Editing Spinor (section)

=== Spinor fields in physics ===
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 [[fiber bundle]], the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the [[Dirac equation]], or the [[Weyl equation]] on the fiber bundle. These equations (Dirac or Weyl) have solutions that are [[plane wave]]s, having symmetries characteristic of the fibers, ''i.e.'' having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called [[fermion]]s; 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.{{cn|date=February 2024}}

It appears that all [[fundamental particle]]s in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the [[neutrino]]. There does not seem to be any ''a priori'' reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of {{math|Cℓ<sub>2,2</sub>(<math>\Reals</math>)}}, the [[Majorana spinor]].<ref>Named after [[Ettore Majorana]].</ref> There also does not seem to be any particular prohibition to having [[Weyl spinor]]s appear in nature as fundamental particles.

The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.<ref>{{cite journal |first1=Matthew R. |last1=Francis |first2=Arthur |last2=Kosowsky |title=The construction of spinors in geometric algebra |journal=Annals of Physics |orig-year=20 March 2004 |year=2005 |volume=317 |issue=2 |pages=383–409 |doi=10.1016/j.aop.2004.11.008 |arxiv=math-ph/0403040|bibcode=2005AnPhy.317..383F |s2cid=119632876 }}</ref> Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

Weyl spinors are insufficient to describe massive particles, such as [[electron]]s, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the [[Dirac equation]] is needed. The initial construction of the [[Standard Model]] of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the [[Higgs mechanism]] gives electrons a mass; the classical [[neutrino]] remained massless, and was thus an example of a Weyl spinor.{{efn|More precisely, the electron starts out as two massless Weyl spinors, left and right-handed. Upon symmetry breaking, both gain a mass, and are coupled to form a Dirac spinor.}} However, because of observed [[neutrino oscillation]], it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors.<ref>{{cite journal|last=Wilczek|first=Frank|author-link=Frank Wilczek|title=Majorana returns|journal=Nature Physics|volume=5|issue=9|year=2009|doi=10.1038/nphys1380|pages=614–618|publisher=[[Macmillan Publishers]] |issn=1745-2473|bibcode = 2009NatPh...5..614W }}</ref> It is not known whether Weyl spinor fundamental particles exist in nature.

The situation for [[condensed matter physics]] is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from [[semiconductor]]s to far more exotic materials. In 2015, an international team led by [[Princeton University]] scientists announced that they had found a [[quasiparticle]] that behaves as a Weyl fermion.<ref>{{cite journal|last=Xu|first=Yang-Su|title=Discovery of a Weyl Fermion semimetal and topological Fermi arcs|journal=Science Magazine|publisher=[[American Association for the Advancement of Science|AAAS]]|issn=0036-8075|year=2015|doi=10.1126/science.aaa9297|display-authors=etal|arxiv = 1502.03807 |bibcode = 2015Sci...349..613X| volume=349| issue=6248|pages=613–617|pmid=26184916|s2cid=206636457}}</ref>