# Spinor

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In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors. However, spinors transform well under the infinitesimal orthogonal transformations (like infinitesimal rotations or infinitesimal Lorentz transformations). Under the full orthogonal group, however, they do not quite transform well, but only "up to a sign". This means that a 360 degree rotation transforms a spinor into its negative, and so it takes a rotation of 720 degrees for a spinor to be transformed into itself. Specifically, spinors are objects associated to a vector space with a quadratic form (like Euclidean space with the standard metric or Minkowski space with the Lorentz metric), and are realized as elements of representation spaces of Clifford algebras. For a given quadratic form, several different spaces of spinors with extra properties may exist.

Spinors in general were discovered by Élie Cartan in 1913.[1] Later, spinors were adopted by quantum mechanics in order to study the properties of the intrinsic angular momentum of the electron and other fermions. Today spinors enjoy a wide range of physics applications. Classically, spinors in three dimensions are used to describe the spin of the non-relativistic electron and other spin-½ particles. Via the Dirac equation, Dirac spinors are required in the mathematical description of the quantum state of the relativistic electron. In quantum field theory, spinors describe the state of relativistic many-particle systems. In mathematics, particularly in differential geometry and global analysis, spinors have since found broad applications to algebraic and differential topology,[2] symplectic geometry, gauge theory, complex algebraic geometry,[3] index theory,[4] and special holonomy.[5]