Using wavelet bases to separate scales in quantum field theory

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.