spinors

Definition of spinors and their relation to Clifford algebras

The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional space-time. To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4-dimensional space-time (Minkowski space).
Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with \(SO(3,1)\) symmetry, and then builds the spin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fibre bundle, the fibers of which are affine spaces transforming under the spin group. This is the so-called spinor bundle.

After constructing the fiber bundle, one may then consider dynamical systems, such as the __Dirac equation__, or the __Weyl equation__ **on the fiber bundle.** These equations (Dirac or Weyl) have solutions that are plane waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebras / spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called __fermions__; fermions have the algebraic qualities of spinors. By general convention, the terms “fermion” and “spinor” are often used interchangeably in physics, as synonyms of one-another. It appears that all __fundamental particles__ in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the neutrino.