Relativistic Quantum Mechanics

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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

Course Curriculum

Introduction, The Klein-Gordon equation Details 57:57
Particles and antiparticles, Two component framework Details 58:12
Coupling to electromagnetism, Solution of the Coulomb problem Details 56:58
Bohr-Sommerfeld semiclassical solution of the Coulomb problem Details 56:26
Dirac matrices, Covariant form of the Dirac equation Details 57:38
Electromagnetic interactions, Gyromagnetic ratio Details 57:14
The Hydrogen atom problem, Symmetries, Parity, Separation of variables Details 57:14
The Frobenius method solution, Energy levels and wavefunctions Details 56:39
Non-relativistic reduction, The Foldy-Wouthuysen transformation Details 56:20
Interpretation of relativistic corrections, Reflection from a potential barrier Details 55:11
The Klein paradox, Pair creation process and examples Details 57:10
Zitterbewegung, Hole theory and antiparticles Details 56:50
Charge conjugation symmetry, Chirality, Projection operators Details 55:44
Weyl and Majorana representations of the Dirac equation Details 56:59
Time reversal symmetry, The PCT invariance Details 57:3
Arrow of time and particle-antiparticle asymmetry, Band theory for graphene Details 57:55
Dirac equation structure of low energy graphene states, Details 57:46
Groups and symmetries, The Lorentz and Poincare groups Details 58:22
Group representations, generators and algebra, Translations, rotations and boosts Details 57:45
The spinor representation of SL(2,C), The spin-statistics theorem Details 57:35
Finite dimensional representations of the Lorentz group, Euclidean and Galilean groups Details 57:16
Classification of one particle states, The little group, Mass, spin and helicity Details 55:33
Massive and massless one particle states Details 55:53
P and T transformations, Lorentz covariance of spinors Details 57:58
Lorentz group classification of Dirac operators, Orthogonality Details 57:38
Propagator theory, Non-relativistic case and causality Details 58:15
Relativistic case, Particle and antiparticle contributions, Feynman prescription Details 58:49
Interactions and formal perturbative theory, The S-matrix and Feynman diagrams Details 57:58
Trace theorems for products of Dirac matrices Details 55:59
Photons and the gauge symmetry Details 1:49
Abelian local gauge symmetry, The covariant derivative and invariants Details 1:32
Charge quantisation, Photon propagator, Current conservation and polarisations Details 59:7
Feynman rules for Quantum Electrodynamics, Nature of perturbative expansion Details 59:5
Dyson’s analysis of the perturbation series, Singularities of the S-matrix Details 59:13
The T-matrix, Coulomb scattering Details 1:9
Mott cross-section, Compton scattering Details 59:1
Klein-Nishina result for cross-section Details 59:5
Photon polarisation sums, Pair production through annihilation Details 1:21
Unpolarised and polarised cross-sections Details 0:59
Helicity properties, Bound state formation Details 1:1:51
Bound state decay, Non-relativistic potentials Details 59:46
Lagrangian formulation of QED, Divergences in Green’s functions Details 1:1:17
Infrared divergences due to massless particles, Renormalisation Details 1:3:53
Symmetry constraints on Green’s functions, Furry’s theorem, Ward-Takahashi identity Details 57:34
Status of QED, Organisation of perturbative expansion, Precision tests Details 51:32

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