauge field? 126

14.2 Electromagnetism is the simplest gauge theory 129

14.3 Canonical quantization of the electromagnetic field 131

Exercises 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 143

16 Propagators and Green’s functions 144

16.1 What is a Green’s function? 144

16.2 Propagators in quantum mechanics 146

16.3 Turning it around: quantum mechanics from the propagator and a first look at perturbation theory 149

16.4 The many faces of the propagator 151

Exercises 152

17 Propagators and fields 154

17.1 The field propagator in outline 155

17.2 The Feynman propagator 156

17.3 Finding the free propagator for scalar field theory 158

17.4 Yukawa’s force-carrying particles 159

17.5 Anatomy of the propagator 162

Exercises 163

18 The S-matrix 165

18.1 The S-matrix: a hero for our times 166

18.2 Some new machinery: the interaction representation 167

18.3 The interaction picture applied to scattering 168

18.4 Perturbation expansion of the S-matrix 169

18.5 Wick’s theorem 171

Exercises 174

19 Expanding the S-matrix: Feynman diagrams 175

19.1 Meet some interactions 176

19.2 The example of φ 4 theory 177

19.3 Anatomy of a diagram 181

19.4 Symmetry factors 182

19.5 Calculations in p-space 183

19.6 A first look at scattering 186

Exercises 187

20 Scattering theory 188

20.1 Another theory: Yukawa’s ψ † ψφ interactions 188

20.2 Scattering in the ψ † ψφ theory 190

20.3 The transition matrix and the invariant amplitude 192

20.4 The scattering cross-section 193

Exercises 194

V Interlude: wisdom from statistical physics 195

21 Statistical physics: a crash course 196

21.1 Statistical mechanics in a nutshell 196

21.2 Sources in statistical physics 197

21.3 A look ahead 198

Exercises 199

22 The generating functional for fields 201

22.1 How to find Green’s functions 201

22.2 Linking things up with the Gell-Mann–Low theorem 203

22.3 How to calculate Green’s functions with diagrams 204

22.4 More facts about diagrams 206

Exercises 208

VI Path integrals 209

23 Path integrals: I said to him, ‘You’re crazy’ 210

23.1 How to do quantum mechanics using path integrals 210

23.2 The Gaussian integral 213

23.3 The propagator for the simple harmonic oscillator 217

Exercises 220

24 Field integrals 221

24.1 The functional integral for fields 221

24.2 Which field integrals should you do? 222

24.3 The generating functional for scalar fields 223

Exercises 226

25 Statistical field theory 228

25.1 Wick rotation and Euclidean space 229

25.2 The partition function 231

25.3 Perturbation theory and Feynman rules 233

Exercises 236

26 Broken symmetry 237

26.1 Landau theory 237

26.2 Breaking symmetry with a Lagrangian 239

26.3 Breaking a continuous symmetry: Goldstone modes 240

26.4 Breaking a symmetry in a gauge theory 242

26.5 Order in reduced dimensions 244

Exercises 245

27 Coherent states 247

27.1 Coherent states of the harmonic oscillator 247

27.2 What do coherent states look like? 249

27.3 Number, phase and the phase operator 250

27.4 Examples of coherent states 252

Exercises 253

28 Grassmann numbers: coherent states and the path integral for fermions 255

28.1 Grassmann numbers 255

28.2 Coherent states for fermions 257

28.3 The path integral for fermions 257

Exercises 258

VII Topological ideas 259

29 Topological objects 260

29.1 What is topology? 260

29.2 Kinks 262

29.3 Vortices 264

Exercises 266

30 Topological field theory 267

30.1 Fractional statistics à la Wilczek: the strange case of anyons 267

30.2 Chern–Simons theory 269

30.3 Fractional statistics from Chern–Simons theory 271

Exercises 272

VIII Renormalization: taming the infinite 273

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 285

32.1 The problem is divergences 285

32.2 The solution is counterterms 287

32.3 How to tame an integral 288

32.4 What counterterms mean 290

32.5 Making renormalization even simpler 292

