From Invariant Decomposition to Spinors
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911745657798656 |
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| author | Roelfs, Martin Eelbode, David De Keninck, Steven |
| author_facet | Roelfs, Martin Eelbode, David De Keninck, Steven |
| contents | Plane-based Geometric Algebra (PGA) has revealed points in a $d$-dimensional pseudo-Euclidean space $\mathbb{R}_{p,q,1}$ to be represented by $d$-blades rather than vectors. This discovery allows points to be factored into $d$ orthogonal hyperplanes, establishing points as pseudoscalars of a local geometric algebra $\mathbb{R}_{pq}$. Astonishingly, the non-uniqueness of this factorization reveals the existence of a local $\text{Spin}(p,q)$ geometric gauge group at each point. Moreover, a point can alternatively be factored into a product of the elements of the Cartan subalgebra of $\mathfrak{spin}(p,q)$, which are traditionally used to label spinor representations. Therefore, points reveal previously hidden geometric foundations for some of quantum field theory's mysteries. This work outlines the impact of PGA on the study of spinor representations in any number of dimensions, and is the first in a research programme exploring the consequences of this insight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01142 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Invariant Decomposition to Spinors Roelfs, Martin Eelbode, David De Keninck, Steven Mathematical Physics High Energy Physics - Theory Plane-based Geometric Algebra (PGA) has revealed points in a $d$-dimensional pseudo-Euclidean space $\mathbb{R}_{p,q,1}$ to be represented by $d$-blades rather than vectors. This discovery allows points to be factored into $d$ orthogonal hyperplanes, establishing points as pseudoscalars of a local geometric algebra $\mathbb{R}_{pq}$. Astonishingly, the non-uniqueness of this factorization reveals the existence of a local $\text{Spin}(p,q)$ geometric gauge group at each point. Moreover, a point can alternatively be factored into a product of the elements of the Cartan subalgebra of $\mathfrak{spin}(p,q)$, which are traditionally used to label spinor representations. Therefore, points reveal previously hidden geometric foundations for some of quantum field theory's mysteries. This work outlines the impact of PGA on the study of spinor representations in any number of dimensions, and is the first in a research programme exploring the consequences of this insight. |
| title | From Invariant Decomposition to Spinors |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2401.01142 |