Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects similar to vectorss, but which change sign under a rotation of 2π radians. Spinors were invented by Wolfgang Pauli and Paul Dirac to describe the physical property of spin. The word "spinor" was coined by Paul Ehrenfest. (The mathematics of spinors is said to have been anticipated by Elie Cartan as early as 1913.)

An n-dimensional spinor of a certain type is an element of a specific projective representation of the rotation group SO(n,R), or more generally of the group SO(p,q,R), where p + q = n for spinors in a space of nontrivial signature. This is equivalent to an ordinary (non-projective) representation of the universal cover of SO(p,q,R), which is a real Lie group called the spinor group Spin(p,q) (in the exceptional cases where the SO group has infinite cyclic fundamental group, the Spin group is taken as the double cover - see anyon).

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra C(p,q), into which Spin(p,q) may be embedded. In even dimensions, this representation is reducible when taken as a representation of Spin(p,q) and may be decomposed into two: the left-handed and right-handed Weyl spinor representations. These may be distinguished only by the action of parity transformations (not part of Spin(p,q), but present in C(p,q)). In addition, sometimes the non-complexified version of C(p,q) has a smaller real representation, the Majorana spinor representation. If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana-Weyl spinor representations. Of all these, only the Dirac representation exists in all dimensions. Dirac spinors are complex representations while Majorana spinors are real representations.

A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2ⁿ complex numbers. (See Special unitary group.)

In the early 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute created games such as Tangloids to teach and model the calculus of spinors.