In geometry and physics,spinors (pronounced "spinner" IPA/spɪnər/) are elements of acomplexvector space that can be associated withEuclidean space.^[b] A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation,^[c] but unlikegeometric vectors andtensors, 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" ofsections ofvector bundles – in the case of theexterior algebra bundle of thecotangent bundle, they thus become "square roots" ofdifferential forms).

It is also possible to associate a substantially similar notion of spinor toMinkowski space, in which case theLorentz transformations ofspecial relativity play the role of rotations. Spinors were introduced in geometry byÉlie Cartan in 1913.^[1][d] In the 1920s physicists discovered that spinors are essential to describe theintrinsic angular momentum, or "spin", of theelectron and other subatomic particles.^[e]

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 therotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as illustrated by thebelt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. Thespin group is the group of all rotations keeping track of the class.^[f] 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 groupact as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.^[g] In mathematical terms, spinors are described by a double-valuedprojective representation of the rotation groupSO(3).

Although spinors can be defined purely as elements of a representation space of the spin group (or itsLie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of theClifford algebra. The Clifford algebra is anassociative 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.^[h] A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.^[3] After choosing anorthonormal basis of Euclidean space, a representation of the Clifford algebra is generated bygamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are thecolumn vectors on which these matrices act. In three Euclidean dimensions, for instance, thePauli spin matrices are a set of gamma matrices,^[i] 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^[j]) column vectors will either beirreducible 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.^[k]

A gradual rotation can be visualized as a ribbon in space.^[l] Two gradual rotations with different classes, one through 360° and one through 720° are illustrated here in thebelt 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 explicithomotopy in the rotation group between the 720° rotation and the 0° identity rotation.

What characterizes spinors and distinguishes them fromgeometric vectors 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 undergothe same rotation as the coordinates. More broadly, anytensor associated with the system (for instance, thestress 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 (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 themsensitive 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 ("topologically") inequivalentgradual (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called thehomotopy class of the gradual rotation. Thebelt 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 2π and the other through an angle of 4π, 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 ofCartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice ofPauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices withcomplex entries, and the two-component complexcolumn vectors on which these matrices act bymatrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2unitary matrices withdeterminant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves,^[m] realizing it as a group of rotations among them,^[n] but it also acts on the column vectors (that is, the spinors).

More generally, a Clifford algebra can be constructed from any vector spaceV equipped with a (nondegenerate)quadratic form, such asEuclidean space with its standard dot product orMinkowski space with its standard Lorentz metric. Thespace of spinors is the space of column vectors with${\displaystyle 2^{\lfloor \dim V/2\rfloor }}$ 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.^[o] Depending on the dimension andmetric signature, this realization of spinors as column vectors may beirreducible or it may decompose into a pair of so-called "half-spin" or Weyl representations.^[p] When the vector spaceV is four-dimensional, the algebra is described by thegamma matrices.

The space of spinors is formally defined as thefundamental representation of theClifford algebra. (This may or may not decompose into irreducible representations.) The space of spinors may also be defined as aspin representation of theorthogonal 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-dimensionalgroup representation of thespin group on which thecenter acts non-trivially.

There are essentially two frameworks for viewing the notion of a spinor: therepresentation theoretic point of view and thegeometric point of view.

From arepresentation theoretic point of view, one knows beforehand that there are some representations of theLie algebra of theorthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled thespin representations, and their constituentsspinors. From this view, a spinor must belong to arepresentation of thedouble cover of therotation groupSO(n,${\displaystyle \mathbb{R}}$), or more generally of a double cover of thegeneralized special orthogonal groupSO⁺(p, q,${\displaystyle \mathbb{R}}$) on spaces with ametric signature of(p, q). These double covers areLie groups, called thespin groupsSpin(n) orSpin(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-valuedprojective representations 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${\displaystyle (V,\text{Spin}(p,q),\rho )}$ where${\displaystyle V}$ is a vector space over${\displaystyle K=\mathbb{R}}$ or${\displaystyle \mathbb{C}}$ and${\displaystyle \rho }$ is a homomorphism${\displaystyle \rho :\text{Spin}(p,q)\to \text{GL}(V)}$, aspinor is an element of the vector space${\displaystyle V}$.

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 asFierz identities, are needed.

