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Relativistic Quantum Mechanics

Relativistic Quantum Mechanics. 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. (from nptel.ac.in)

Lecture 29 - Trace Theorems for Products of Dirac Matrices

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Klein-Gordon and Dirac Equations

Lecture 01 - Introduction, the Klein-Gordon Equation

Lecture 02 - Particles and Antiparticles, Two Component Framework

Lecture 03 - Coupling to Electromagnetism, Solution of the Coulomb Problem

Lecture 04 - Bohr-Sommerfeld Semi Classical Solution of the Coulomb Problem, The Dirac Equation and the Clifford Algebra

Lecture 05 - Dirac Matrices, Covariant Form of the Dirac Equation, Equations of Motion, Spin, Free Particle Solutions

Lecture 06 - Electromagnetic Interactions, Gyromagnetic Ratio, Lorentz Force, Larmor Precession

Lecture 07 - The Hydrogen Atom Problem, Symmetries, Parity, Separation of Variables

Lecture 08 - The Frobenius Method Solution, Energy Levels and Wavefunctions

Lecture 09 - Non-relativistic Reduction, The Foldy-Wouthuysen Transformation

Lecture 10 - Interpretation of Relativistic Corrections, Reflection from a Potential Barrier

Lecture 11 - The Klein Paradox, Pair Creation Process and Examples

Lecture 12 - Zitterbewegung, Hole Theory and Antiparticles

Lecture 13 - Charge Conjugation Symmetry, Chirality, Projection Operators, The Weyl Equation

Lecture 14 - Weyl and Majorana Representations of the Dirac Equation, Unitary and Antiunitary Symmetries

Lecture 15 - Time Reversal Symmetry, The PCT Invariance

Lecture 16 - Arrow of Time and Particle-antiparticle Asymmetry, Band Theory for Graphene

Lecture 17 - Dirac Equation Structure of Low Energy Graphene States, Relativistic Signatures in Graphene Properties

Lorentz and Poincare Groups

Lecture 18 - Groups and Symmetries, the Lorentz and Poincare Groups

Lecture 19 - Group Representations, Generators and Algebra, Translations, Rotations and Boosts

Lecture 20 - The Spinor Representation of SL (2, C), The Spin-statistics Theorem

Lecture 21 - Finite Dimensional Representations of the Lorentz Group, Euclidean and Galilean Groups

Lecture 22 - Classification of One Particle States, The Little Group, Mass, Spin and Helicity

Lecture 23 - Massive and Massless One Particle States

Lecture 24 - P and T Transformations, Lorentz Covariance of Spinors

Lecture 25 - Lorentz Group Classification of Dirac Operators, Orthogonality and Completeness of Dirac Spinors, Projection Operators

Quantum Electrodynamics

Lecture 26 - Propagator Theory, Non-relativistic Case and Causality

Lecture 27 - Relativistic Case, Particle and Antiparticle Contributions, Feynman Prescription and the Propagator

Lecture 28 - Interactions and Formal Perturbative Theory, The S-matrix and Feynman Diagrams

Lecture 29 - Trace Theorems for Products of Dirac Matrices

Lecture 30 - Photons and the Gauge Symmetry

Lecture 31 - Abelian Local Gauge Symmetry, the Covariant Derivative and Invariants

Lecture 32 - Charge Quantization, Photon Propagator, Current Conservation and Polarizations

Lecture 33 - Feynman Rules for Quantum Electrodynamics, Nature of the Perturbative Expansion

Lecture 34 - Dyson's Analysis of Perturbation Series, Singularities of the S-matrix, Elementary QED Processes

Lecture 35 - The T-matrix, Coulomb Scattering

Lecture 36 - Mott Cross-section, Compton Scattering

Lecture 37 - Klein-Nishina Result for Cross-Section

Lecture 38 - Photon Polarisation Sums, Pair Production through Annihilation

Lecture 39 - Unpolarised and Polarised Cross-Sections

Lecture 40 - Helicity Properties, Bound State Formation

Lecture 41 - Bound State Decay, Non-relativistic Potentials

Lecture 42 - Lagrangian Formulation of QED, Divergences in Green's Functions, Superficially Divergent 1-loop Diagrams and Regularization

Lecture 43 - Infrared Divergences due to Massless Particles, Renormalization and Finite Physical Results

Lecture 44 - Symmetry Constraints on Green's Functions, Furry's Theorem, Ward-Takahashi Identity, Spontaneous Breaking of Gauge Symmetry and Superconductivity

Lecture 45 - Status of QED, Organization of Perturbative Expansion, Precision Tests