Spinor

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The term "spinor" refers to a certain kind of mathematical object similar to a vector, but which changes sign under a rotation of 360 degrees. Spinors were originally invented by Paul Dirac to describe the wave function of the electron. An n-dimensional spinor of a certain type is an element of a specific projective representation of the rotation group SO(n), or more generally SO(p,q) where p + q = n for spinors in a space of nontrivial signature. By <insert name of theorem here>, this is equivalent to an ordinary representation of the universal cover of SO(p,q), which is called Spin(p,q).


The most typical type of spinor, the Dirac spinor, is a member 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, which 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.


A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2n complex numbers.

Ok, now just tell us how to represent such thing on a computer and we'll be happy. --Taw