The language ofClifford algebras^[5] (sometimes calledgeometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via theclassification of Clifford algebras. It largely removes the need forad hoc constructions.

In detail, letV be a finite-dimensional complex vector space with nondegenerate symmetric bilinear formg. The Clifford algebraCℓ(V, g) is the algebra generated byV along with the anticommutation relationxy +yx = 2g(x, y). It is an abstract version of the algebra generated by thegamma orPauli matrices. IfV =${\displaystyle {\mathbb{C}}^n}$, with the standard formg(x, y) =x^Ty =x₁y₁ + ... +x_ny_n we denote the Clifford algebra by Cℓ_n(${\displaystyle \mathbb{C}}$). 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 ifdim_{${\displaystyle \mathbb{C}}$}(V) =n. Ifn = 2k is even,Cℓ_n(${\displaystyle \mathbb{C}}$) is isomorphic as an algebra (in a non-unique way) to the algebraMat(2^k, ${\displaystyle \mathbb{C}}$) of2^k × 2^k complex matrices (by theArtin–Wedderburn theorem and the easy to prove fact that the Clifford algebra iscentral simple). Ifn = 2k + 1 is odd,Cℓ_2k+1(${\displaystyle \mathbb{C}}$) is isomorphic to the algebraMat(2^k, ${\displaystyle \mathbb{C}}$) ⊕ Mat(2^k, ${\displaystyle \mathbb{C}}$) of two copies of the2^k × 2^k complex matrices. Therefore, in either caseCℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simpleClifford module), commonly denoted by Δ, of dimension 2^[n/2]. Since the Lie algebraso(V, g) is embedded as a Lie subalgebra inCℓ(V, g) equipped with the Clifford algebracommutator as Lie bracket, the space Δ is also a Lie algebra representation ofso(V, g) called aspin representation. Ifn is odd, this Lie algebra representation is irreducible. Ifn is even, it splits further^{[clarification needed]} into two irreducible representationsΔ = Δ₊ ⊕ Δ₋ called the Weyl orhalf-spin representations.

Irreducible representations over the reals in the case whenV is a real vector space are much more intricate, and the reader is referred to theClifford algebra article for more details.

Spinors form avector space, usually over thecomplex numbers, equipped with a lineargroup representation of thespin group that does not factor through a representation of the group of rotations (see diagram). The spin group is thegroup 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 notsimply connected, but the simply connected spin group is itsdouble cover. So for every rotation there are two elements of the spin group that represent it.Geometric vectors and othertensors cannot feel the difference between these two elements, but they produceopposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group ashomotopy 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 thebelt 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 aquadratic form such asEuclidean space with its standarddot product, orMinkowski space with itsLorentz metric. In the latter case, the "rotations" include theLorentz boosts, but otherwise the theory is substantially similar.^{[citation needed]}

The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensionalspace-time. To obtain the spinors of physics, such as theDirac spinor, one extends the construction to obtain aspin structure on 4-dimensional space-time (Minkowski space). Effectively, one starts with thetangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds thespin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of afiber 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 theDirac equation, or theWeyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that areplane waves, 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 calledfermions; 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.^{[citation needed]}

It appears that allfundamental particles in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of theneutrino. There does not seem to be anya priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version ofCℓ_2,2(${\displaystyle \mathbb{R}}$), theMajorana spinor.^[6] There also does not seem to be any particular prohibition to havingWeyl spinors 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.^[7] Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

Weyl spinors are insufficient to describe massive particles, such aselectrons, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, theDirac equation is needed. The initial construction of theStandard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; theHiggs mechanism gives electrons a mass; the classicalneutrino remained massless, and was thus an example of a Weyl spinor.^[q] However, because of observedneutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors.^[8] It is not known whether Weyl spinor fundamental particles exist in nature.

The situation forcondensed matter physics is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging fromsemiconductors to far more exotic materials. In 2015, an international team led byPrinceton University scientists announced that they had found aquasiparticle that behaves as a Weyl fermion.^[9]

One major mathematical application of the construction of spinors is to make possible the explicit construction oflinear representations of theLie algebras of thespecial orthogonal groups, 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 theAtiyah–Singer index theorem, and to provide constructions in particular fordiscrete series representations ofsemisimple groups.

