Editing Dirac spinor (section)

==Completeness==
The four rest-frame spinors <math>u^{(s)}\left(\vec{0}\right),</math> <math>\;v^{(s)}\left(\vec{0}\right)</math> 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
<math display="block">({p\!\!\!/} - m)u^{(s)}\left(\vec{p}\right) = 0</math>
and
<math display="block">({p\!\!\!/} + m)v^{(s)}\left(\vec{p}\right) = 0</math>

This follows because
<math display="block"> {p\!\!\!/}{p\!\!\!/} = p^\mu p_\mu = m^2 </math>
which in turn follows from the anti-commutation relations for the [[gamma matrices]]:
<math display="block">\left\{\gamma^\mu, \gamma^\nu\right\} = 2\eta^{\mu\nu}</math>
with <math>\eta^{\mu\nu}</math> the [[metric tensor]] in flat space (in curved space, the gamma matrices can be viewed as being a kind of [[vielbein]], although this is beyond the scope of the current article). It is perhaps useful to note that the Dirac equation, written in the rest frame, takes the form
<math display="block">\left(\gamma^0 - 1\right)u^{(s)}\left(\vec{0}\right) = 0</math>
and
<math display="block">\left(\gamma^0 + 1\right)v^{(s)}\left(\vec{0}\right) = 0</math>
so that the rest-frame spinors can correctly be interpreted as solutions to the Dirac equation. There are four equations here, not eight. 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. That is, one could take each of the four vectors <math>u^{(s)}\left(\vec{0}\right),</math> <math>\;v^{(s)}\left(\vec{0}\right)</math> and multiply each by a distinct global phase <math>e^{i\eta}.</math> This phase changes nothing; it can be interpreted as a kind of global gauge freedom. This is not to say that "phases don't matter", as of course they do; the Dirac equation must be written in complex form, and the phases couple to electromagnetism.  Phases even have a physical significance, as the [[Aharonov–Bohm effect]] implies: the Dirac field, coupled to electromagnetism, is a [[U(1)]] [[fiber bundle]] (the [[circle bundle]]), and the Aharonov–Bohm effect demonstrates the [[holonomy]] of that bundle. All this has no direct impact on the counting of the number of distinct components of the Dirac field. In any setting, there are only four real, distinct components.

With an appropriate choice of the gamma matrices, it is possible to write the Dirac equation in a purely real form, having only real solutions: this is the [[Majorana equation]]. However, it has only two linearly independent solutions. 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. But of course, this makes sense: coupling to electromagnetism requires taking a real field, and making it complex.  With some effort, the Dirac equation can be interpreted as the "complexified" Majorana equation. This is most easily demonstrated in a generic geometrical setting, outside the scope of this article.