Editing Spinor

{{short description|Non-tensorial representation of the spin group}}
[[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.{{efn|Spinors in three dimensions are points of a [[line bundle]] over a [[conic]] in the [[projective plane]]. In this picture, which is associated to spinors of a three-dimensional [[pseudo-Euclidean space]] of signature (1,2), the conic is an ordinary real conic (here the circle), the line bundle is the Möbius bundle, and the spin group is {{math|SL<sub>2</sub>(<math>\Reals</math>)}}. In Euclidean signature, the projective plane, conic and line bundle are over the complex instead, and this picture is just a real slice.}}]]

In [[geometry]] and [[physics]], '''spinors''' (pronounced "spinner" IPA {{IPAc-en|s|p|ɪ|n|ɚ}}) are elements of a [[complex numbers|complex]] [[vector space]] that can be associated with [[Euclidean space]].{{efn|Spinors can always be defined over the complex numbers. However, in some signatures there exist real spinors. Details can be found in [[spin representation]].}} A spinor transforms linearly when the Euclidean space is subjected to a slight ([[infinitesimal transformation|infinitesimal]]) rotation,{{efn|A formal definition of spinors at this level is that the space of spinors is a [[linear representation]] of the [[Lie algebra]] of [[infinitesimal rotation]]s of a [[spin representation|certain kind]].}} but unlike [[Euclidean vector|geometric vectors]] and [[tensor]]s, a spinor transforms to its negative when the 
space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of [[Section (fiber bundle)|sections]] of [[vector bundle]]s – in the case of the [[exterior algebra]] bundle of the [[cotangent bundle]], they thus become "square roots" of [[differential form]]s).

It is also possible to associate a substantially similar notion of spinor to [[Minkowski space]], in which case the [[Lorentz transformation]]s of [[special relativity]] play the role of rotations. Spinors were introduced in geometry by [[Élie Cartan]] in 1913.<ref>{{Harvnb|Cartan|1913}}.</ref>{{efn|"Spinors were first used under that name, by physicists, in the field of Quantum Mechanics. In their most general form, spinors were discovered in 1913 by the author of this work, in his investigations on the linear representations of simple groups*; they provide a linear representation of the group of 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">Quote from Elie Cartan: ''The Theory of Spinors'', Hermann, Paris, 1966, first sentence of the Introduction section at the beginning of the book, before page numbers start.</ref> The star (*) refers to Cartan (1913).}} In the 1920s physicists discovered that spinors are essential to describe the [[intrinsic angular momentum]], or "spin", of the [[electron]] and other subatomic particles.{{efn|More precisely, it is the [[fermion]]s of [[spin-1/2]] that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2-component spinors transforming under 3-dimensional infinitesimal rotations. The relativistic [[Dirac equation]] for the electron is an equation for 4-component spinors transforming under infinitesimal Lorentz transformations, for which a substantially similar theory of spinors exists.}}

Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the [[rotation group]]). There are two topologically distinguishable classes ([[homotopy class]]es) of paths through rotations that result in the same overall rotation, as illustrated by the [[belt trick]] puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The [[spin group]] is the group of all rotations keeping track of the class.{{efn|Formally, the spin group is the group of [[relative homotopy class]]es with fixed endpoints in the rotation group.}} It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) [[linear representation]] of the spin group, meaning that elements of the spin group [[Group action (mathematics)|act]] as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.{{efn|More formally, the space of spinors can be defined as an ([[irreducible representation|irreducible]]) representation of the spin group that does not factor through a representation of the rotation group (in general, the connected component of the identity of the [[orthogonal group]]).}} In mathematical terms, spinors are described by a double-valued [[projective representation]] of the rotation group [[SO(3)]].

Although spinors can be defined purely as elements of a representation space of the spin group (or its [[Lie algebra]] of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the [[Clifford algebra]]. The Clifford algebra is an [[associative algebra]] that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with.{{efn|[[Geometric algebra]] is a name for the Clifford algebra in an applied setting.}} A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.<ref>{{cite journal |author=Rukhsan-Ul-Haq |url=https://www.ias.ac.in/article/fulltext/reso/021/12/1105-1117 |title=Geometry of Spin: Clifford Algebraic Approach |journal=Resonance |date=December 2016 |volume=21 |issue=12 |pages=1105–1117|doi=10.1007/s12045-016-0422-5 |s2cid=126053475 |url-access=subscription }}</ref> After choosing an [[Orthonormality|orthonormal]] basis of Euclidean space, a representation of the Clifford algebra is generated by [[gamma matrices]], matrices that satisfy a set of canonical anti-commutation relations. The spinors are the [[column vectors]] on which these matrices act. In three Euclidean dimensions, for instance, the [[Pauli spin matrices]] are a set of gamma matrices,{{efn|The Pauli matrices correspond to [[angular momentum operator|angular momenta]] operators about the three coordinate axes. This makes them slightly atypical gamma matrices because in addition to their anticommutation relation they also satisfy commutation relations.}} and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex{{efn|The [[metric signature]] relevant as well if we are concerned with real spinors. See [[spin representation]].}}) column vectors will either be [[irreducible representation|irreducible]] if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.{{efn|Whether the representation decomposes depends on whether they are regarded as representations of the spin group (or its Lie algebra), in which case it decomposes in even but not odd dimensions, or the Clifford algebra when it is the other way around. Other structures than this decomposition can also exist; precise criteria are covered at [[spin representation]] and [[Clifford algebra]].}}

== Introduction ==
{{multiple image
 | image1=Belt trick 2.gif
 | image2=Belt trick 1.gif
 | total_width=404
 | footer=A gradual rotation can be visualized as a ribbon in space.{{efn|The [[TNB frame]] of the ribbon defines a rotation continuously for each value of the arc length parameter.}} Two gradual rotations with different classes, one through 360° and one through 720° are illustrated here in the [[belt trick]] puzzle. A solution of the puzzle is a continuous manipulation of the belt, fixing the endpoints, that untwists it. This is impossible with the 360° rotation, but possible with the 720° rotation. A solution, shown in the second animation, gives an explicit [[homotopy]] in the rotation group between the 720° rotation and the 0° identity rotation.
}}
[[File:Belt Trick.ogv|thumb|An object attached to belts or strings can spin continuously without becoming tangled. Notice that after the cube completes a 360° rotation, the spiral is reversed from its initial configuration. The belts return to their original configuration after spinning a full 720°.]]
[[File:Antitwister.ogv|thumb|A more extreme example demonstrating that this works with any number of strings. In the limit, a piece of solid continuous space can rotate in place like this without tearing or intersecting itself]]

What characterizes spinors and distinguishes them from [[geometric vector]]s and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo ''the same'' rotation as the coordinates. More broadly, any [[tensor]] associated with the system (for instance, the [[Cauchy stress tensor|stress]] of some medium) also has coordinate descriptions that adjust to compensate for changes to the coordinate system itself.

Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually ([[continuous function|continuously]]) rotated between some initial and final configuration. For any of the familiar and intuitive ("tensorial") quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration. Spinors, on the other hand, are constructed in such a way that makes them ''sensitive'' to how the gradual rotation of the coordinates arrived there: They exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are actually two ("[[topology|topologically]]") inequivalent ''gradual'' (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called the [[homotopy class]] of the gradual rotation. The [[belt trick]] (shown, in which both ends of the rotated object are physically tethered to an external reference) demonstrates two different rotations, one through an angle of 2{{math|π}} and the other through an angle of 4{{math|π}}, having the same final configurations but different classes. Spinors actually exhibit a sign-reversal that genuinely depends on this homotopy class. This distinguishes them from vectors and other tensors, none of which can feel the class.

Spinors can be exhibited as concrete objects using a choice of [[Cartesian coordinates]]. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of [[Pauli spin matrices]] corresponding to ([[angular momentum operator|angular momenta]] about) the three coordinate axes. These are 2×2 matrices with [[complex number|complex]] entries, and the two-component complex [[column vector]]s on which these matrices act by [[matrix multiplication]] are the spinors. In this case, the spin group is isomorphic to the group of 2×2 [[unitary matrix|unitary matrices]] with [[determinant]] one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves,{{efn|This is the set of 2×2 complex [[Trace (linear algebra)|traceless]] [[hermitian matrix|hermitian matrices]].}} realizing it as a group of rotations among them,{{efn|Except for a [[kernel (algebra)|kernel]] of <math>\{\pm 1\}</math> corresponding to the two different elements of the spin group that go to the same rotation.<ref>For details, see {{cite journal |author-link=William Frederick Eberlein|first=W. F. |last=Eberlein |title=The Spin Model of Euclidean 3-Space |journal=The American Mathematical Monthly |volume=69 |issue=7 |pages=587–598 |doi=10.2307/2310821  |year=1962|jstor=2310821 }}</ref>}} but it also acts on the column vectors (that is, the spinors).

More generally, a Clifford algebra can be constructed from any vector space ''V'' equipped with a (nondegenerate) [[quadratic form]], such as [[Euclidean space]] with its standard dot product or [[Minkowski space]] with its standard Lorentz metric. The [[spin group|space of spinors]] is the space of column vectors with <math>2^{\lfloor\dim V/2\rfloor}</math> components. The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group.{{efn|So the ambiguity in identifying the spinors themselves persists from the point of view of the group theory, and still depends on choices.}} Depending on the dimension and [[metric signature]], this realization of spinors as column vectors may be [[irreducible representation|irreducible]] or it may decompose into a pair of so-called "half-spin" or Weyl representations.{{efn|The Clifford algebra can be given an even/odd [[graded algebra|grading]] from the parity of the degree in the gammas, and the spin group and its Lie algebra both lie in the even part. Whether here by "representation" we mean representations of the spin group or the Clifford algebra will affect the determination of their reducibility. Other structures than this splitting can also exist; precise criteria are covered at [[spin representation]] and [[Clifford algebra]].}} When the vector space ''V'' is four-dimensional, the algebra is described by the [[gamma matrices]].

== 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>}}

== History ==
The most general mathematical form of spinors was discovered by [[Élie Cartan]] in 1913.<ref>{{Harvnb|Cartan|1913}}</ref> The word "spinor" was coined by [[Paul Ehrenfest]] in his work on [[quantum physics]].<ref>{{harvnb|Tomonaga|1998|p=129}}</ref>

Spinors were first applied to [[mathematical physics]] by [[Wolfgang Pauli]] in 1927, when he introduced his [[Pauli matrices|spin matrices]].<ref>{{Harvnb|Pauli|1927}}.</ref> The following year, [[Paul Dirac]] discovered the fully [[special relativity|relativistic]] theory of [[electron]] [[Spin (physics)|spin]] by showing the connection between spinors and the [[Lorentz group]].<ref>{{Harvnb|Dirac|1928}}.</ref> By the 1930s, Dirac, [[Piet Hein (Denmark)|Piet Hein]] and others at the [[Niels Bohr Institute]] (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as [[Tangloids]] to teach and model the calculus of spinors.

Spinor spaces were represented as [[left ideal]]s of a matrix algebra in 1930, by [[Gustave Juvett]]<ref>{{cite journal |last1=Juvet |first1=G. |year=1930 |title=Opérateurs de Dirac et équations de Maxwell |language=fr |journal=[[Commentarii Mathematici Helvetici]] |volume=2 |pages=225–235 |doi=10.1007/BF01214461|s2cid=121226923 }}</ref> and by [[Fritz Sauter]].<ref>{{cite journal |first=F. |last=Sauter |title=Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren |journal=Zeitschrift für Physik |volume=63 |issue=11–12 |pages=803–814 |doi=10.1007/BF01339277 |bibcode=1930ZPhy...63..803S |year=1930|s2cid=122940202 }}</ref><ref name="lounesto-1995-p151">Pertti Lounesto: ''[[Albert Crumeyrolle|Crumeyrolle]]'s bivectors and spinors'', pp.&nbsp;137–166, In: Rafał Abłamowicz, Pertti Lounesto (eds.): ''Clifford algebras and spinor structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992)'', {{isbn|0-7923-3366-7}}, 1995, [https://books.google.com/books?id=DnyUDg483kEC&pg=PA151 p. 151]</ref> More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a [[minimal ideal|minimal left ideal]] in {{math|Mat(2, <math>\Complex</math>)}}.{{efn|The matrices of dimension ''N'' × ''N'' in which only the elements of the left column are non-zero form a ''left ideal'' in the ''N'' × ''N'' matrix algebra {{math|Mat(''N'', <math>\Complex</math>)}} – multiplying such a matrix ''M'' from the left with any ''N'' × ''N'' matrix ''A'' gives the result ''AM'' that is again an ''N'' × ''N'' matrix in which only the elements of the left column are non-zero. Moreover, it can be shown that it is a ''minimal left ideal''.<ref>See also: Pertti Lounesto: ''Clifford algebras and spinors'', London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, {{ISBN|978-0-521-00551-7}}, p.&nbsp;52</ref>}}<ref name="lounesto-2001-p148f-p327f">Pertti Lounesto: ''Clifford algebras and spinors'', London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, {{ISBN|978-0-521-00551-7}}, p.&nbsp;148&nbsp;f. and [https://books.google.com/books?id=DTecU6UpkSgC&pg=PA327 p. 327 f.]</ref>