The spin representations of the special orthogonal Lie algebras are distinguished from thetensor representations given byWeyl's construction by theweights. 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 thespin representation article.

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".^[10] Stated differently:

Spinors ... provide a linear representation of the group ofrotations in a space with any number${\displaystyle n}$ of dimensions, each spinor having${\displaystyle 2^{\nu }}$ components where${\displaystyle n=2\nu +1}$ or${\displaystyle 2\nu }$.^[2]

Several ways of illustrating everyday analogies have been formulated in terms of theplate trick,tangloids and other examples oforientation entanglement.

Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated byMichael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo:

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 thesquare root of −1 took centuries, the same might be true of spinors.^[11]

The most general mathematical form of spinors was discovered byÉlie Cartan in 1913.^[12] The word "spinor" was coined byPaul Ehrenfest in his work onquantum physics.^[13]

Spinors were first applied tomathematical physics byWolfgang Pauli in 1927, when he introduced hisspin matrices.^[14] The following year,Paul Dirac discovered the fullyrelativistic theory ofelectronspin by showing the connection between spinors and theLorentz group.^[15] By the 1930s, Dirac,Piet Hein and others at theNiels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such asTangloids to teach and model the calculus of spinors.

Spinor spaces were represented asleft ideals of a matrix algebra in 1930, byGustave Juvett^[16] and byFritz Sauter.^[17][18] 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 aminimal left ideal inMat(2, ${\displaystyle \mathbb{C}}$).^[r][20]

In 1947Marcel Riesz constructed spinor spaces as elements of a minimal left ideal ofClifford algebras. In 1966/1967,David Hestenes^[21][22] replaced spinor spaces by theeven subalgebra Cℓ⁰_1,3(${\displaystyle \mathbb{R}}$) of thespacetime algebra Cℓ_1,3(${\displaystyle \mathbb{R}}$).^[18][20] As of the 1980s, the theoretical physics group atBirkbeck College aroundDavid Bohm andBasil Hiley has been developingalgebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.

Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebraCℓ_p, q(${\displaystyle \mathbb{R}}$). This is an algebra built up from an orthonormal basis ofn =p + q mutually orthogonal vectors under addition and multiplication,p of which have norm +1 andq of which have norm −1, with the product rule for the basis vectors

The Clifford algebra Cℓ_2,0(${\displaystyle \mathbb{R}}$) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors,σ₁ andσ₂, and one unitpseudoscalari =σ₁σ₂. From the definitions above, it is evident that(σ₁)² = (σ₂)² = 1, and(σ₁σ₂)(σ₁σ₂) = −σ₁σ₁σ₂σ₂ = −1.

The even subalgebra Cℓ⁰_2,0(${\displaystyle \mathbb{R}}$), spanned byeven-graded basis elements of Cℓ_2,0(${\displaystyle \mathbb{R}}$), determines the space of spinors via its representations. It is made up of real linear combinations of 1 andσ₁σ₂. As a real algebra, Cℓ⁰_2,0(${\displaystyle \mathbb{R}}$) is isomorphic to the field ofcomplex numbers ${\displaystyle \mathbb{C}}$. As a result, it admits a conjugation operation (analogous tocomplex conjugation), sometimes called thereverse of a Clifford element, defined by$${\displaystyle (a+b{\sigma }_1{\sigma }_2{)}^{\ast }=a+b{\sigma }_2{\sigma }_1}$$which, by the Clifford relations, can be written$${\displaystyle (a+b{\sigma }_1{\sigma }_2{)}^{\ast }=a+b{\sigma }_2{\sigma }_1=a-b{\sigma }_1{\sigma }_2.}$$

The action of an even Clifford elementγ ∈ Cℓ⁰_2,0(${\displaystyle \mathbb{R}}$) on vectors, regarded as 1-graded elements of Cℓ_2,0(${\displaystyle \mathbb{R}}$), is determined by mapping a general vectoru =a₁σ₁ +a₂σ₂ to the vector$${\displaystyle \gamma (u)=\gamma u{\gamma }^{\ast },}$$where${\displaystyle {\gamma }^{\ast }}$ is the conjugate of${\displaystyle \gamma }$, and the product is Clifford multiplication. In this situation, aspinor^[s] is an ordinary complex number. The action of${\displaystyle \gamma }$ on a spinor${\displaystyle \varphi }$ is given by ordinary complex multiplication:$${\displaystyle \gamma (\varphi )=\gamma \varphi .}$$

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:$${\displaystyle \gamma (u)=\gamma u{\gamma }^{\ast }={\gamma }^{2}u.}$$

On the other hand, in comparison with its action on spinors${\displaystyle \gamma (\varphi )=\gamma \varphi }$, the action of${\displaystyle \gamma }$ on ordinary vectors appears as thesquare of its action on spinors.

Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle ofθ corresponds toγ² = exp(θ σ₁σ₂), so that the corresponding action on spinors is viaγ = ± exp(θ σ₁σ₂/2). In general, because oflogarithmic 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, byabuse 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, tocomputer graphics) they make sense.

• The even-graded element$${\displaystyle \gamma =\frac{1}{\sqrt{2}}(1-{\sigma }_1{\sigma }_2)}$$ corresponds to a vector rotation of 90° fromσ₁ around towardsσ₂, which can be checked by confirming that$${\displaystyle \frac{1}{2}(1-{\sigma }_1{\sigma }_2)\{a_1{\sigma }_1+a_2{\sigma }_2\}(1-{\sigma }_2{\sigma }_1)=a_1{\sigma }_2-a_2{\sigma }_1}$$ It corresponds to a spinor rotation of only 45°, however:$${\displaystyle \frac{1}{\sqrt{2}}(1-{\sigma }_1{\sigma }_2)\{a_1+a_2{\sigma }_1{\sigma }_2\}=\frac{a_1+a_2}{\sqrt{2}}+\frac{-a_1+a_2}{\sqrt{2}}{\sigma }_1{\sigma }_2}$$
• Similarly the even-graded elementγ = −σ₁σ₂ corresponds to a vector rotation of 180°:$${\displaystyle (-{\sigma }_1{\sigma }_2)\{a_1{\sigma }_1+a_2{\sigma }_2\}(-{\sigma }_2{\sigma }_1)=-a_1{\sigma }_1-a_2{\sigma }_2}$$ but a spinor rotation of only 90°:$${\displaystyle (-{\sigma }_1{\sigma }_2)\{a_1+a_2{\sigma }_1{\sigma }_2\}=a_2-a_1{\sigma }_1{\sigma }_2}$$
• Continuing on further, the even-graded elementγ = −1 corresponds to a vector rotation of 360°:$${\displaystyle (-1)\{a_1{\sigma }_1+a_2{\sigma }_2\}(-1)=a_1{\sigma }_1+a_2{\sigma }_2}$$ but a spinor rotation of 180°.

The Clifford algebra Cℓ_3,0(${\displaystyle \mathbb{R}}$) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors,σ₁,σ₂ andσ₃, the three unit bivectorsσ₁σ₂,σ₂σ₃,σ₃σ₁ and thepseudoscalari =σ₁σ₂σ₃. It is straightforward to show that(σ₁)² = (σ₂)² = (σ₃)² = 1, and(σ₁σ₂)² = (σ₂σ₃)² = (σ₃σ₁)² = (σ₁σ₂σ₃)² = −1.

The sub-algebra of even-graded elements is made up of scalar dilations,$${\displaystyle u^{\prime }={\rho }^{\left(\frac{1}{2}\right)}u{\rho }^{\left(\frac{1}{2}\right)}=\rho u,}$$and vector rotations$${\displaystyle u^{\prime }=\gamma u{\gamma }^{\ast },}$$where

1

corresponds to a vector rotation through an angleθ about an axis defined by a unit vectorv =a₁σ₁ + a₂σ₂ + a₃σ₃.

As a special case, it is easy to see that, ifv =σ₃, this reproduces theσ₁σ₂ rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in theσ₃ direction invariant, since

$${\displaystyle \left[\cos \left(\frac{\theta }{2}\right)-i{\sigma }_3\sin \left(\frac{\theta }{2}\right)\right]{\sigma }_3\left[\cos \left(\frac{\theta }{2}\right)+i{\sigma }_3\sin \left(\frac{\theta }{2}\right)\right]=\left[{\cos }^2\left(\frac{\theta }{2}\right)+{\sin }^2\left(\frac{\theta }{2}\right)\right]{\sigma }_3={\sigma }_3.}$$

The bivectorsσ₂σ₃,σ₃σ₁ andσ₁σ₂ are in factHamilton'squaternionsi,j, andk, discovered in 1843:

With the identification of the even-graded elements with the algebra${\displaystyle \mathbb{H}}$ of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.^[t] Thus the (real^[u]) 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 angleθ,the angle appearing in γ was halved. Thus the spinor rotationγ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the spinorψ 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(180° + θ/2) in place ofθ/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.

