Editing Spinor (section)

== Mathematical definition ==
{{hatnote|For a more elementary definition, see also: [[spinors in three dimensions]]}}

The space of spinors is formally defined as the [[fundamental representation]] of the [[Clifford algebra]]. (This may or may not decompose into irreducible representations.) The space of spinors may also be defined as a [[spin representation]] of the [[orthogonal Lie algebra]]. These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional [[group representation]] of the [[spin group]] on which the [[center of a group|center]] acts non-trivially.

=== Overview ===
There are essentially two frameworks for viewing the notion of a spinor: the ''representation theoretic point of view'' and the ''geometric point of view''.

====Representation theoretic point of view====
From a [[representation theory|representation theoretic]] point of view, one knows beforehand that there are some representations of the [[Lie algebra]] of the [[orthogonal group]] that cannot be formed by the usual tensor constructions. These missing representations are then labeled the '''[[spin representation]]s''', and their constituents ''spinors''. From this view, a spinor must belong to a [[group representation|representation]] of the [[covering space|double cover]] of the [[special orthogonal group|rotation group]] {{math|SO(''n'',<math>\Reals</math>)}}, or more generally of a double cover of the [[generalized special orthogonal group]] {{math|SO<sup>+</sup>(''p'', ''q'', <math>\Reals</math>)}} on spaces with a [[metric signature]] of {{math|(''p'', ''q'')}}. These double covers are [[Lie groups]], called the [[spin group]]s {{math|Spin(''n'')}} or {{math|Spin(''p'', ''q'')}}. All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued [[projective representation]]s of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.)

In summary, given a representation specified by the data <math>(V,\text{Spin}(p,q), \rho)</math> where <math>V</math> is a vector space over <math>K = \mathbb{R}</math> or <math>\mathbb{C}</math> and <math>\rho</math> is a homomorphism <math>\rho:\text{Spin}(p,q)\rightarrow \text{GL}(V)</math>, a '''spinor''' is an element of the vector space <math>V</math>.

====Geometric point of view====
From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as [[Fierz identity|Fierz identities]], are needed.

=== Clifford algebras ===
{{further|Clifford algebra}}

The language of [[Clifford algebra]]s<ref>Named after [[William Kingdon Clifford]],</ref> (sometimes called [[geometric algebra]]s) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the [[classification of Clifford algebras]]. It largely removes the need for ''ad hoc'' constructions.

In detail, let ''V'' be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form ''g''. The Clifford algebra {{math|Cℓ(''V'', ''g'')}} is the algebra generated by ''V'' along with the anticommutation relation {{math|1=''xy'' + ''yx'' = 2''g''(''x'', ''y'')}}. It is an abstract version of the algebra generated by the [[gamma matrices|gamma]] or [[Pauli matrices]]. If ''V'' = <math>\Complex^n</math>, with the standard form {{math|1=''g''(''x'', ''y'') = ''x''<sup>T</sup>''y'' = ''x''<sub>1</sub>''y''<sub>1</sub> + ... + ''x''<sub>''n''</sub>''y''<sub>''n''</sub>}} we denote the Clifford algebra by Cℓ<sub>''n''</sub>(<math>\Complex</math>). Since by the choice of an orthonormal basis every complex vector space with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if {{math|1=dim<sub><math>\Complex</math></sub>(''V'') = ''n''}}. If {{math|1=''n'' = 2''k''}} is even, {{math|Cℓ<sub>''n''</sub>(<math>\Complex</math>)}} is isomorphic as an algebra (in a non-unique way) to the algebra {{math|Mat(2<sup>''k''</sup>, <math>\Complex</math>)}} of {{math|2<sup>''k''</sup> × 2<sup>''k''</sup>}} complex matrices (by the [[Artin–Wedderburn theorem]] and the easy to prove fact that the Clifford algebra is [[central simple algebra|central simple]]). If {{math|1=''n'' = 2''k'' + 1}} is odd, {{math|Cℓ<sub>2''k''+1</sub>(<math>\Complex</math>)}} is isomorphic to the algebra {{math|Mat(2<sup>''k''</sup>, <math>\Complex</math>) ⊕ Mat(2<sup>''k''</sup>, <math>\Complex</math>)}} of two copies of the {{math|2<sup>''k''</sup> × 2<sup>''k''</sup>}} complex matrices. Therefore, in either case {{math|Cℓ(''V'', ''g'')}}  has a unique (up to isomorphism) irreducible representation (also called simple [[Clifford module]]), commonly denoted by Δ, of dimension 2<sup>[''n''/2]</sup>. Since the Lie algebra {{math|'''so'''(''V'', ''g'')}} is embedded as a Lie subalgebra in {{math|Cℓ(''V'', ''g'')}} equipped with the Clifford algebra [[commutator]] as Lie bracket, the space Δ is also a Lie algebra representation of {{math|'''so'''(''V'', ''g'')}} called a [[spin representation]]. If ''n'' is odd, this Lie algebra representation is irreducible. If ''n'' is even, it splits further{{clarification needed|date=July 2022}} into two irreducible representations {{math|1=Δ = Δ<sub>+</sub> ⊕ Δ<sub>−</sub>}} called the Weyl or ''half-spin representations''.

