Editing Dirac spinor (section)

{{Short description|Complex four-component spinor}}
In [[quantum field theory]], the '''Dirac spinor''' is the [[spinor]] that describes all known [[fundamental particle]]s that are [[fermion]]s, with the possible exception of [[neutrino]]s. It appears in the [[Plane wave|plane-wave]] solution to the [[Dirac equation]], and is a certain combination of two [[Weyl spinor]]s, specifically, a [[bispinor]] that transforms "spinorially" under the action of the [[Lorentz group]].

Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in [[nature]]; this includes the [[electron]] and the [[quark]]s. Algebraically they behave, in a certain sense, as the "square root" of a [[vector (mathematics and physics)|vector]]. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to [[geometry]]. For example, effectively all [[Riemannian manifold]]s can have spinors and [[spin connection]]s built upon them, via the [[Clifford algebra]].<ref>{{cite book |first=Jürgen |last=Jost |year=2002 |title=Riemannian Geometry and Geometric Analysis |edition=3rd |location= |publisher=Springer |chapter=Riemannian Manifolds |pages=1–39 |doi=10.1007/978-3-642-21298-7_1 }} ''See section 1.8.''</ref> The Dirac spinor is specific to that of [[Minkowski spacetime]] and [[Lorentz transformation]]s; the general case is quite similar.

This article is devoted to the Dirac spinor in the '''Dirac representation'''. This corresponds to a specific representation of the [[gamma matrices]], and is best suited for demonstrating the positive and negative energy solutions of the Dirac equation. There are other representations, most notably the [[bispinor|chiral representation]], which is better suited for demonstrating the [[chiral symmetry]] of the solutions to the Dirac equation. The chiral spinors may be written as linear combinations of the Dirac spinors presented below; thus, nothing is lost or gained, other than a change in perspective with regards to the [[discrete symmetries]] of the solutions.

The remainder of this article is laid out in a pedagogical fashion, using notations and conventions specific to the standard presentation of the Dirac spinor in textbooks on quantum field theory. It focuses primarily on the algebra of the plane-wave solutions. The manner in which the Dirac spinor transforms under the action of the Lorentz group is discussed in the article on [[bispinor]]s.