In 1947 [[Marcel Riesz]] constructed spinor spaces as elements of a minimal left ideal of [[Clifford algebra]]s. In 1966/1967, [[David Hestenes]]<ref>D. Hestenes: ''Space–Time Algebra'', Gordon and Breach, New York, 1966, 1987, 1992</ref><ref>{{cite journal |first=D. |last=Hestenes |title=Real spinor fields |journal=[[Journal of Mathematical Physics|J. Math. Phys.]] |volume=8 |year=1967 |issue=4 |pages=798–808 |doi=10.1063/1.1705279 |bibcode=1967JMP.....8..798H |s2cid=13371668 |doi-access=free |url=https://davidhestenes.net/geocalc/pdf/RealSpinorFields.pdf }}</ref> replaced spinor spaces by the [[even subalgebra]] Cℓ<sup>0</sup><sub>1,3</sub>(<math>\Reals</math>) of the [[spacetime algebra]] Cℓ<sub>1,3</sub>(<math>\Reals</math>).<ref name="lounesto-1995-p151" /><ref name="lounesto-2001-p148f-p327f" /> As of the 1980s, the theoretical physics group at [[Birkbeck College]] around [[David Bohm]] and [[Basil Hiley]] has been developing [[Basil Hiley#Implicate orders, pre-space and algebraic structures|algebraic approaches to quantum theory]] that build on Sauter and Riesz' identification of spinors with minimal left ideals.

== Examples ==
Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra {{math|Cℓ<sub>''p'', ''q''</sub>(<math>\Reals</math>)}}. This is an algebra built up from an orthonormal basis of {{math|1=''n'' = ''p'' + ''q''}} mutually orthogonal vectors under addition and multiplication, ''p'' of which have norm +1 and ''q'' of which have norm −1, with the product rule for the basis vectors
<math display="block">e_ie_j = \begin{cases}
+1 & i=j, \, i \in (1, \ldots, p) \\
-1 & i=j, \, i \in (p+1, \ldots, n)  \\
-e_j e_i & i \neq j.
\end{cases}</math>

=== Two dimensions ===
The Clifford algebra Cℓ<sub>2,0</sub>(<math>\Reals</math>) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, ''σ''<sub>1</sub> and ''σ''<sub>2</sub>, and one unit [[pseudoscalar]] {{math|1=''i'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>}}. From the definitions above, it is evident that {{math|1=(''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>2</sub>)<sup>2</sup> = 1}}, and {{math|1=(''σ''<sub>1</sub>''σ''<sub>2</sub>)(''σ''<sub>1</sub>''σ''<sub>2</sub>) = −''σ''<sub>1</sub>''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>2</sub> = −1}}.

The even subalgebra Cℓ<sup>0</sup><sub>2,0</sub>(<math>\Reals</math>), spanned by ''even-graded'' basis elements of Cℓ<sub>2,0</sub>(<math>\Reals</math>), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and ''σ''<sub>1</sub>''σ''<sub>2</sub>. As a real algebra, Cℓ<sup>0</sup><sub>2,0</sub>(<math>\Reals</math>) is isomorphic to the field of [[complex numbers]] {{math| <math>\Complex</math>}}. As a result, it admits a conjugation operation (analogous to [[complex conjugate|complex conjugation]]), sometimes called the ''reverse'' of a Clifford element, defined by
<math display="block">(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1</math>
which, by the Clifford relations, can be written
<math display="block">(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2.</math>

The action of an even Clifford element {{math|''γ'' ∈ Cℓ<sup>0</sup><sub>2,0</sub>(<math>\Reals</math>)}} on vectors, regarded as 1-graded elements of Cℓ<sub>2,0</sub>(<math>\Reals</math>), is determined by mapping a general vector {{math|1=''u'' = ''a''<sub>1</sub>''σ''<sub>1</sub> + ''a''<sub>2</sub>''σ''<sub>2</sub>}} to the vector
<math display="block">\gamma(u) = \gamma u \gamma^*,</math>
where <math>\gamma^*</math> is the conjugate of <math>\gamma</math>, and the product is Clifford multiplication. In this situation, a '''spinor'''{{efn|These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via {{math|1=''γ''(''ϕ'') = ''{{overline|γ}}ϕ''}}. The Majorana spinors are the common underlying real representation for the Weyl representations.}} is an ordinary complex number. The action of <math>\gamma</math> on a spinor <math>\phi</math> is given by ordinary complex multiplication:
<math display="block"> \gamma(\phi) = \gamma\phi.</math>

An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:
<math display="block"> \gamma(u) = \gamma u \gamma^* = \gamma^2 u.</math>

On the other hand, in comparison with its action on spinors <math> \gamma(\phi) = \gamma\phi</math>, the action of <math>\gamma</math> on ordinary vectors appears as the ''square'' of its action on spinors.

Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of ''θ'' corresponds to {{math|1=''γ''<sup>2</sup> = exp(''θ σ''<sub>1</sub>''σ''<sub>2</sub>)}}, so that the corresponding action on spinors is via {{math|1=''γ'' = ± exp(''θ σ''<sub>1</sub>''σ''<sub>2</sub>/2)}}. In general, because of [[branch cut|logarithmic branching]], it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.

In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by [[abuse of notation|abuse of language]], the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to [[computer graphics]]) they make sense.

====Examples====

* The even-graded element <math display="block">\gamma = \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2)</math> corresponds to a vector rotation of 90° from ''σ''<sub>1</sub> around towards ''σ''<sub>2</sub>, which can be checked by confirming that <math display="block">\tfrac{1}{2} (1 - \sigma_1 \sigma_2) \{a_1\sigma_1+a_2\sigma_2\}(1 - \sigma_2 \sigma_1) = a_1\sigma_2 - a_2\sigma_1</math> It corresponds to a spinor rotation of only 45°, however: <math display="block">\tfrac{1}{\sqrt{2}}(1-\sigma_1 \sigma_2)\{a_1+a_2\sigma_1\sigma_2\}=\frac{a_1+a_2}{\sqrt{2}} + \frac{-a_1+a_2}{\sqrt{2}}\sigma_1\sigma_2</math>
* Similarly the even-graded element {{math|1=''γ'' = −''σ''<sub>1</sub>''σ''<sub>2</sub>}} corresponds to a vector rotation of 180°: <math display="block">(- \sigma_1 \sigma_2)\{a_1\sigma_1 + a_2\sigma_2\} (- \sigma_2 \sigma_1) = - a_1\sigma_1 -a_2\sigma_2</math> but a spinor rotation of only 90°:<math display="block">(- \sigma_1 \sigma_2) \{a_1 + a_2\sigma_1\sigma_2\} = a_2 - a_1\sigma_1\sigma_2</math>
* Continuing on further, the even-graded element {{math|1=''γ'' = −1}} corresponds to a vector rotation of 360°: <math display="block"> (-1) \{a_1\sigma_1+a_2\sigma_2\} \, (-1) = a_1\sigma_1+a_2\sigma_2</math> but a spinor rotation of 180°.