Irreducible representations over the reals in the case when ''V'' is a real vector space are much more intricate, and the reader is referred to the [[Clifford algebra#Spinors|Clifford algebra]] article for more details.

=== Spin groups ===
[[File:Spin representations do not lift.svg|thumb|The spin representation Δ is a vector space equipped with a representation of the spin group that does not factor through a representation of the (special) orthogonal group. The vertical arrows depict a [[short exact sequence]].]]
Spinors form a [[vector space]], usually over the [[complex numbers]], equipped with a linear [[group representation]] of the [[spin group]] that does not factor through a representation of the group of rotations (see diagram). The spin group is the [[special orthogonal group|group of rotations]] keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not [[simply connected]], but the simply connected spin group is its [[Double covering group|double cover]]. So for every rotation there are two elements of the spin group that represent it. [[Geometric vector]]s and other [[tensor]]s cannot feel the difference between these two elements, but they produce ''opposite'' signs when they affect any spinor under the representation. Thinking of the elements of the spin group as [[homotopy classes]] of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the [[belt trick]] puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a [[quadratic form]] such as [[Euclidean space]] with its standard [[dot product]], or [[Minkowski space]] with its [[Lorentz metric]]. In the latter case, the "rotations" include the [[Lorentz boost]]s, but otherwise the theory is substantially similar.{{Citation needed|date=October 2023}}

=== 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>

=== Spinors in representation theory ===
{{Main|Spin representation}}

One major mathematical application of the construction of spinors is to make possible the explicit construction of [[linear representation]]s of the [[Lie algebra]]s of the [[special orthogonal group]]s, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the [[Atiyah–Singer index theorem]], and to provide constructions in particular for [[discrete series]] representations of [[semisimple group]]s.

The spin representations of the special orthogonal Lie algebras are distinguished from the [[tensor]] representations given by [[Young symmetrizer|Weyl's construction]] by the [[weight (representation theory)|weights]]. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the [[spin representation]] article.

=== Attempts at intuitive understanding ===
The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space".<ref>Jean Hladik: ''Spinors in Physics'', translated by J. M. Cole, Springer 1999, {{isbn|978-0-387-98647-0}}, p. 3</ref> Stated differently:
{{blockquote|Spinors ... provide a linear representation of the group of [[Rotation (mathematics)|rotations]] in a space with any number <math>n</math> of dimensions, each spinor having <math>2^\nu</math> components where <math>n = 2\nu+1</math> or <math>2\nu</math>.<ref name="cartan-1966-quote" />}}

Several ways of illustrating everyday analogies have been formulated in terms of the [[plate trick]], [[tangloids]] and other examples of [[orientation entanglement]].

Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by [[Michael Atiyah]]'s statement that is recounted by Dirac's biographer Graham Farmelo:
{{blockquote|No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the "square root" of geometry and, just as understanding the [[square root of −1]] took centuries, the same might be true of spinors.<ref>{{cite book |first=Graham |last=Farmelo |title=The Strangest Man: The hidden life of Paul Dirac, quantum genius |publisher=Faber & Faber |year=2009 |isbn=978-0-571-22286-5 |page=430}}</ref>}}