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

Description

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