A space of spinors can be constructed explicitly with concrete and abstract constructions. Theequivalence 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, seespinors in three dimensions.

Given a vector spaceV and a quadratic formg an explicit matrix representation of the Clifford algebraCℓ(V, g) can be defined as follows. Choose an orthonormal basise¹ ...eⁿ forV i.e.g(e^μe^ν) =η^μν whereη^μμ = ±1 andη^μν = 0 forμ ≠ν. Letk = ⌊n/2⌋. Fix a set of2^k × 2^k matricesγ¹ ...γⁿ such thatγ^μγ^ν +γ^νγ^μ = 2η^μν1 (i.e. fix a convention for thegamma matrices). Then the assignmente^μ →γ^μ extends uniquely to an algebra homomorphismCℓ(V, g) → Mat(2^k, ${\displaystyle \mathbb{C}}$) by sending the monomiale^μ₁ ⋅⋅⋅e^μ_k in the Clifford algebra to the productγ^μ₁ ⋅⋅⋅γ^μ_k of matrices and extending linearly. The space${\displaystyle \mathrm{\Delta }={\mathbb{C}}^{2^k}}$ 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 thePauli sigma matrices gives rise to the familiar two component spinors used in non relativisticquantum mechanics. Likewise using the4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativisticquantum field theory. In general, in order to define gamma matrices of the required kind, one can use theWeyl–Brauer matrices.

In this construction the representation of the Clifford algebraCℓ(V, g), the Lie algebraso(V, g), and the Spin groupSpin(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^k complex numbers and is denoted with spinor indices (usuallyα, β, γ). In the physics literature, suchindices are often used to denote spinors even when an abstract spinor construction is used.

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 ofCℓ(V, g) on itself. These are subspaces of the Clifford algebra of the formCℓ(V, g)ω, admitting the evident action ofCℓ(V, g) by left-multiplication:c :xω →cxω. There are two variations on this theme: one can either find a primitive elementω that is anilpotent element of the Clifford algebra, or one that is anidempotent. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.^[23] 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 ofV, and then specify the action of the Clifford algebraexternally to that vector space.

In either approach, the fundamental notion is that of anisotropic subspaceW. 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 ofV is given.

As above, we let(V, g) be ann-dimensional complex vector space equipped with a nondegenerate bilinear form. IfV is a real vector space, then we replaceV by itscomplexification${\displaystyle V{\otimes }_{\mathbb{R}}\mathbb{C}}$ and letg denote the induced bilinear form on${\displaystyle V{\otimes }_{\mathbb{R}}\mathbb{C}}$. LetW be a maximal isotropic subspace, i.e. a maximal subspace ofV such thatg|_W = 0. Ifn =  2k is even, then letW′ be an isotropic subspace complementary toW. Ifn =  2k + 1 is odd, letW′ be a maximal isotropic subspace withW ∩ W′ = 0, and letU be the orthogonal complement ofW ⊕ W′. In both the even- and odd-dimensional casesW andW′ have dimensionk. In the odd-dimensional case,U is one-dimensional, spanned by a unit vectoru.

SinceW′ is isotropic, multiplication of elements ofW′ insideCℓ(V, g) isskew. Hence vectors inW′ anti-commute, andCℓ(W′, g|_W′) = Cℓ(W′, 0) is just theexterior algebra Λ^∗W′. Consequently, thek-fold product ofW′ with itself,W′^k, is one-dimensional. Letω be a generator ofW′^k. In terms of a basisw′₁, ...,w′_k of inW′, one possibility is to set$${\displaystyle \omega =w_1^{\prime }{w}_2^{\prime }\cdots w_k^{\prime }.}$$

Note thatω² = 0 (i.e.,ω is nilpotent of order 2), and moreover,w′ω = 0 for allw′ ∈W′. The following facts can be proven easily:

1. Ifn = 2k, then the left idealΔ = Cℓ(V, g)ω is a minimal left ideal. Furthermore, this splits into the two spin spacesΔ₊ = Cℓ^evenω andΔ₋ = Cℓ^oddω on restriction to the action of the even Clifford algebra.
2. Ifn = 2k + 1, then the action of the unit vectoru on the left idealCℓ(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 thatn is even. Suppose thatI is a non-zero left ideal contained inCℓ(V, g)ω. We shall show thatI must be equal toCℓ(V, g)ω by proving that it contains a nonzero scalar multiple ofω.

