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 relates particles which behave differently under rotations. 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 appealing in constructing models of particle physics, and likely a necessary ingredient to unify quantum physics with gravity.
From a theoretical perspective, supersymmetry is useful because it allows one to compute certain physical quantities exactly, even when the interactions between particles are strong and non-linear, making standard methods of little use. At the same time, supersymmetry opens a bridge towards various areas of mathematics, most notably geometry. Very non-trivial mathematical predictions have been obtained by physicists studying supersymmetric quantum field theories. Several of them were later proven rigourously by mathematicians.
Although it was originally devised in the contexts of quantum field theory and string theory, supersymmetry already shows its power in the simpler context of quantum mechanics, where it appears unexpectedly in the description of various physical systems such as electrons in external magnetic fields. Many solvable models in quantum mechanics can be constructed by assuming supersymmetry: one can exactly determine the wave functions of bound states, or transmission and reflection probabilities in scattering processes, even if the potential looks at first sight complicated. Even more interestingly, the 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.
Recommended pre- and co-requisites
- Mathematical Physics II
- Quantum Mechanics III / Foundations of Physics 3A
If you are interested in the project but lack some prerequisites, please get in touch and we will see what can be done.
Resources
Context:
Wikipedia article on Supersymmetric Quantum Mechanics.
Introductory reading:
F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, 2001.
F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and Quantum Mechanics.
L. E. Gendenshtein, I. V. Krive, Supersymmetry in quantum mechanics.
R. de Lima Rodrigues, The quantum mechanics SUSY algebra: an introductory review.
Advanced reading:
D. Skinner, Supersymmetry (chapter 3).
P. Argyres, An Introduction to Global Supersymmetry.
E. Witten, Dynamical breaking of supersymmetry (section 6).
E. Witten, Constraints on supersymmetry breaking.
K. Hori et al., Mirror Symmetry (chapter 10).
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