Quantum Field Theory.. A Diagrammatic Approach [2019 draft]

Introductory remarks
	Preface
	Basic tools
		Units and fundamental units
		Planck units
		Charges
		Conventions
	The P4 Hall of Fame
	Exercise
QFT in zero dimensions
	Introduction
	Probabilistic considerations
		Quantum field and action
		Green's functions, sources and the path integral
		Connected Green's functions
		The free theory
		The 4 model and perturbation theory
		The Schwinger-Dyson equation
		The Schwinger-Dyson equation for the field function
	Diagrammatics
		Feynman diagrams
		Feynman rules
		Symmetries and multiplicities
		Vacuum bubbles
		An equation for connected graphs
		Semi-connected graphs and the SDe
		The path integral as a set of diagrams
		Dyson summation
	Planck's constant
		The loop expansion, and reverse engineering
		The classical limit
		On second quantisation
		Instanton contributions
	The effective action
		The effective action as a Legendre transform
		Diagrams for the effective action
		Computing the effective action
	Exercises
On renormalization
	Doing physics : mentality against reality
		Physics vs. Mathematics
		The renormalisation program : an example
	A handle on loop divergences
		A toy : the dot model
		Nonrenormalizable theories
	Scale dependence
		Scale-inpendent scale dependence
		Low-order approximation to the renormalised coupling
		Scheme dependence
		Theories with more parameters
		Failure of the dot model
	Asymptotics of renormalisation in 4 theory
	The method of counterterms
		Counterterms in the action
		Return to the dot model, and a preview
	Exercises
More fields in zero dimensions
	The action and the path integral
	Connected Green's functions and field functions
	The Schwinger-Dyson equation
	The sum rules revisited
	A zero-dimensional toy for QED
		Fields and sources
		Bald, furry and quenched toys
	Exercises
QFT in Euclidean spaces
	Introduction
	One-dimensional discrete theory
		An infinite number of fields
		Introducing the propagator
		Computing the propagator
		Figments of the imagination : a sermon
	One-dimensional continuum theory
		The continuum limit for the propagator
		The continuum limit for the action
		The continuum limit of the classical equation
		The continuum Feynman rules and SDe
		Field configurations in one dimension
	The momentum representation
		Fourier transforming the SDe
	Doing it in momentum space
		The Feynman rules
		Some example diagrams
	More-dimensional theories
		The more-dimensional continuum
		The propagator, explicitly
	Exercises
QFT in Minkowski space
	Moving to Minkowski space : making time
		Distance in Minkowski space
		Farewell probability, hello SDe
		A close look at almost nothing: i and -
		The need for quantum transition amplitudes
		Feynman rules for Minkowskian theories
		The propagator, explicitly
	Moving in Minkowski space : particles
		The Klein-Gordon equation
		Enter the particle !
		Unstable particles, i and the flow of time
		The Yukawa potential
		Kinematics and Newton's First Law
		Antimatter
		Counting states : the phase-space integration element
	Exercises
Scattering processes
	Introduction
	Incursion into the scattering process
		Diagrammatic picture of scattering
		The argument for connectedness
	Building predictions
		General formulæ for decay widhts and cross sections
		The truncation bootstrap
		A check on dimensionalities
		Crossing symmetry
	Unitarity issues
		Unitarity of the S matrix
		The cutting rules
		Infrared cancellations in QED
	Some example calculations
		The FEE model
		Two-body phase space
		A decay process
		A scattering process
	The one-loop cookbook
		The one-loop calculation
		Dispersion relations
	Exercises
Dirac particles
	Pimp my propagator
		Down with dyads !
		The spin interpretation
	The Dirac algebra
		The Dirac matrices
		The Clifford algebra
		Trace identities
		Pauli and Chisholm, their identities
		Hermite no, Dirac yes
		Flipping in and flipping out
		A Fierz identity
	Dirac spinors
		Chirality spinors
		Chirality spinors are massless
		Phase conventions, spinor products and Weyl
		General spinors and their dyads
		General spinors and Dirac spinors
		Particular general spinors
	Dirac particles
		Casimir does tricks
		The Dirac propagator, and a convention
		Truncating Dirac particles : external Dirac lines
		Lorentz transformations in Dirac space
		The spin of Dirac particles
		Massless Dirac particles ; helicity states
		The parity transform
	The Feynman rules for Dirac particles
		Dirac loops…
		… and Dirac loops only
		Interchange signs
		The Pauli principle
	The Dirac equation
		The classical limit
		The free Dirac action
	Exercises
Helicity techniques for Dirac particles
	The standard form for spinors
		Opting for helicities, opting for antisymmetry
		The standard form for helicity spinors
		Some useful identities
		How to compute spinor products
		The standard form for massive particles
		The standard form for complex momenta
	Summary of tools for spinor techniques
	Fermionic decays : the Fermi model
		The amplitude for muon decay
		Three-body phase space
		The muon decay width
		Observable distributions in muon decay
		Charged pion decay: helicity suppression
	Exercises
Vector particles
	Massive vector particles
		The propagator
		The Feynman rules for external vector particles
