DOI

10.17077/etd.c55i0it1

Document Type

Dissertation

Date of Degree

Spring 2017

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Polyzou, Wayne

First Committee Member

Jorgensen, Palle

Second Committee Member

Muhly, Paul

Third Committee Member

Stewart, David

Fourth Committee Member

Chifan, Ionut

Abstract

This thesis investigates the use of Daubechies wavelets to separate scales in local quantum field theory. Field theories have an infinite number of degrees of freedom on all distance scales. Quantum field theories are believed to describe the physics of subatomic particles. These theories have no known mathematically convergent approximation methods. Daubechies wavelet bases can be used separate degrees of freedom on different distance scales. Volume and resolution truncations lead to mathematically well-defined truncated theories that can be treated using established methods. This work demonstrates that flow equation methods can be used to block diagonalize truncated field theoretic Hamiltonians by scale. This eliminates the fine scale degrees of freedom. This may lead to approximation methods and provide an understanding of how to formulate well-defined fine resolution limits.

Public Abstract

Wavelets are mathematical functions used in digital photography to create JPEG files from raw images. They significantly reduce the amount computer memory needed to store a photograph without much loss in resolution. Wavelets are fractal valued functions, making them different from most other mathematical functions. This means that they are like snowflakes, where the same structure is repeated on arbitrarily small distance scales. The fractal property makes wavelets ideal candidates for modeling a large class of problems that simultaneously involve structures on all distance scales. Photographs have this property. The class of problems with many scales is one of the most difficult to treat in science. Quantum field theory is one of these problems. Quantum field theories are believed to govern three of the four fundamental forces of nature but have defied mathematical solution for almost 100 years. The goal of this thesis is to use wavelets to decompose quantum theories into degrees of freedom on all scales and then to decouple the short and long distance degrees of freedom. The application of a technique called the flow equation is used to do this. The flow equation is designed to continuously decouple degrees of freedom in quantum field theory on different distance scales. This method is tested on a free field theory, since it is one of the few solvable field theories still involving degrees of freedom on all distance scales. This work is limited to decoupling two distance scales. In this case the flow equation successfully decoupled the two scale degrees of freedom both by resolution and energy scale. This is an important first step in understanding how to decouple all distance scales.

Keywords

flow equation, quantum field theory, wavelets

Pages

ix, 69 pages

Bibliography

Includes bibliographical references (pages 65-69).

Copyright

Copyright © 2017 Tracie L. Michlin

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