Overview

ABOUT THE COURSE:Ordinary quantum mechanics deals with particles in various potentials in the non-relativistic regime. In the relativistic regime the same techniques lead to several difficulties leading to unphysical solutions unless some drastically new physical principles are proposed and new techniques are invented. The present course details with quantum mechanics in the relativistic regime covering aspects like Klien Gordon Equation and solutions, Dirac Equations and solutions, Dirac Equation with potentials, Klien Paradox, Zitterwitterwegung, anti particles etc. Dirac and Majorana fermions. Applications to High Energy Physics and Condensed Matter physics. It closes with quantization of the Electromagnetic field and photons.INTENDED AUDIENCE: Masters Students and Ph. D studentsPREREQUISITES: Quantum Mechanics I, Quantum Mechanics II, Classical Mechanics, Special Theory of Relativity

Syllabus

Module 1:Motivation and Background:
When are particles relativistic and why relativistic quantum mechanics?
Recap of Quantum mechanics, Schroedinger equation, vector spaces, simple calculations in ordinary quantum mechanics, spin/angular momentum, potentials. Hydrogen Atom,Hamiltonian Picture, Lagrangian Picture.
Module 2:Path Integral Picture of Quantum Mechanics:
Classical canonical transformations to Path Integrals, Path Integrals and action principle in quantum mechanics, Free particle, harmonic oscillator, spin-statistics theorem and Grassman numbers
Module 3:Special Theory of Relativity :
STR postulates and Lorentz Transformations, Tensors, Vectors and Scalars, Lorentz Group,
Poincare Group, Representations of Lorentz and Poincare groups, Particle classification in terms of spin and mass.
Module 4:Klein-Gordon :
Klein-Gordon equation, analogy with Harmonic oscillator picture, The particle interpretation of the KG field in terms of the harmonic oscillators for each momentum mode. The non-relativistic reduction of the Klein Gordon Equation, the (re) interpretation of the probability density in terms of charge and current density. Problems with Single particle interpretation.
Module 5:Non-Relativistic Many body Physics :
Multiparticle quantum mechanics. Some many body physics examples, Why it fails in relativistic case, Particle non-conservation in Relativistic case.
Module 6:Dirac Equation:
A linear dispersion relation, alpha and beta matrices. The Dirac equation, properties of the gamma matrices and the metric. Gamma algebra, Weyl and Majorana representation.
Module 7:Dirac equation : Solutions
Solutions of the Dirac Equation, plane wave solutions, Particle and Anti-Particle solutions, Dirac Sea. Spinor algebra, completeness relations, orthonormal relations and their derivations, non-relativistic reduction. Pauli Equation. Building (Lorentz) scalars out of combinations of fermions. Bilinear covariants. Lorentz covariance of Dirac Equation. Projection Operators. Introduction to FeynCalc Mathematica Package.
Module 8:Dirac equation :Interpretations:
Dirac Hole Theory Holes, electrons, Klein Paradox , Zitterbitterwegung, Applications of Dirac Hole Theory. Renormalisation of electric charge.
Module 9:Dirac equation: Other solutions:
Weyl Fermions. Applications in High Energy physics: Neutrinos
Majorana Fermions: Applications in Condensed Matter: Graphene.
Module 10:Covariance of Dirac equation :
Transformation of Dirac Fermions under Lorentz Transformation, Spinor transformations under Lorentz Transformation. Showing the covariance of Dirac equation. C, P and T.
Module 11:Second Quantisation of EM Theory:
Harmonic Oscillators for second quantisation, Photons and outlook.

Module 12:Introduction to QED :
Non relativistic Reduction of Dirac equation, Applications, Hamiltonian with EM potential
Klein paradox revisit. QED interpretation. Gauge Invariance and Lorentz invariance. Lamb Shift
References:Ramamurthy Shankar, Merzbachher, Schiff, Feynman HibbsRaymond,Zinn-Justin (Path Integrals)Weinberg QFT Vol1 , Wu-Ki-Tung, Group Theory, Mukunda and Mukhi.Bjorken and Drell Vol 1Griffiths, Griener, Itzykson and ZuberRamamurti Shankar, Salam’s paper, Cheng and LiRyder, itzkyson and Zuber other QFT 1 books.
Additional Material:Atomic Spectra. Quankonia and Leptonia.