		The spin of vector particles
		Polarisation vectors for helicity states
		The Proca equation
	The spin-statistics theorem
		Spinorial form of vector polarisations
		Proof of the spin-statistics theorem
	Massless vector particles
		Polarisations of massless vector particles
		Current conservation from the polarisation
		Handlebar condition for massive vector particles
		Helicity states for massless vectors
		The massless propagator : the axial gauge
		Gauge vector shift
	Exercises
Quantum Electrodynamics
	Introduction
	Constructing QED
		The QED vertex
		Handlebars : a first look
		Handlebar diagrammatics
		The Ward-Takahashi identity
		The charged Dirac equation
		The Gordon decomposition
		Furry's theorem
	Some QED processes
		A classic calculation : muon pair production
		Compton and Thomson scattering
		Electron-positron annihilation
		Bhabha scattering
		Bremsstrahlung in Mœller scattering
	Scalar electrodynamics
		The vertices
		Proof of current conservation in sQED
	The Coulomb potential
	Electrons in external fields : g=2
		The charged Klein-Gordon equation
		The relativistic Pauli equation
		A constant magnetic field
	Selected topics in QED
		Three-photon production
		The Thomson limit : scalar vs spinor
		The Landau-Yang theorem
	Exercises
Loop effects in QED
	One-loop effects in QED
	The photon self-energy
		Current conservation
		Using the optical theorem
		Getting the divergence
		The vacuum polarization
		Hadronic vacuum polarization
	The fermion self-energy
		A look at gauge invariance
		Summing the self-energies
		The loop calculation
		The curious incident of the divergences in the nighttime
	Infrared singularities in Bremsstrahlung
	The vertex correction
	Exercises
Quantum Chromodynamics
	Introduction: coloured quarks and gluons
	Quarks and gluons : first Feynman rules
		The propagators
		The quark-gluon vertex
		A closer look at the T matrices
		The Fierz identity for T matrices
	The three-gluon interaction
		The need for three-gluon vertices
		Furry's failure
		The ggg vertex and its handlebar
		On coupling quantisation
	The four-gluon interaction
		Colourful manipulations
		A purely gluonic process
	Current conservation in QCD
		More vertices ?
		The Antkaz
		Proof of current conservation
	Selected topics in QCD
		White and coloured states
		The QCD Coulomb interaction
		The process q"7016qgg
		The Landau-Yang theorem revisited
	Exercises
Electroweak theory
	Muon decay
		The Fermi coupling constant
		Failure of the Fermi model in -e-e
	The W particle
		The IVB strategy
		The cross section for -e-e revisited
		The WW vertex
	The Z particle
		W pair production
		The weak mixing angle for couplings
		W,Z and  four-point interactions
	The Higgs sector
		The Higgs hypothesis
		Predictions from the Higgs hypothesis
		W,Z and H four-point interactions
		Higgs-fermion couplings
		Higgs self-interactions
	About anomalies
	Conclusions and remarks
	A look at non-minimal models
		Non-minimal Higgs sector
	Exercises
Example computations
	Neutrino production in e+e- scattering
		The cross section
		Unitarity considerations
	W pair production in e+e- scattering
		Setting up the amplitude
		Momenta and polarisations
		Working out the amplitudes
		W pair production at very high energy
	Higgs coupling to massless vectors
		The H vertex
		The ggH amplitude and the Next Generation
Appendices
	Perturbative (non)convergence issues
		Punishment at the singular point
		Borel summation
	More on symmetry factors
		The origin of symmetry factors
		Explicit computation of symmetry factors
	Derivation of the diagrammatic sum rules
	Alternative solutions to the Schwinger-Dyson equation
		Alternative contours for general theories
		Alternative contours for 3 theory
		Alternative contours for 4 theory
	Diagram counting
		Tree graphs and asymptotics
		Counting one-loop diagrams
	Concavity of the effective action
	Functional derivatives
	Frustrated and unusual actions
		Frustrating your neighbours
		Increasing frustration
	Newton's First Law revisited
		Introduction : the matter of sources
		Slow, fast and abrupt
		Conclusion : general effect of the sources
	Unitarity bounds
		Resonances
		Preliminaries : decay widths
		The rôle of angular momentum conservation
		The unitarity bound
	The fundamental theorem for Dirac matrices
		Proof of the fundamental theorem
		The charge conjugation matrix
	States of higher spin
		The spin algebra for integer spins
		Rank one for spin one
		Rank-2 tensors
		Rank-3 tensors
		Massless particles : surviving states
		The Bermuda triangle
		Massless propagators
		Spin of the Kalb-Ramond state
		Spin 1 from Dirac particles
		Spin 3/2 particles
		The spin algebra for Dirac particles
	Generating three-particle kinematics
	The CPT theorem
		Transforming spinors
		CPT transformation on sandwiches
		CPT transformation on diagrams
		How to kill CPT, and what it costs
	Mathematical Miscellanies
		The Gaussian doubling trick
		Stirling's approximation for n!
		Manipulating asymptotic series
		Gamma, Digamma and Bernoulli
		The Dirac delta distribution
		The principal-value distribution
		Generating the Bell numbers
		The exponential integral E1 and the Bessel K functions
		The Kramers-Kronig relation
		The dilogarithm function
		Some values of the  function
		The Lagrange expansion
		Determinants from traces