@phdthesis{discovery10197514,
           title = {Foundations for an Elementary Algebraic Theory of Systems with
Arbitrary Non-Relativistic Spin},
            year = {2024},
           month = {September},
          school = {UCL (University College London)},
            note = {Copyright {\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.},
          author = {Bradshaw, Peter Thomas Joseph},
             url = {https://discovery.ucl.ac.uk/id/eprint/10197514/},
        keywords = {Quantum Mechanics, Quantum Foundations, Spin, Algebra, Algebraic Methods},
        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.}
}