Student(s):
Anna Lieb
Dates of Involvement:
Summer 2009 – Present
Faculty Advisor(s):
Steve McCormick
Graduate Mentor:
Christian Kettelsen
Numerical Methods for Quantum Chromodynamics
Introduction
I am working with Christian Kettelsen and Steve McCormick on numerical methods for solving problems in quantum chromodynamics (QCD). The Dirac equation (the relativistic analogue to the Schrodinger equation), which governs the time evolution of particle systems, is in most cases not analytically solvable, necessitating the use of numerical algorithms to produce solutions that allow for the comparison of theory with experiment. A variety of challenges are inherent in the process of discretizing and inverting the Dirac operator. Any discretization must preserve certain fundamental physical properties such as chiral symmetry and gauge invariance. Furthermore, many traditional techniques lead to a non-physical phenomenon known as species doubling. Even a physically acceptable discretization may prove difficult or impossible to solve on meaningful grid sizes. We are working to develop a spectral discretization technique that will both preserve the necessary physical symmetries and render the system computationally tractable.
Currently, I am working on using Matlab's fast Fourier transform to build the discrete Dirac operator for a simple two-dimensional system in quantum electrodynamics called the Schwinger Model. We hope to use this model to work out the major numerical issues before proceeding to the full, four-dimensional model for QCD. I will soon be working on checking that the spectral discretization method preserves physical symmetries. After this process is complete, we can move on to generalizing our methods for the full QCD system. If our efforts prove to be successful, it would allow for more realistic and higher fidelity quantum mechanical simulations.
About Anna Lieb:
Anna Lieb is currently a junior Applied Mathematics major. She is interested in physics (especially quantum mechanics) and biology. She hopes someday she can teach people that math is not scary -- its beautiful!
