Relativistic Quantum Mechanics

Overview

KLEIN-GORDON AND DIRAC EQUATIONS:Introduction, The Klein-Gordon equation - Particles and antiparticles, Two component framework - Coupling to electromagnetism, Solution of the Coulomb problem - Bohr-Sommerfeld semiclassical solution of the Coulomb problem, The Dirac equation and the Clifford algebra - Dirac matrices, Covariant form of the Dirac equation, Equations of motion, Spin, Free particle solutions - Electromagnetic interactions, Gyromagnetic ratio - The Hydrogen atom problem, Symmetries, Parity, Separation of variables - The Frobenius method solution, Energy levels and wavefunctions - Non-relativistic reduction, The Foldy-Wouthuysen transformation - Interpretation of relativistic corrections, Reflection from a potential barrier - The Klein paradox, Pair creation process and examples - Zitterbewegung, Hole theory and antiparticles - Charge conjugation symmetry, Chirality, Projection operators, The Weyl equation - Weyl and Majorana representations of the Dirac equation, Unitary and antiunitary symmetries - Time reversal symmetry, The PCT invariance - Arrow of time and particle-antiparticle asymmetry, Band theory for graphene - Dirac equation structure of low energy graphene states, Relativistic signatures in graphene properties;LORENTZ AND POINCARE GROUPS:Groups and symmetries, The Lorentz and Poincare groups - Group representations, generators and algebra, Translations, rotations and boosts - The spinor representation of SL(2,C), The spin-statistics theorem - Finite dimensional representations of the Lorentz group, Euclidean and Galilean groups - Classification of one particle states, The little group, Mass, spin and helicity - Massive and massless one particle states - P and T transformations, Lorentz covariance of spinors - Lorentz group classification of Dirac operators, Orthogonality and completeness of Dirac spinors, Projection operators

QUANTUM ELECTRODYNAMICS:Propagator theory, Non-relativistic case and causality - Relativistic case, Particle and antiparticle contributions, Feynman prescription and the propagator - Interactions and formal perturbative theory, The S-matrix and Feynman diagrams - Trace theorems for products of Dirac matrices - Photons and the gauge symmetry - Abelian local gauge symmetry, The covariant derivative and invariants - Charge quantisation, Photon propagator, Current conservation and polarisations - Feynman rules for Quantum Electrodynamics, Nature of perturbative expansion - Dyson's analysis of the perturbation series, Singularities of the S-matrix, Elementary QED processes - The T-matrix, Coulomb scattering - Mott cross-section, Compton scattering - Klein-Nishina result for cross-section - Photon polarisation sums, Pair production through annihilation - Unpolarised and polarised cross-sections - Helicity properties, Bound state formation - Bound state decay, Non-relativistic potentials - Lagrangian formulation of QED, Divergences in Green's functions, Superficially divergent 1-loop diagrams and regularisation - Infrared divergences due to massless particles, Renormalisation and finite physical results - Symmetry constraints on Green's functions, Furry's theorem, Ward-Takahashi identity, Spontaneous breaking of gauge symmetry and superconductivity - Status of QED, Organisation of perturbative expansion, Precision tests