User:Jambaugh/Sandbox/Spinors

Current Intro

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.

My Attempt

Spinors are elements of an abstract vector space on which the actions of a group of rotations have a projective representation. A linear representation of a group is a mapping from that group to linear operators such that the group product gets mapped isomorphically to the operator product. In a projective representation group products need only correspond to operator products give or take a sign change. For spinors this means that while two 180 degree rotations in the same plane combine to yield the identity element, i.e. a 360degree rotation is the same as taking no action at all, the effect on the spinors is that a 360 degree rotation equates to multiplication by -1. The spinor has reversed directions (in its abstract vector space).

Since the spin representation does not quite respect the product we have in fact a distinct group called the spin group which forms a double covering of the corresponding rotation group. One may alternatively think in terms of the spin group itself being represented