Abstract
In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.
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Acknowledgements
This research was funded in part by the Fetzer Franklin Memorial Trust and the University College London Impact Scheme. The author would like to thank the organisers of the “13th International Conference on Clifford Algebras and Their Applications in Mathematical Physics” for the opportunity to present this work there. He would also like to thank B. J. Hiley, P. Van Reeth, M. Hajtanian, and D. Nellist for their insightful conversations and support.
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Bradshaw, P.T.J. A Relationship Between Spin and Geometry. Adv. Appl. Clifford Algebras 34, 26 (2024). https://doi.org/10.1007/s00006-024-01322-1
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DOI: https://doi.org/10.1007/s00006-024-01322-1