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Project

Spectroscopic and Geometric Algebra Methods for Lattice Gauge Theory

The best theory currently available to describe the formation of hadronic particles like protons and neutrons is Quantum Chromodynamics (QCD). It predicts the existence of quarks and gluons, although neither of these particles have been observed directly. The theory predicts this is due to a phenomenon called confinement, which forbids quarks and gluons to travel unaccompanied. However, they are believed to be real physical entities, as the predictions of this model are in excellent agreement with a wide range of experiments. But perhaps one of the most exciting predictions of this theory has not yet been observed: glueballs. Glueballs are neutral particles consisting purely of gluons. Some glueball candidates have been identified in particle accelerators, but physicist cannot be sure yet whether these are really the glueballs predicted by QCD. This is due to the great difficulty of separating them from other particles which have similar masses. Therefore more accurate predictions of the glueball mass and other spectral properties are needed.

Another type of particle which is unobservable, is the Faddeev-Popov ghost. These particles are introduced to gauge fix the theory: a mathematical necessity to allow the calculation of scattering amplitudes. As the Faddeev-Popov ghosts are obtained by rewriting the mathematical machinery of gauge fixing in the language of particle physics, they are not just unobservable; they are unphysical.

The primary goal of this thesis was to use lattice QCD to calculate the spectral properties of gluons, glueballs, and Faddeev-Popov ghosts. This was done by finding the Källén-Lehmann spectral representation from the propagators of these particles. Additionally, a method for calculating the Faddeev-Popov ghosts in linear covariant gauges on the lattice was proposed.

As QCD is a local gauge theory over the non-abelian group SU(3), a new set of mathematical tools to study SU(3) was also developed. These tools find their origin in geometric algebra, but have also been translated to the more traditional matrix representation of SU(3). The results are presented using both the geometric algebra and the matrix representation.

Lastly, the exploration of geometric algebra lead to the discovery of a novel technique for the numerical computation of derivatives of holomorphic functions, using complex quaternions. This chapter stands somewhat apart from the others, although the complex quaternions are isomorphic to SU(2) x U(1), which is the gauge group of the electroweak force. Therefore this paper does briefly describe a different way of understanding SU(2) x U(1) compared to the usual matrix formalism, although it then quickly digresses into the derivation of holomorphic functions.

Date:7 Feb 2017 →  2 Jun 2021
Keywords:Lattice QCD
Disciplines:Other biological sciences, Other natural sciences
Project type:PhD project