Peter Bradshaw

Abstract

Algebraic approaches to physics are an active and growing area of research. Such theories often capture properties of a physical system in a simple and intuitive way [1], and often can yield new insights into established phenomena [2]. Their study also often avoids the need for explicit matrix or analytical representations, providing an elementary, flexible, and extensible way to describe a system coordinate-free and isolated from unnecessary additional structure. As such, they offer deep insights into the essential foundations of physical phenomena.

In this talk, we will discuss a new algebraic theory to describe the spin of arbitrary non-relativistic systems, as derived in [3]. These algebras are real and written in terms of, and entirely characterised by, the complete set of physically distinct observables of the system, offering a novel characterisation of spin based on its physical properties instead of eigenvalues. This construction is realised directly from the symmetries of Euclidean three-space without: dynamical notions like angular momentum and time; nor additional mathematical structure, such as complex numbers. This indicates that spin is more fundamentally related to geometry than dynamics; this connection will be made more concrete by realising the spin of a non-relativistic system within a natural non-commutative geometry of position operators, without the need for "internal" degrees of freedom, following [4]. 

References

[1] C. Doran, A. Lasenby, "Geometric Algebra for Physicists", Cambridge University Press, Cambridge, 2003.

[2] N. Furey, M. J. Hughes, "Division algebraic symmetry breaking", Physics Letters B, 831, 2022, 137186.

[3] P. T. J. Bradshaw, "An algebraic theory of non-relativistic spin", Physica Scripta, 99, 2024, 036102.

[4] P. T. J. Bradshaw, "A Relationship Between Spin and Geometry", Adv. Appl. Clifford Algebras 34, 26 (2024).