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Elementary particles are classified according to their spin either as bosons, obeying Bose-Einstein statistics, or as fermions, obeying Fermi-Dirac statistics. In quantum mechanics, the Schrödinger equation describes the non-relativistic evolution of a boson, and the Klein-Gordon equation is an analogous relativistic formulation. The Dirac equation describes the evolution of a fermion while also obeying special relativity, and the Lévy-Leblond equation is its adaptation to non-relativistic settings. Although the two kinds of particles behave very differently, there’s a proposed symmetry that associates pairs of bosons and fermions differing only in their spin, called supersymmetry.
In this project we introduce the idea of supersymmetry and the mathematical tools it had engendered. We begin with a method to factorize the one-dimensional time-independent Schrödinger equation, and obtain a symmetry between pairs of Hamiltonian operators with matching spectra. After looking at a supersymmetric description of harmonic oscillations, we construct and present the Lie algebraic structure of the symmetries that leave the solutions of the time dependent free-particle Schrödinger and Lévy-Leblond equations in one spatial dimension invariant. We then construct a d+s “superspace” combining d space-time dimensions and s anticommuting dimensions, and introduce a single supersymmetric superfield in the 2+2 dimensional superspace from which both fermionic and bosonic equations of motions can be extracted. Lastly, we show that using a Laplace transformation the symmetry group of the non-relativistic equations can be extended to represent the symmetries of the relativistic three-dimensional bosonic and fermionic fields obeying the Klein-Gordon and Dirac equations for a free massless particle, while the supersymmetric algebra can be extended to the quite extensive algebra behind the massless relativistic superfield in a 3+2 dimensional superspace.
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Neznansky, Mark, "Supersymmetry and Non-Relativistic Quantum Mechanics" (2013). Senior Projects Spring 2013. 343.
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