Quantum Field Theory For The Gifted Amateur

Cover
Preface
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
Part I 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
	2 Simple harmonic oscillators
		2.1 Introduction
		2.2 Mass on a spring
		2.3 A trivial generalization
		2.4 Phonons
		Exercises
	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
	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
		4.5 The Hubbard model
		Exercises
Part II Writing down Lagrangians
	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
	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
	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
Part III The need for quantum fields
	8 The passage of time
		8.1 Schr¨odinger’s 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
	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
	10 Symmetry
		10.1 Invariance and conservation
		10.2 Noether’s theorem
		10.3 Spacetime translation
		10.4 Other symmetries
		Exercises
	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
	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
	13 Fields with many components and massive electromagnetism
		13.1 Internal symmetries
		13.2 Massive electromagnetism
		13.3 Polarizations and projections
		Exercises
	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
	15 Discrete transformations
		15.1 Charge conjugation
		15.2 Parity
		15.3 Time reversal
		15.4 Combinations of discrete and continuous transformations
		Exercises
Part 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
	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
	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
	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
	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
Part V 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
	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
Part VI Path integrals
	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
	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
	25 Statistical field theory
		25.1 Wick rotation and Euclidean space
		25.2 The partition function
		25.3 Perturbation theory and Feynman rules
		Exercises
	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
	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
	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
Part VII Topological ideas
	29 Topological objects
		29.1 What is topology?
		29.2 Kinks
		29.3 Vortices
		Exercises
	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
Part VIII Renormalization: taming the infinite
	31 Renormalization, quasiparticles and the Fermi surface
		31.1 Recap: interacting and non-interacting theories
		31.2 Quasiparticles
		31.3 The propagator for a dressed particle
		31.4 Elementary quasiparticles in a metal
		31.5 The Landau Fermi liquid
		Exercises
	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
	33 Renormalization in action: propagators and Feynman diagrams
		33.1 How interactions change the propagator in perturbation theory
		33.2 The role of counterterms: renormalization conditions
		33.3 The vertex function
		Exercises
	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
	35 Ferromagnetism: a renormalization group tutorial
		35.1 Background: critical phenomena and scaling
		35.2 The ferromagnetic transition and critical phenomena
		Exercises
Part IX Putting a spin on QFT
	36 The Dirac equation
		36.1 The Dirac equation
		36.2 Massless particles: left- and right-handed wave functions
		36.3 Dirac and Weyl spinors
		36.4 Basis states for superpositions
		36.5 The non-relativistic limit of the Dirac equation
		Exercises
	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
	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
	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
	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
	41 The renormalization of QED and two great results
		41.1 Renormalizing the photon propagator: dielectric vacuum
		41.2 The renormalization group and the electric charge
		41.3 Vertex corrections and the electron g-factor
		Exercises
Part X Some applications from the world of condensed matter
	42 Superfluids
		42.1 Bogoliubov’s hunting license
		42.2 Bogoliubov’s transformation
		42.3 Superfluids and fields
		42.4 The current in a superfluid
		Exercises
	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
	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
	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
Part XI Some applications from the world of particle physics
	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
	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
	48 Majorana fermions
		48.1 The Majorana solution
		48.2 Field operators
		48.3 Majorana mass and charge
		Exercises
	49 Magnetic monopoles
		49.1 Dirac’s monopole and the Dirac string
		49.2 The ’t Hooft–Polyakov monopole
		Exercises
	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
A Further reading
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
Index