It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks.
Algebraically they behave, in a certain sense, as the "square root" of a 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 manifolds can have spinors and spin connections built upon them, via the Clifford algebra.
[1] The Dirac spinor is specific to that of Minkowski spacetime and Lorentz transformations; the general case is quite similar.
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.
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.
The manner in which the Dirac spinor transforms under the action of the Lorentz group is discussed in the article on bispinors.
(which is called the inverse reduced Compton wavelength) in ordinary units.
In order to derive an expression for the four-spinor ω, the matrices α and β must be given in concrete form.
Assembling these pieces, the full positive energy solution is conventionally written as
In the laboratory frame, both are considered to have positive mass and positive energy, although they are still very much dual to each other, with the flipped sign on the antiparticle plane-wave suggesting that it is "travelling backwards in time".
The interpretation of "backwards-time" is a bit subjective and imprecise, amounting to hand-waving when one's only evidence are these solutions.
It does gain stronger evidence when considering the quantized Dirac field.
A more precise meaning for these two sets of solutions being "opposite to each other" is given in the section on charge conjugation, below.
since these form an orthonormal basis with respect to a (complex) inner product.
indicate that there are four distinct, real, linearly independent solutions to the Dirac equation.
That they are indeed solutions can be made clear by observing that, when written in momentum space, the Dirac equation has the form
It is perhaps useful to note that the Dirac equation, written in the rest frame, takes the form
so that the rest-frame spinors can correctly be interpreted as solutions to the Dirac equation.
Although 4-spinors are written as four complex numbers, thus suggesting 8 real variables, only four of them have dynamical independence; the other four have no significance and can always be parameterized away.
This phase changes nothing; it can be interpreted as a kind of global gauge freedom.
All this has no direct impact on the counting of the number of distinct components of the Dirac field.
These solutions do not couple to electromagnetism; they describe a massive, electrically neutral spin-1/2 particle.
Apparently, coupling to electromagnetism doubles the number of solutions.
This is most easily demonstrated in a generic geometrical setting, outside the scope of this article.
Thus, a positive-energy solution (dropping the spin superscript to avoid notational overload)
Charge conjugation is very much a pairing of Dirac spinor to its "exact opposite".