=== Three dimensions ===
{{Main|Spinors in three dimensions|Quaternions and spatial rotation}}

The Clifford algebra Cℓ<sub>3,0</sub>(<math>\Reals</math>) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, [[Pauli matrices|''σ''<sub>1</sub>, ''σ''<sub>2</sub> and ''σ''<sub>3</sub>]], the three unit bivectors ''σ''<sub>1</sub>''σ''<sub>2</sub>, ''σ''<sub>2</sub>''σ''<sub>3</sub>, ''σ''<sub>3</sub>''σ''<sub>1</sub> and the [[pseudoscalar]] {{math|1=''i'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>3</sub>}}. It is straightforward to show that {{math|1=(''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>2</sub>)<sup>2</sup> = (''σ''<sub>3</sub>)<sup>2</sup> = 1}}, and {{math|1=(''σ''<sub>1</sub>''σ''<sub>2</sub>)<sup>2</sup> = (''σ''<sub>2</sub>''σ''<sub>3</sub>)<sup>2</sup> = (''σ''<sub>3</sub>''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>3</sub>)<sup>2</sup> = −1}}.

The sub-algebra of even-graded elements is made up of scalar dilations,
<math display="block">u' = \rho^{\left(\frac{1}{2}\right)} u \rho^{\left(\frac{1}{2}\right)} = \rho u,</math>
and vector rotations
<math display="block">u' = \gamma u\gamma^*,</math>
where
{{NumBlk||<math display="block">\left.\begin{align}
  \gamma &= \cos\left(\frac{\theta}{2}\right) - \{a_1\sigma_2\sigma_3 + a_2\sigma_3\sigma_1 + a_3\sigma_1\sigma_2\} \sin\left(\frac{\theta}{2}\right) \\
         &= \cos\left(\frac{\theta}{2}\right) - i\{a_1\sigma_1 + a_2\sigma_2 + a_3\sigma_3\} \sin\left(\frac{\theta}{2}\right) \\
         &= \cos\left(\frac{\theta}{2}\right) - iv\sin\left(\frac{\theta}{2}\right)
\end{align}\right\}</math>|{{EquationRef|1}}}}
corresponds to a vector rotation through an angle ''θ'' about an axis defined by a unit vector {{math|1=''v'' = ''a''<sub>1</sub>''σ''<sub>1</sub> + ''a''<sub>2</sub>''σ''<sub>2</sub> + ''a''<sub>3</sub>''σ''<sub>3</sub>}}.

As a special case, it is easy to see that, if {{math|1=''v'' = ''σ''<sub>3</sub>}}, this reproduces the ''σ''<sub>1</sub>''σ''<sub>2</sub> rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the ''σ''<sub>3</sub> direction invariant, since

<math display="block">
  \left[\cos\left(\frac{\theta}{2}\right) - i\sigma_3 \sin\left(\frac{\theta}{2}\right)\right]
    \sigma_3 \left[\cos\left(\frac{\theta}{2}\right) + i \sigma_3 \sin\left(\frac{\theta}{2}\right)\right] =
  \left[\cos^2\left(\frac{\theta}{2}\right) + \sin^2\left(\frac{\theta}{2}\right)\right] \sigma_3 =
  \sigma_3.
</math>

The bivectors ''σ''<sub>2</sub>''σ''<sub>3</sub>, ''σ''<sub>3</sub>''σ''<sub>1</sub> and ''σ''<sub>1</sub>''σ''<sub>2</sub> are in fact [[William Rowan Hamilton|Hamilton's]] [[quaternion]]s '''i''', '''j''', and '''k''', discovered in 1843:

<math display="block">\begin{align}
  \mathbf{i} &= -\sigma_2 \sigma_3 = -i \sigma_1 \\
  \mathbf{j} &= -\sigma_3 \sigma_1 = -i \sigma_2 \\
  \mathbf{k} &= -\sigma_1 \sigma_2 = -i \sigma_3
\end{align}</math>

With the identification of the even-graded elements with the algebra <math>\mathbb{H}</math> of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.{{efn|Since, for a [[skew field]], the kernel of the representation must be trivial. So inequivalent representations can only arise via an [[automorphism]] of the skew-field. In this case, there are a pair of equivalent representations: {{math|1=''γ''(''ϕ'') = ''γϕ''}}, and its quaternionic conjugate {{math|1=''γ''(''ϕ'') = ''ϕ{{overline|γ}}''}}.}} Thus the (real{{efn|The complex spinors are obtained as the representations of the [[tensor product]] {{math|1=<math>\mathbb{H} \otimes_\Reals \Complex</math> = Mat<sub>2</sub>(<math>\Complex</math>)}}. These are considered in more detail in [[spinors in three dimensions]].}}) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

Note that the expression (1) for a vector rotation through an angle {{mvar|θ}}, ''the angle appearing in γ was halved''. Thus the spinor rotation {{math|1=''γ''(''ψ'') = ''γψ''}} (ordinary quaternionic multiplication) will rotate the spinor {{mvar|ψ}} through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with {{math|(180° + ''θ''/2)}} in place of ''θ''/2 will produce the same vector rotation, but the negative of the spinor rotation.

The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.

== Explicit constructions ==

A space of spinors can be constructed explicitly with concrete and abstract constructions. The
equivalence of these constructions is a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see [[spinors in three dimensions]].

