An algebraic theory of non-relativistic spin

Bradshaw, Peter T J

doi:10.1088/1402-4896/ad033e

Cited by 1 publication

(6 citation statements)

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“…This paper is primarily concerned with developing non-commutative position operator algebras for systems of arbitrary spin in Euclidean three-space; this is because real algebraic descriptions for the structure of an arbitrary spin system (by which we mean their algebras of spin generators) are known in this case [6]. Despite this, many of the constructions given here are valid for spacetimes with non-degenerate signatures.…”

Section: Scope Of This Papermentioning

confidence: 99%

“…by the two-sided ideal (2.10), for some κ ∈ R. In this paper, we shall abuse notation slightly and denote the product of P κ (V, g) by ⊗, as this is consistent with the developments of [6].…”

Section: The Indefinite-spin Position Algebramentioning

confidence: 99%

“…1.3, to construct non-commutative position operators algebras which subsume the structure of an arbitrary spin system, we will specialise our arguments to Euclidean three-space (E, δ). Doing so will enable us to complete this task by utilising the real arbitrary spin algebras developed in [6]. For clarity, we will discuss the construction and composition of these arbitrary spin algebras before proceeding further.…”

Section: The Algebraic Structure Of Arbitrary Spin Systemsmentioning

confidence: 99%

“…We will abuse notation for a third time by also denoting its product by ⊗. This is to ensure consistency with [6].…”

Section: The Universal Enveloping Algebra U (So(3 R)) Of So(3 R)mentioning

confidence: 99%

“…(Adjoint action) (2.17) Doing so naturally identifies the "multipoles", Definition 2.27. (Multipoles) For all k ∈ N, the multipole tensors [6] are defined recursively,…”

Section: The Universal Enveloping Algebra U (So(3 R)) Of So(3 R)mentioning

confidence: 99%

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A Relationship Between Spin and Geometry

Bradshaw

2024

Adv. Appl. Clifford Algebras

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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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Section: Scope Of This Papermentioning

confidence: 99%

Section: The Indefinite-spin Position Algebramentioning

confidence: 99%

Section: The Algebraic Structure Of Arbitrary Spin Systemsmentioning

confidence: 99%

“…We will abuse notation for a third time by also denoting its product by ⊗. This is to ensure consistency with [6].…”

Section: The Universal Enveloping Algebra U (So(3 R)) Of So(3 R)mentioning

confidence: 99%

“…(Adjoint action) (2.17) Doing so naturally identifies the "multipoles", Definition 2.27. (Multipoles) For all k ∈ N, the multipole tensors [6] are defined recursively,…”

Section: The Universal Enveloping Algebra U (So(3 R)) Of So(3 R)mentioning

confidence: 99%

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