Publication Date

Summer 7-2023

Abstract

In this dissertation I will explore the question whether various entities commonly used in quantum field theory can be “constructed". In particular, can spacetime be “constructed" out of building blocks, and can Berezin integral be “constructed" in terms of Riemann integrals.

As far as “constructing" spacetime out of building blocks, it has been attempted by multiple scientific communities and various models were proposed. But the common downfall is they break the principles of relativity. I will explore the ways of doing so in such a way that principles of relativity are respected. One of my approaches is to replace points with some entities that have a direction. In one of the sections, those entities are tangent vectors, in the other section they are pairs of points. In both cases, the success of respecting relativity is partial. On the one hand, I successfully removed preferred time axis. But, on the other hand, I still did not restore the symmetry under Lorentz group (or some modification of Lorentz group). For this reason, I also explored a different approach where I designed a matrix model for spacetime. In this case, I successfully restored Eucledian symmetry. Restoring the MInkowksian symmetry proved to be iii more difficult, and, depending on the models, came with the price. In both cases, though, the qestion of whether the resulting quantum field theory approximates the one that we know is left as an open question. There are reasons to hope that it does, but rigorously finding out whether that is the case is left for the future research.

As far as constructing Berezin integral in terms of Riemann integrals, this was not attempted nearly as much (although there was one attempt in 1983 by Rabin, and possibly a couple of other isolated attempts). I have attempted to address it in my joint paper with Thomas Scanlon where we supplemented the anticommutting wedge product with Clifford algebra. This essentially amounts to “geometric algebra" model of Doran and Lasenby. Our contribution to that model is a design of specific sequence of integrals that would, in a limit, lead to Berezin integral.

Finally, in the last chapter of the dissertation, I tie up those two projects by developing a graph-like interpretation of the Clifford structure that I used with Scanlon for Berezin integral. This implies that Berezin integral, itself, can also be modeled in terms of graphs. Those graphs are separate from the graphs used in the model of spacetime. A model of fermionic fields living in the spacetime would involve tensor products between those two types of graphs.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Terry Loring (Chair)

Second Committee Member

Dimiter Vassilev (Co-chair)

Third Committee Member

Kevin Cahill (Department of Physics, UNM)

Fourth Committee Member

Thomas Scanlon (Math, UC Berkeley)

Language

English

Keywords

Constructivism, Locality, Symmetry, Ontology, Discretization, Realism, Interpretation, Quantum, Spacetime, Subriemannian, Matrix models

Document Type

Dissertation

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