=== Component spinors ===

Given a vector space ''V'' and a quadratic form ''g'' an explicit matrix representation of the Clifford algebra {{math|Cℓ(''V'', ''g'')}} can be defined as follows. Choose an orthonormal basis {{math|''e''<sup>1</sup> ... ''e''<sup>''n''</sup>}} for ''V'' i.e. {{math|1=''g''(''e''<sup>''μ''</sup>''e''<sup>''ν''</sup>) = ''η''<sup>''μν''</sup>}} where {{math|1=''η''<sup>''μμ''</sup> = ±1}} and {{math|1=''η''<sup>''μν''</sup> = 0}} for {{math|''μ'' ≠ ''ν''}}. Let {{math|1=''k'' = ⌊''n''/2⌋}}. Fix a set of {{math|2<sup>''k''</sup> × 2<sup>''k''</sup>}} matrices {{math|''γ''<sup>1</sup> ... ''γ''<sup>''n''</sup>}} such that {{math|1=''γ''<sup>''μ''</sup>''γ''<sup>''ν''</sup> + ''γ''<sup>''ν''</sup>''γ''<sup>''μ''</sup> = 2''η''<sup>''μν''</sup>1}} (i.e. fix a convention for the [[gamma matrices]]). Then the assignment {{math|''e''<sup>''μ''</sup> → ''γ''<sup>''μ''</sup>}} extends uniquely to an algebra homomorphism {{math|Cℓ(''V'', ''g'') → Mat(2<sup>''k''</sup>, <math>\Complex</math>)}} by sending the monomial {{math|''e''<sup>''μ''<sub>1</sub></sup> ⋅⋅⋅ ''e''<sup>''μ''<sub>''k''</sub></sup>}} in the Clifford algebra to the product {{math|''γ''<sup>''μ''<sub>1</sub></sup> ⋅⋅⋅ ''γ''<sup>''μ''<sub>''k''</sub></sup>}} of matrices and extending linearly. The space <math>\Delta = \Complex^{2^k}</math> on which the gamma matrices act is now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the [[Pauli matrices|Pauli sigma matrices]] gives rise to the familiar two component spinors used in non relativistic [[quantum mechanics]]. Likewise using the {{math|4 × 4}} Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic [[quantum field theory]]. In general, in order to define gamma matrices of the required kind, one can use the [[Weyl–Brauer matrices]].

In this construction the representation of the Clifford algebra {{math|Cℓ(''V'', ''g'')}}, the Lie algebra {{math|'''so'''(''V'', ''g'')}}, and the Spin group {{math|Spin(''V'', ''g'')}}, all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2<sup>''k''</sup> complex numbers and is denoted with spinor indices (usually ''α'', ''β'', ''γ''). In the physics literature, such [[abstract indices|indices]] are often used to denote spinors even when an abstract spinor construction is used.

=== Abstract spinors ===
There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of {{math|Cℓ(''V'', ''g'')}} on itself. These are subspaces of the Clifford algebra of the form {{math|Cℓ(''V'', ''g'')''ω''}}, admitting the evident action of {{math|Cℓ(''V'', ''g'')}} by left-multiplication: {{math|''c'' : ''xω'' → ''cxω''}}. There are two variations on this theme: one can either find a primitive element {{math|''ω''}} that is a [[nilpotent]] element of the Clifford algebra, or one that is an [[idempotent]]. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.<ref>This construction is due to Cartan (1913). The treatment here is based on {{Harvtxt|Chevalley|1996}}.</ref> In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of {{math|''V''}}, and then specify the action of the Clifford algebra ''externally'' to that vector space.

In either approach, the fundamental notion is that of an [[isotropic subspace]] {{math|''W''}}. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of {{math|''V''}} is given.

As above, we let {{math|(''V'', ''g'')}} be an {{math|''n''}}-dimensional complex vector space equipped with a nondegenerate bilinear form. If {{math|''V''}} is a real vector space, then we replace {{math|''V''}} by its [[complexification]] <math>V \otimes_\Reals \Complex</math> and let {{math|''g''}} denote the induced bilinear form on <math>V \otimes_\Reals \Complex</math>. Let {{math|''W''}} be a maximal isotropic subspace, i.e. a maximal subspace of {{math|''V''}} such that {{math|1=''g''{{!}}<sub>''W''</sub> = 0}}. If {{math|1=''n'' =  2''k''}} is even, then let {{math|''W''{{′}}}} be an isotropic subspace complementary to {{math|''W''}}. If {{math|1=''n'' =  2''k'' + 1}} is odd, let {{math|''W''{{′}}}} be a maximal isotropic subspace with {{math|1=''W'' ∩ ''W''{{′}} = 0}}, and let {{math|''U''}} be the orthogonal complement of {{math|''W'' ⊕ ''W''{{′}}}}. In both the even- and odd-dimensional cases {{math|''W''}} and {{math|''W''{{′}}}} have dimension {{math|''k''}}. In the odd-dimensional case, {{math|''U''}} is one-dimensional, spanned by a unit vector {{math|''u''}}.

=== Minimal ideals ===
Since ''W''{{′}} is isotropic, multiplication of elements of ''W''{{′}} inside {{math|Cℓ(''V'', ''g'')}} is [[alternative algebra|skew]]. Hence vectors in ''W''{{′}} anti-commute, and {{math|1=Cℓ(''W''{{′}}, ''g''{{!}}<sub>''W''{{′}}</sub>) = Cℓ(''W''{{′}}, 0)}} is just the [[exterior algebra]] Λ<sup>∗</sup>''W''{{′}}. Consequently, the ''k''-fold product of ''W''{{′}} with itself, ''W''{{′}}<sup>''k''</sup>, is one-dimensional. Let ''ω'' be a generator of ''W''{{′}}<sup>''k''</sup>. In terms of a basis {{math|''w''{{′}}<sub>1</sub>, ..., ''w''{{′}}<sub>k</sub>}} of in ''W''{{′}}, one possibility is to set
<math display="block">\omega = w'_1 w'_2 \cdots w'_k.</math>

Note that {{math|1=''ω''<sup>2</sup> = 0}} (i.e., ''ω'' is nilpotent of order 2), and moreover, {{math|1=''w''{{′}}''ω'' = 0}} for all {{math|''w''{{′}} ∈ ''W''{{′}}}}. The following facts can be proven easily:
# If {{math|1=''n'' = 2''k''}}, then the left ideal {{math|1=Δ = Cℓ(''V'', ''g'')''ω''}} is a minimal left ideal. Furthermore, this splits into the two spin spaces {{math|1=Δ<sub>+</sub> = Cℓ<sup>even</sup>''ω''}} and {{math|1=Δ<sub>−</sub> = Cℓ<sup>odd</sup>''ω''}} on restriction to the action of the even Clifford algebra.
# If {{math|1=''n'' = 2''k'' + 1}}, then the action of the unit vector ''u'' on the left ideal {{math|Cℓ(''V'', ''g'')''ω''}} decomposes the space into a pair of isomorphic irreducible eigenspaces (both denoted by Δ), corresponding to the respective eigenvalues +1 and −1.

In detail, suppose for instance that ''n'' is even. Suppose that ''I'' is a non-zero left ideal contained in {{math|Cℓ(''V'', ''g'')''ω''}}. We shall show that ''I'' must be equal to {{math|Cℓ(''V'', ''g'')''ω''}} by proving that it contains a nonzero scalar multiple of ''ω''.

