Why No Tensor Is Spin-1/2: The Double Cover of SO(3) and the Clifford Algebras Behind Spinors

We prove that the unit quaternions double-cover the rotation group SO(3), deduce that every irreducible representation of SO(3) has odd dimension so that no tensor transforms as spin-1/2, and classify the real Clifford algebras by a table of period eight, identifying spinors with minimal left ideals. The algebraic part of the double-cover theorem is machine-checked in Lean 4 against Mathlib commit 10156080d0e7997b71f3ef86bd6f0211b42ab90d with Lean 4.36.0-rc1. The record contains the paper, the Lean file and the project files needed to rebuild it. See also the authors' note: https://doi.org/10.5281/zenodo.23407287

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-04
DOI
https://doi.org/10.5281/zenodo.23936209
Primary Topic
Algebraic and Geometric Analysis
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
preprint

Why No Tensor Is Spin-1/2: The Double Cover of SO(3) and the Clifford Algebras Behind Spinors

Vikas Sah, Nanik Tirath Mulchandani
Zenodo (CERN European Organization for Nuclear Research)
Algebraic and Geometric Analysis
preprint

Why No Tensor Is Spin-1/2: The Double Cover of SO(3) and the Clifford Algebras Behind Spinors

Vikas Sah, Nanik Tirath Mulchandani
preprint en

Abstract

We prove that the unit quaternions double-cover the rotation group SO(3), deduce that every irreducible representation of SO(3) has odd dimension so that no tensor transforms as spin-1/2, and classify the real Clifford algebras by a table of period eight, identifying spinors with minimal left ideals. The algebraic part of the double-cover theorem is machine-checked in Lean 4 against Mathlib commit 10156080d0e7997b71f3ef86bd6f0211b42ab90d with Lean 4.36.0-rc1. The record contains the paper, the Lean file and the project files needed to rebuild it. See also the authors' note: https://doi.org/10.5281/zenodo.23407287

Zenodo (CERN European Organization for Nuclear Research)
Algebraic and Geometric Analysis
DOIOpen Access PDFOpenAlex
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

Laika access coming soon
Why No Tensor Is Spin-1/2: The Double Cover of SO(3) and the Clifford Algebras Behind Spinors — Vikas Sah, Nanik Tirath Mulchandani · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS