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.
Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. Mathematically, spinors are elements of spaces carrying representations of the spin group or of the associated Clifford algebra. After choosing a matrix realization of the Clifford algebra, spinors may be represented concretely as column vectors on which the corresponding gamma matrices act.
