spinor

thumb|upright=1.5|A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.

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Image: Spinor on the circle.pdf

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Commons category: Spinors
Stack Exchange tag: mathoverflow.net/tags/spinor

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~47 min read

Article

32 sections

Contents

Introduction
Mathematical definition
Overview
Representation theoretic point of view
Geometric point of view
Clifford algebras
Spin groups
Spinor fields in physics
Spinors in representation theory
Attempts at intuitive understanding
History
Examples
Two dimensions
Examples
Three dimensions
Explicit constructions
Component spinors
Abstract spinors
Minimal ideals
Exterior algebra construction
Hermitian vector spaces and spinors
Clebsch–Gordan decomposition
Even dimensions
Odd dimensions
Consequences
Summary in low dimensions
Reality type and signature
See also
Notes
References
Works cited
Further reading

In geometry and physics, spinors (pronounced "spinner"; ) are elements of a complex vector space that can be associated with Euclidean space. Spinors can be thought of as companion geometric objects to Euclidean space that, like Euclidean vectors, respond when the Euclidean space is subjected to a rotation. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. Spinors are therefore often described heuristically as "square roots" of (geometric) vectors, and a geometric vector can be constructed quadratically from a spinor.