In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), a spinor /ˈspɪnər/ is an element of a complex vector space, an irreducible representation space of the Clifford algebra associated to a vector space with a quadratic form (such as Euclidean space with the standard metric or Minkowski space with the Lorentz metric). Like vectors, spinors are representations of the special orthogonal Lie algebra i.e. they transform under infinitesimal orthogonal transformations (such as the infinitesimal rotations or infinitesimal Lorentz transformations). Spinors in general were discovered by Élie Cartan in 1913. Soon after, spinors turned out to be essential in quantum physics to describe the electron and other spin-½ particles. Like vectors and tensors, the transformation properties of spinors are built in their definition. However, unlike spatial vectors, spinors only transform "up to a sign" under the full orthogonal group. This means that a 360 degree rotation transforms a spinor into its negative, and so it takes a rotation of 720 degrees to re-obtain the original spinor. Moreover, unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors. Depending on the quadratic form, several different but closely related spaces of spinors with extra properties may exist. In physics, spinors have a wide range of applications. In quantum mechanics, spinors in three dimensions are used to describe the intrinsic angular momentum, the spin, of the non-relativistic electron and other fermions. Dirac spinors, spinors of the Lorentz metric in dimension 4, are required in the mathematical description of the quantum state of the relativistic electron via the Dirac equation. 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, symplectic geometry, gauge theory, complex algebraic geometry, index theory, and special holonomy.
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