Fix a basis ''w''<sub>''i''</sub> of ''W'' and a complementary basis ''w''<sub>''i''</sub>′ of ''W''{{′}} so that
{{block indent | em = 1.5 | text = ''w''<sub>''i''</sub>''w''<sub>''j''</sub>′ +''w''<sub>''j''</sub>′''w''<sub>''i''</sub> = ''δ''<sub>''ij''</sub>, and}}
{{block indent | em = 1.5 | text = (''w''<sub>''i''</sub>)<sup>2</sup> = 0, (''w''<sub>''i''</sub>′)<sup>2</sup> = 0.}}

Note that any element of ''I'' must have the form ''αω'', by virtue of our assumption that {{math|''I'' ⊂ Cℓ(''V'', ''g'') ''ω''}}. Let {{math|''αω'' ∈ ''I''}} be any such element. Using the chosen basis, we may write
<math display="block">\alpha = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} + \sum_j B_j w'_j</math>
where the ''a''<sub>''i''<sub>1</sub>...''i''<sub>''p''</sub></sub> are scalars, and the ''B''<sub>''j''</sub> are auxiliary elements of the Clifford algebra. Observe now that the product
<math display="block">\alpha\omega = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p}\omega.</math>
Pick any nonzero monomial ''a'' in the expansion of ''α'' with maximal homogeneous degree in the elements ''w''<sub>i</sub>:
<math display="block">a = a_{i_1\dots i_\text{max}}w_{i_1}\dots w_{i_\text{max}}</math> (no summation implied),
then
<math display="block">w'_{i_\text{max}}\cdots w'_{i_1}\alpha\omega = a_{i_1\dots i_\text{max}}\omega</math>
is a nonzero scalar multiple of ''ω'', as required.

Note that for ''n'' even, this computation also shows that
<math display="block">\Delta = \mathrm{C}\ell(W)\omega = \left(\Lambda^* W\right)\omega</math>
as a vector space. In the last equality we again used that ''W'' is isotropic. In physics terms, this shows that Δ is built up like a [[Fock space]] by [[creation and annihilation|creating]] spinors using anti-commuting creation operators in ''W'' acting on a vacuum ''ω''.

=== Exterior algebra construction ===
The computations with the minimal ideal construction suggest that a spinor representation can
also be defined directly using the [[exterior algebra]] {{math|1=Λ<sup>∗</sup> ''W'' = ⊕<sub>''j''</sub> Λ<sup>''j''</sup> ''W''}} of the isotropic subspace ''W''.
Let {{math|1=Δ = Λ<sup>∗</sup> ''W''}} denote the exterior algebra of ''W'' considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.<ref>One source for this subsection is {{Harvtxt|Fulton|Harris|1991}}.</ref><ref>Jurgen Jost, "Riemannian Geometry and Geometric Analysis" (2002) Springer-Verlag Universitext {{ISBN|3-540-42627-2}}. ''See chapter 1.''</ref>

The action of the Clifford algebra on Δ is defined first by giving the action of an element of ''V'' on Δ, and then showing that this action respects the Clifford relation and so extends to a [[homomorphism]] of the full Clifford algebra into the [[endomorphism ring]] End(Δ) by the [[Clifford algebra#Universal property and construction|universal property of Clifford algebras]]. The details differ slightly according to whether the dimension of ''V'' is even or odd.

When dim({{mvar|V}}) is even, {{math|1=''V'' = ''W'' ⊕ ''W''{{′}}}} where ''W''{{′}} is the chosen isotropic complement. Hence any {{math|''v'' ∈ ''V''}} decomposes uniquely as {{math|1=''v'' = ''w'' + ''w''{{′}}}} with {{math|''w'' ∈ ''W''}} and {{math|''w''{{′}} ∈ ''W''{{′}}}}. The action of {{mvar|v}} on a spinor is given by
<math display="block">c(v) w_1 \wedge\cdots\wedge w_n = \left(\epsilon(w) + i\left(w'\right)\right)\left(w_1 \wedge\cdots\wedge w_n\right)</math>
where ''i''(''w''{{′}}) is [[interior product]] with ''w''{{′}} using the nondegenerate quadratic form to identify ''V'' with ''V''<sup>∗</sup>, and ''ε''(''w'') denotes the [[exterior product]]. This action is sometimes called the '''Clifford product'''. It may be verified that
<math display="block">c(u)\,c(v) + c(v)\,c(u) = 2\,g(u,v)\,,</math>
and so {{mvar|c}} respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).

The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group<ref>Via the even-graded Clifford algebra.</ref> (the half-spin representations, or Weyl spinors) via
<math display="block">\Delta_+ = \Lambda^\text{even} W,\, \Delta_- = \Lambda^\text{odd} W.</math>

When dim(''V'') is odd, {{math|1=''V'' = ''W'' ⊕ ''U'' ⊕ ''W''{{′}}}}, where ''U'' is spanned by a unit vector ''u'' orthogonal to ''W''. The Clifford action ''c'' is defined as before on {{math|''W'' ⊕ ''W''′}}, while the Clifford action of (multiples of) ''u'' is defined by
<math display="block">c(u)\alpha = \begin{cases}
   \alpha & \hbox{if } \alpha \in \Lambda^\text{even} W \\
  -\alpha & \hbox{if } \alpha \in \Lambda^\text{odd} W
\end{cases}</math>
As before, one verifies that ''c'' respects the Clifford relations, and so induces a homomorphism.

=== Hermitian vector spaces and spinors ===
If the vector space ''V'' has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.

The main example is the case that the real vector space ''V'' is a [[hermitian form|hermitian vector space]] {{math|(''V'', ''g'')}}, i.e., ''V'' is equipped with a [[Linear complex structure|complex structure]] ''J'' that is an [[orthogonal transformation]] with respect to the inner product ''g'' on ''V''. Then <math>V \otimes_\Reals \Complex</math> splits in the {{math|±''i''}} eigenspaces of ''J''. These eigenspaces are isotropic for the complexification of ''g'' and can be identified with the complex vector space {{math|(''V'', ''J'')}} and its complex conjugate {{math|(''V'', −''J'')}}. Therefore, for a hermitian vector space {{math|(''V'', ''g'')}} the vector space <math>\Lambda^\cdot_\Complex \bar V</math> (as well as its complex conjugate <math>\Lambda^\cdot_\Complex V</math>) is a spinor space for the underlying real euclidean vector space.