32.6 Which theories are renormalizable? 293

Exercises 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 302

34.1 The problem 302

34.2 Flows in parameter space 304

34.3 The renormalization group method 305

34.4 Application 1: asymptotic freedom 307

34.5 Application 2: Anderson localization 308

34.6 Application 3: the Kosterlitz–Thouless transition 309

Exercises 312

35 Ferromagnetism: a renormalization group tutorial 313

35.1 Background: critical phenomena and scaling 313

35.2 The ferromagnetic transition and critical phenomena 315

Exercises 320

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 336

37.1 Spinors aren’t vectors 336

37.2 Rotating spinors 337

37.3 Boosting spinors 337

37.4 Why are there four components in the Dirac equation? 339

Exercises 340

38 The quantum Dirac field 341

38.1 Canonical quantization and Noether current 341

38.2 The fermion propagator 343

38.3 Feynman rules and scattering 345

38.4 Local symmetry and a gauge theory for fermions 346

Exercises 347

39 A rough guide to quantum electrodynamics 348

39.1 Quantum light and the photon propagator 348

39.2 Feynman rules and a first QED process 349

39.3 Gauge invariance in QED 351

Exercises 353

40 QED scattering: three famous cross-sections 355

40.1 Example 1: Rutherford scattering 355

40.2 Example 2: Spin sums and the Mott formula 356

40.3 Example 3: Compton scattering 357

40.4 Crossing symmetry 358

Exercises 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 370

42.1 Bogoliubov’s hunting license 370

42.2 Bogoliubov’s transformation 372

42.3 Superfluids and fields 374

42.4 The current in a superfluid 377

Exercises 379

43 The many-body problem and the metal 380

43.1 Mean-field theory 380

43.2 The Hartree–Fock ground state energy of a metal 383

43.3 Excitations in the mean-field approximation 386

43.4 Electrons and holes 388

43.5 Finding the excitations with propagators 389

43.6 Ground states and excitations 390

43.7 The random phase approximation 393

Exercises 398

44 Superconductors 400

44.1 A model of a superconductor 400

44.2 The ground state is made of Cooper pairs 402

44.3 Ground state energy 403

44.4 The quasiparticles are bogolons 405

44.5 Broken symmetry 406

44.6 Field theory of a charged superfluid 407

Exercises 409

45 The fractional quantum Hall fluid 411

45.1 Magnetic translations 411

45.2 Landau Levels 413

45.3 The integer quantum Hall effect 415

45.4 The fractional quantum Hall effect 417

Exercises 421

XI Some applications from the world of particle physics 423

46 Non-abelian gauge theory 424

46.1 Abelian gauge theory revisited 424

46.2 Yang–Mills theory 425

46.3 Interactions and dynamics of W µ 428

46.4 Breaking symmetry with a non-abelian gauge theory 430

Exercises 432

47 The Weinberg–Salam model 433

47.1 The symmetries of Nature before symmetry breaking 434

47.2 Introducing the Higgs field 437

47.3 Symmetry breaking the Higgs field 438

47.4 The origin of electron mass 439

47.5 The photon and the gauge bosons 440

Exercises 443

48 Majorana fermions 444

48.1 The Majorana solution 444

48.2 Field operators 446

48.3 Majorana mass and charge 447

Exercises 450

49 Magnetic monopoles 451

49.1 Dirac’s monopole and the Dirac string 451

49.2 The ’t Hooft–Polyakov monopole 453

Exercises 456

50 Instantons, tunnelling and the end of the world 457

50.1 Instantons in quantum particle mechanics 458

50.2 A particle in a potential well 459

50.3 A particle in a double well 460

50.4 The fate of the false vacuum 463

Exercises 466

A Further reading 467

B Useful complex analysis 473

B.1 What is an analytic function? 473

B.2 What is a pole? 474

B.3 How to find a residue 474

B.4 Three rules of contour integrals 475

B.5 What is a branch cut? 477

B.6 The principal value of an integral 478

Index

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