Description

In their quest for a unified description of all the forces in Nature, in the 1970s physicists discovered supersymmetry, a hypothetical symmetry of Nature that unifies bosons and fermions, particles with different spacetime properties. It was soon realized that supersymmetry is the largest symmetry that quantum field theories (the physical theories that describe elementary particles and their interactions) can have, and that theories which enjoy supersymmetry have a remarkably simpler physical behaviour. These features make supersymmetry very appealing in the construction of models of particle physics, and likely a necessary ingredient to unify quantum mechanics with gravity.

From a theoretical perspective, supersymmetry is useful because it allows one to solve certain subsectors of quantum field theory, even when the interactions between particles are strong and non-linear and standard methods are of little use. At the same time, supersymmetry opens a bridge towards various areas of mathematics, most notably geometry. Many highly non-trivial mathematical predictions have been obtained by physicists studying supersymmetric quantum field theories. Several of them have then been proven rigourously by mathematicians.

Although it was originally devised in the context of quantum field theory and string theory, supersymmetry already shows its full power in the simpler context of quantum mechanics. Many solvable models of quantum mechanics can be obtained by assuming supersymmetry: one can exactly determine the wave functions of bound states, or transmission and reflection probabilities in scattering processes, even if the Schroedinger potential looks very complicated. Even more interestingly, the physical study of ground states of supersymmetric quantum mechanical models leads to strikingly simple and powerful mathematical results in geometry and topology. The aim of this project will be to explore the physics and mathematics of supersymmetric quantum mechanics.

Pre- and co-requisites

2H Mathematical Physics

3H Quantum Mechanics.

Resources

For some context:

• Wikipedia article on Supersymmetric Quantum Mechanics.

Reading material:

• F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, 2001.
  F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and Quantum Mechanics.
• E. Witten, Dynamical breaking of supersymmetry (section 6).
• K. Hori et al., Mirror Symmetry (chapter 10, advanced).