With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an [[almost Hermitian manifold]] and is the reason why every [[almost complex manifold]] (in particular every [[symplectic manifold]]) has a [[Spin-c structure|Spin<sup>c</sup> structure]]. Likewise, every complex vector bundle on a manifold carries a Spin<sup>c</sup> structure.<ref>{{Harvnb|Lawson|Michelsohn|1989|loc=Appendix D}}.</ref>

== Clebsch–Gordan decomposition ==
A number of [[Clebsch–Gordan coefficients|Clebsch–Gordan decompositions]] are possible on the [[tensor product]] of one spin representation with another.<ref>{{Harvnb|Brauer|Weyl|1935}}.</ref> These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.

For the real or complex case, the alternating representations are
* {{math|1=Γ<sub>''r''</sub> = Λ<sup>''r''</sup>''V''}}, the representation of the orthogonal group on skew tensors of rank ''r''.

In addition, for the real orthogonal groups, there are three [[character theory|characters]] (one-dimensional representations)
* ''σ''<sub>+</sub> : O(''p'', ''q'') → {−1, +1} given by {{math|1=''σ''<sub>+</sub>(R) = −1}}, if ''R'' reverses the spatial orientation of ''V'', +1, if ''R'' preserves the spatial orientation of ''V''. (''The spatial character''.)
* ''σ''<sub>−</sub> : O(''p'', ''q'') → {−1, +1} given by {{math|1=''σ''<sub>−</sub>(R) = −1}}, if ''R'' reverses the temporal orientation of ''V'', +1, if ''R'' preserves the temporal orientation of ''V''. (''The temporal character''.)
* ''σ'' = ''σ''<sub>+</sub>''σ''<sub>−</sub> . (''The orientation character''.)

The Clebsch–Gordan decomposition allows one to define, among other things:
* An action of spinors on vectors.
* A [[Hermitian metric]] on the complex representations of the real spin groups.
* A [[Dirac operator]] on each spin representation.

=== Even dimensions ===
If {{math|1=''n'' = 2''k''}} is even, then the tensor product of Δ with the [[contragredient representation]] decomposes as
<math display="block">\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^n \Gamma_p \cong \bigoplus_{p=0}^{k-1} \left(\Gamma_p\oplus\sigma\Gamma_p\right) \oplus \Gamma_k</math>
which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements {{math|''αω'' ⊗ ''βω''′}}. The rightmost formulation follows from the transformation properties of the [[Hodge star operator]]. Note that on restriction to the even Clifford algebra, the paired summands {{math|Γ<sub>''p''</sub> ⊕ ''σ''Γ<sub>''p''</sub>}} are isomorphic, but under the full Clifford algebra they are not.

There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:
<math display="block" display="block">(\alpha\omega)^* = \omega\left(\alpha^*\right).</math>
So {{math|Δ ⊗ Δ}} also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose
<math display="block">\begin{align}
  \Delta_+\otimes\Delta^*_+ \cong \Delta_-\otimes\Delta^*_- &\cong \bigoplus_{p=0}^k \Gamma_{2p}\\
  \Delta_+\otimes\Delta^*_- \cong \Delta_-\otimes\Delta^*_+ &\cong \bigoplus_{p=0}^{k-1} \Gamma_{2p+1}
\end{align}</math>

For the complex representations of the real Clifford algebras, the associated [[reality structure]] on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate {{overline|Δ}} of the representation Δ, and the following isomorphism is seen to hold:
<math display="block">\bar{\Delta} \cong \sigma_-\Delta^*</math>

In particular, note that the representation Δ of the orthochronous spin group is a [[unitary representation]]. In general, there are Clebsch–Gordan decompositions
<math display="block">\Delta \otimes\bar{\Delta} \cong \bigoplus_{p=0}^k\left(\sigma_-\Gamma_p \oplus \sigma_+\Gamma_p\right).</math>

In metric signature {{math|(''p'', ''q'')}}, the following isomorphisms hold for the conjugate half-spin representations
* If ''q'' is even, then <math>\bar{\Delta}_+ \cong \sigma_- \otimes \Delta_+^*</math> and <math>\bar{\Delta}_- \cong \sigma_- \otimes \Delta_-^*.</math>
* If ''q'' is odd, then <math>\bar{\Delta}_+ \cong \sigma_- \otimes \Delta_-^*</math> and <math>\bar{\Delta}_- \cong \sigma_- \otimes \Delta_+^*.</math>
Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations {{math|Δ<sub>±</sub> ⊗ {{overline|Δ}}<sub>±</sub>}}.

=== Odd dimensions ===
If {{math|1=''n'' = 2''k'' + 1}} is odd, then
<math display="block">\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^k \Gamma_{2p}.</math>
In the real case, once again the isomorphism holds
<math display="block">\bar{\Delta} \cong \sigma_-\Delta^*.</math>
Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by
<math display="block">\Delta \otimes \bar{\Delta} \cong \sigma_-\Gamma_0\oplus\sigma_+\Gamma_1\oplus\dots\oplus\sigma_\pm\Gamma_k</math>

=== Consequences ===
There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are
* A manner of regarding the product of two spinors ''{{overline|ϕ}}ψ'' as a scalar. In physical terms, a spinor should determine a [[probability amplitude]] for the [[quantum state]].
* A manner of regarding the product ''ψ{{overline|ϕ}}'' as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space.
* A manner of regarding a spinor as acting upon a vector, by an expression such as ''ψv{{overline|ψ}}''. In physical terms, this represents an [[electric current]] of Maxwell's [[electromagnetic theory]], or more generally a [[probability current]].

== Summary in low dimensions ==
* In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a [[real representation|real]] 1-dimensional representation that does not transform.
* In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component [[complex representation]]s, i.e. complex numbers that get multiplied by ''e''<sup>±''iφ''/2</sup> under a rotation by angle ''φ''.
* In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and [[quaternionic representation|quaternionic]]. The existence of spinors in 3 dimensions follows from the isomorphism of the [[group (mathematics)|groups]] {{math|SU(2) ≅ Spin(3)}} that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as [[Pauli matrices]].
* In 4 Euclidean dimensions, the corresponding isomorphism is {{math|Spin(4) ≅ SU(2) × SU(2)}}. There are two inequivalent [[quaternionic representation|quaternionic]] 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
* In 5 Euclidean dimensions, the relevant isomorphism is {{math|Spin(5) ≅ USp(4) ≅ Sp(2)}} that implies that the single spinor representation is 4-dimensional and quaternionic.
* In 6 Euclidean dimensions, the isomorphism {{math|Spin(6) ≅ SU(4)}} guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
* In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.
* In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of [[Spin(8)]] called [[triality]].
* In {{math|''d'' + 8}} dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in ''d'' dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See [[Bott periodicity]].
* In spacetimes with ''p'' spatial and ''q'' time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the {{math|(''p'' + ''q'')}}-dimensional Euclidean space, but the reality projections mimic the structure in {{math|{{abs|''p'' − ''q''}}}} Euclidean dimensions. For example, in {{math|3 + 1}} dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism {{math|SL(2, <math>\Complex</math>) ≅ Spin(3,1)}}.

