20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy: Difference between revisions

Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time).  

For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the special orthogonal Lie groups of rotations <math> SO(n,\mathbb{R}) </math> acting on linear n-dimensional space admit double coverings <math> Spin(n) </math> such that two elements of the spin group correspond to a single rotation. Five algebraic structures comprise the story here: an n-dimensional real vector space, the n-choose-2 dimensional rotation group acting on it, the spin group, another vector space acted upon by the spin group to be constructed, and the Clifford algebra of the first real vector space. To distinguish between the first vector space and the second, the elements of the former are simply referred to as vectors and the latter as spinors. This may be confusing because mathematically both are vector spaces whose elements are vectors, however physically the vectors of the first space have a more basic meaning as directions in physical coordinate space.

::The Spin group is then found within the Clifford algebra, and because the fiber of the double cover map <math> Spin(n)\rightarrow SO(n, \mathbb{R})</math> is discrete, it is of the same dimension n-choose-2.