An Overview of Computational Mathematical Physics: A Deep Dive on Gauge Theories

• Main Author: Simoneau, Andre
• Format: Others
• Published: Scholarship @ Claremont 2019
• Subjects: math; gauge theory; hamiltonian; Other Applied Mathematics
• Online Access: https://scholarship.claremont.edu/cmc_theses/2182; https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=3263&context=cmc_theses

Over the course of a college mathematics degree, students are inevitably exposed to elementary physics. The derivation of the equations of motion are the classic examples of applications of derivatives and integrals. These equations of motion are easy to understand, however they can be expressed in other ways that students aren't often exposed to. Using the Lagrangian and the Hamiltonian, we can capture the same governing dynamics of Newtonian mechanics with equations that emphasize physical quantities other than position, velocity, and acceleration like Newton's equations do. Building o of these alternate interpretations of mechanics and understanding gauge transformations, we begin to understand some of the mathematical physics relating to gauge theories. In general, gauge theories are eld theories that can have gauge transformations applied to them in such a way that the meaningful physical quantities remain invariant. This paper covers the buildup to gauge theories, some of their applications, and some computational approaches to understanding them.