{| class="wikitable" style="margin:1em auto; text-align:center;"
! rowspan=2 | [[Metric signature]]
! colspan=2 | Weyl, complex
! rowspan=2 | Conjugacy
! rowspan=2 | Dirac, <br />complex
! colspan=2 | Majorana–Weyl, real
! rowspan=2 | Majorana, <br />real
|-
! Left-handed
! Right-handed
! Left-handed
! Right-handed
|-
|(2,0)||1||1||Mutual||2||–||–||2
|-
|(1,1)||1||1||Self||2||1||1||2
|-
|(3,0)||–||–||–||2||–||–||–
|-
|(2,1)||–||–||–||2||–||–||2
|-
|(4,0)||2||2||Self||4||–||–||–
|-
|(3,1)||2||2||Mutual||4||–||–||4
|-
|(5,0)||–||–||–||4||–||–||–
|-
|(4,1)||–||–||–||4||–||–||–
|-
|(6,0)||4||4||Mutual||8||–||–||8
|-
|(5,1)||4||4||Self||8||–||–||–
|-
|(7,0)||–||–||–||8||–||–||8
|-
|(6,1)||–||–||–||8||–||–||–
|-
|(8,0)||8||8||Self||16||8||8||16
|-
|(7,1)||8||8||Mutual||16||–||–||16
|-
|(9,0)||–||–||–||16||–||–||16
|-
|(8,1)||–||–||–||16||–||–||16
|}

== See also ==
{{cols|colwidth=27em}}
* [[Anyon]]
* [[Dirac equation in the algebra of physical space]]
* [[Eigenspinor]]
* [[Einstein–Cartan theory]]
* [[Projective representation]]
* [[Pure spinor]]
* [[Spin-1/2]]
* [[Spinor bundle]]
* [[Supercharge]]
* [[Twistor theory]]
* [[Spacetime algebra]]
{{colend}}

== Notes ==
{{notelist|25em}}

== References ==
{{reflist|25em}}

== Works cited ==
* {{cite journal |last1=Brauer |first1=Richard |author1-link=Richard Brauer |last2=Weyl |first2=Hermann |author2-link=Hermann Weyl |year=1935 |title=Spinors in ''n'' dimensions |journal=American Journal of Mathematics |volume=57 |issue=2 |pages=425–449 |doi=10.2307/2371218 |publisher=The Johns Hopkins University Press |jstor=2371218}}
* {{cite journal |last=Cartan |first=Élie |author-link=Élie Cartan |year=1913 |title=Les groupes projectifs qui ne laissent invariante aucune multiplicité plane |journal=Bull. Soc. Math. Fr. |volume=41 |pages=53–96 |doi=10.24033/bsmf.916 |url=http://archive.numdam.org/article/BSMF_1913__41__53_1.pdf|doi-access=free }}
* {{cite book |last=Chevalley |first=Claude |author-link=Claude Chevalley |year=1996 |orig-year=1954 |title=The Algebraic Theory of Spinors and Clifford Algebras |publisher=Columbia University Press (1954); Springer (1996) |edition=reprint |isbn=978-3-540-57063-9}}
* {{cite journal |last=Dirac |first=Paul M. |author-link=Paul Dirac |year=1928 |title=The quantum theory of the electron |journal=Proceedings of the Royal Society of London A |volume=117 |issue=778 |pages=610–624 |jstor=94981 |doi=10.1098/rspa.1928.0023 |bibcode=1928RSPSA.117..610D |doi-access=free}}
* {{cite book |last1=Fulton |first1=William |author1-link=William Fulton (mathematician) |last2=Harris |first2=Joe |author2-link=Joe Harris (mathematician) |year=1991 |title=Representation Theory: A first course |publisher=[[Springer-Verlag]] |location=New York |series=[[Graduate Texts in Mathematics]], Readings in Mathematics |volume=129 |isbn=0-387-97495-4 |mr=1153249 |doi=10.1007/978-1-4612-0979-9}}
* {{cite book |last1=Lawson |first1= H. Blaine |author1-link=H. Blaine Lawson |last2=Michelsohn |first2=Marie-Louise |author2-link=Marie-Louise Michelsohn |title=Spin Geometry |publisher=Princeton University Press |year=1989 |isbn=0-691-08542-0}}
* {{cite journal |last=Pauli |first=Wolfgang |author-link=Wolfgang Pauli |year=1927 |title=Zur Quantenmechanik des magnetischen Elektrons |journal=Zeitschrift für Physik |volume=43 |issue=9–10 |pages=601–632 |doi=10.1007/BF01397326 |bibcode=1927ZPhy...43..601P|s2cid=128228729 }}
* {{cite book |first=Sin-Itiro |last=Tomonaga |title=The Story of Spin |chapter=Lecture 7: The quantity which is neither vector nor tensor |page=129 |isbn=0-226-80794-0 |publisher=University of Chicago Press |year=1998}}

==Further reading==
* {{cite book |last=Cartan |first=Élie |author-link=Élie Cartan |year=1981 |orig-year=1966 |title=The Theory of Spinors |place=Paris, FR |publisher=Hermann (1966); Dover Publications (1981) |edition=reprint |isbn=978-0-486-64070-9}}
* {{cite book |last=Gilkey |first=Peter B. |author-link=Peter B. Gilkey |year=1984 |title=Invariance Theory: The heat equation, and the Atiyah–Singer index theorem |publisher=Publish or Perish |isbn=0-914098-20-9 |url=http://www.emis.de/monographs/gilkey/index.html}}
* {{cite book |last=Harvey |first=F. Reese |year=1990 |title=Spinors and Calibrations |publisher=Academic Press |isbn=978-0-12-329650-4}}
* {{cite journal |last=Hitchin |first=Nigel J. |author-link=Nigel Hitchin |year=1974 |title=Harmonic spinors |journal=[[Advances in Mathematics]] |volume=14 |pages=1–55 |mr=358873 |doi=10.1016/0001-8708(74)90021-8 |doi-access=free}}
* {{cite book |last1=Penrose|first1= Roger|author1-link=Roger Penrose |last2=Rindler |first2=W. |series=Spinors and Space-Time |volume=2 |title=Spinor and twistor methods in space-time geometry |publisher=Cambridge University Press |year=1988 |isbn=0-521-34786-6}}

{{tensors}}

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[[Category:Rotation in three dimensions]]
[[Category:Quantum mechanics]]
[[Category:Quantum field theory]]