Short Course

A. Shadi Tahvildar-Zadeh, Rutgers University (New Brunswick)
Mathematical Foundations of General-Relativistic Quantum Mechanics

In this short course, we will embark on a mathematically rigorous study of relativistic one-body quantum Hamiltonians for Hydrogenic atoms that include gravitational interactions as well as electromagnetic ones. The first three lectures will provide the necessary background material. In the last two lectures, some recently obtained results in this direction will be presented. A working knowledge of PDEs and Differential Geometry would be helpful. No Physics background will be assumed. Outine of the course:

Lecture 1. General relativity and gravitation. Einsteins vacuum equations. Minkowski, Schwarzschild, and Kerr spacetimes. Black holes and naked singularities. Arnowitt-Deser-Misner mass and angular momentum of asymptotically flat spacetimes. G-dependence of vacuum metrics.

Lecture 2. Linear and nonlinear electromagnetics. Maxwell’s equations. Infinite Self-Energy problem. Born-Infeld theory. Gravity and electromagnetism. Einstein-Maxwell equations. Reissner-Nordstrom and Kerr- Newman spacetimes. The Hoffmann spacetime.

Lecture 3. Special-Relativistic quantum mechanics. Diracs equation for the electron. Born-Oppenheimer approximation. Spectral theory of Dirac operators. The relativistic Coulomb problem. Derivation of Bohr- Sommerfeld Spectrum for special-relativistic Hydrogen. Scattering and bound states. Ground and excited states.

Lecture 4. General-Relativistic quantum mechanics. Dirac equation on curved background spacetimes. Chandrasekhar-Page-Toop separation of variables. Selfadjointness. Non-existence results of Belgiorno et al, and Finster et al. Dirac equation on the Hoffmann background. Spectrum of general-relativistic Hydrogen, I.

Lecture 5. Zero-gravity limit. Branched Riemann spaces. Appel-Sommerfeld fields, Zipoy topology, and duodromy. Dirac equation on the zero-gravity Kerr-Newman background. Spectrum of general-relativistic Hydrogen, II. Hyperfine Splitting and Lamb Shift without QED.

Date & Time: 
Saturday - Wednesday, 25 - 29 July, 2015, 10:00 - 12:00
Room 221, Department of Mathematical Sciences, Sharif University of Technology