Bradshaw, Peter Thomas Joseph;
(2024)
Foundations for an Elementary Algebraic Theory of Systems with
Arbitrary Non-Relativistic Spin.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The description of spin in modern physics is multifaceted, and links together a broad variety of physical concepts, including angular momentum, spinors, quantum mechanics, and special relativity. However, there remain foundational aspects of the existence of spin which are not fully understood: What physical and mathematical structure is strictly necessary for arbitrary spin to exist within a general physical model? Are quantum mechanics, relativity, or notions of angular momentum essential to its existence? What are the physically distinct observables in a physical theory with spin? In this thesis, we will address these questions by presenting a new account for the emergence of spin in non-relativistic physical theories through the mathematical language of non-commutative algebras. The structure of these algebras will fundamentally derive from the geometry of real Euclidean three-space, and reveals a geometric origin for spin which is neither classical nor quantum. We will see that spin’s phenomenology as a form of angular momentum is an emergent prediction of quantum mechanics, and that spin may be a natural source of non-commutative geometry, entailing couplings between the position and spin of a system. To achieve this, we will use limited mathematical structure to: construct a generic methodology for the elementary study of algebraic structures from their minimal polynomials; present an elementary algebraic method to derive real algebras which describe arbitrary spins in terms of the physically distinct observables of the system; and define a family of algebras of position operators whose structures encode both the geometric action of rotations, and the structure of its spin operators in terms of geometrical objects.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Foundations for an Elementary Algebraic Theory of Systems with Arbitrary Non-Relativistic Spin |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2022. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| Keywords: | Quantum Mechanics, Quantum Foundations, Spin, Algebra, Algebraic Methods |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Physics and Astronomy |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10197514 |
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