Overview

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Instructor: Prof. Apoorva D. Patel, Department of Physics, IIT Bangalore.

This course covers topics on relativistic quantum mechanics: Dirac and Klein-Gordon equations, Lorentz and Poincare groups, Fundamental processes of Quantum Electrodynamics.

This is a course on relativistic quantum mechanics. Relativity, specifically special relativity, and quantum mechanics have been two very highly successful theories of our twentieth century. And what this subject amounts to is combining the two theories in a very successful manner and working out the predictions. Relativity essentially follows from the property that speed of light in vacuum is an invariant quantity. And mathematically, that is extended to the principle of the Lorentz transformations. Quantum mechanics tells that nature is discrete at a small scale and its formulation is based on unitary evolution of quantities known as wave functions or states. These two theories have been successful on their own. Relativistic quantum mechanics is just on the border line of merging relativity and quantum mechanics, and it offers many consequences as a result. This course will explore those consequences.

Syllabus

Mod-01 Lec-01 Introduction, The Klein-Gordon equation.
Mod-01 Lec-02 Particles and antiparticles, Two component framework.
Mod-01 Lec-03 Coupling to electromagnetism, Solution of the Coulomb problem.
Mod-01 Lec-04 Bohr-Sommerfeld semiclassical solution of the Coulomb problem.
Mod-01 Lec-05 Dirac matrices, Covariant form of the Dirac equation.
Mod-01 Lec-06 Electromagnetic interactions, Gyromagnetic ratio.
Mod-01 Lec-07 The Hydrogen atom problem, Symmetries, Parity, Separation of variables.
Mod-01 Lec-08 The Frobenius method solution, Energy levels and wavefunctions.
Mod-01 Lec-09 Non-relativistic reduction, The Foldy-Wouthuysen transformation.
Mod-01 Lec-10 Interpretation of relativistic corrections, Reflection from a potential barrier.
Mod-01 Lec-11 The Klein paradox, Pair creation process and examples.
Mod-01 Lec-12 Zitterbewegung, Hole theory and antiparticles.
Mod-01 Lec-13 Charge conjugation symmetry, Chirality, Projection operators.
Mod-01 Lec-14 Weyl and Majorana representations of the Dirac equation.
Mod-01 Lec-15 Time reversal symmetry, The PCT invariance.
Mod-01 Lec-16 Arrow of time and particle-antiparticle asymmetry, Band theory for graphene.
Mod-01 Lec-17 Dirac equation structure of low energy graphene states,.
Mod-02 Lec-18 Groups and symmetries, The Lorentz and Poincare groups.
Mod-02 Lec-19 Group representations, generators and algebra, Translations, rotations and boosts.
Mod-02 Lec-20 The spinor representation of SL(2,C), The spin-statistics theorem.
Mod-02 Lec-21 Finite dimensional representations of the Lorentz group, Euclidean and Galilean groups.
Mod-02 Lec-22 Classification of one particle states, The little group, Mass, spin and helicity.
Mod-02 Lec-23 Massive and massless one particle states.
Mod-02 Lec-24 P and T transformations, Lorentz covariance of spinors.
Mod-02 Lec-25 Lorentz group classification of Dirac operators, Orthogonality.
Mod-03 Lec-26 Propagator theory, Non-relativistic case and causality.
Mod-03 Lec-27 Relativistic case, Particle and antiparticle contributions, Feynman prescription.
Mod-03 Lec-28 Interactions and formal perturbative theory, The S-matrix and Feynman diagrams.
Mod-03 Lec-29 Trace theorems for products of Dirac matrices.
Mod-03 Lec-30 Photons and the gauge symmetry.
Mod-03 Lec-31 Abelian local gauge symmetry, The covariant derivative and invariants.
Mod-03 Lec-32 Charge quantisation, Photon propagator, Current conservation and polarisations.
Mod-03 Lec-33 Feynman rules for Quantum Electrodynamics, Nature of perturbative expansion.
Mod-03 Lec-34 Dyson\'s analysis of the perturbation series, Singularities of the S-matrix.
Mod-03 Lec-35 The T-matrix, Coulomb scattering.
Mod-03 Lec-36 Mott cross-section, Compton scattering.
Mod-03 Lec-37 Klein-Nishina result for cross-section.
Mod-03 Lec-38 Photon polarisation sums, Pair production through annihilation.
Mod-03 Lec-39 Unpolarised and polarised cross-sections.
Mod-03 Lec-40 Helicity properties, Bound state formation.
Mod-03 Lec-41 Bound state decay, Non-relativistic potentials.
Mod-03 Lec-42 Lagrangian formulation of QED, Divergences in Green\'s functions.
Mod-03 Lec-43 Infrared divergences due to massless particles, Renormalisation.
Mod-03 Lec-44 Symmetry constraints on Green\'s functions, Furry\'s theorem, Ward-Takahashi identity.
Mod-03 Lec-45 Status of QED, Organisation of perturbative expansion, Precision tests.