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