Fix a basisw_i ofW and a complementary basisw_i′ ofW′ so that

Note that any element ofI must have the formαω, by virtue of our assumption thatI ⊂ Cℓ(V, g) ω. Letαω ∈I be any such element. Using the chosen basis, we may writewhere thea_i1...ip are scalars, and theB_j are auxiliary elements of the Clifford algebra. Observe now that the productPick any nonzero monomiala in the expansion ofα with maximal homogeneous degree in the elementsw_i:$${\displaystyle a=a_{i_1\dots i_{\text{max}}}{w}_{i_1}\dots w_{i_{\text{max}}}}$$ (no summation implied),then$${\displaystyle w_{i_{\text{max}}}^{\prime }\cdots w_{i_1}^{\prime }\alpha \omega =a_{i_1\dots i_{\text{max}}}\omega }$$is a nonzero scalar multiple ofω, as required.

Note that forn even, this computation also shows that$${\displaystyle \mathrm{\Delta }=\mathrm{C}\ell (W)\omega =\left({\mathrm{\Lambda }}^{\ast }W\right)\omega }$$as a vector space. In the last equality we again used thatW is isotropic. In physics terms, this shows that Δ is built up like aFock space bycreating spinors using anti-commuting creation operators inW acting on a vacuumω.

The computations with the minimal ideal construction suggest that a spinor representation canalso be defined directly using theexterior algebraΛ^∗W = ⊕_j Λ^jW of the isotropic subspaceW.LetΔ = Λ^∗W denote the exterior algebra ofW considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.^[24][25]

The action of the Clifford algebra on Δ is defined first by giving the action of an element ofV on Δ, and then showing that this action respects the Clifford relation and so extends to ahomomorphism of the full Clifford algebra into theendomorphism ring End(Δ) by theuniversal property of Clifford algebras. The details differ slightly according to whether the dimension ofV is even or odd.

When dim(V) is even,V =W ⊕W′ whereW′ is the chosen isotropic complement. Hence anyv ∈V decomposes uniquely asv =w +w′ withw ∈W andw′ ∈W′. The action ofv on a spinor is given by$${\displaystyle c(v)w_1\wedge \cdots \wedge w_n=\left(ϵ(w)+i\left(w^{\prime }\right)\right)\left(w_1\wedge \cdots \wedge w_n\right)}$$wherei(w′) isinterior product withw′ using the nondegenerate quadratic form to identifyV withV^∗, andε(w) denotes theexterior product. This action is sometimes called theClifford product. It may be verified that$${\displaystyle c(u)c(v)+c(v)c(u)=2g(u,v),}$$and soc 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^[26] (the half-spin representations, or Weyl spinors) via$${\displaystyle {\mathrm{\Delta }}_{+}={\mathrm{\Lambda }}^{\text{even}}W,{\mathrm{\Delta }}_{-}={\mathrm{\Lambda }}^{\text{odd}}W.}$$

When dim(V) is odd,V =W ⊕U ⊕W′, whereU is spanned by a unit vectoru orthogonal toW. The Clifford actionc is defined as before onW ⊕W′, while the Clifford action of (multiples of)u is defined byAs before, one verifies thatc respects the Clifford relations, and so induces a homomorphism.

If the vector spaceV 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 spaceV is ahermitian vector space(V, h), i.e.,V is equipped with acomplex structureJ that is anorthogonal transformation with respect to the inner productg onV. Then${\displaystyle V{\otimes }_{\mathbb{R}}\mathbb{C}}$ splits in the±i eigenspaces ofJ. These eigenspaces are isotropic for the complexification ofg and can be identified with the complex vector space(V, J) and its complex conjugate(V, −J). Therefore, for a hermitian vector space(V, h) the vector space${\displaystyle {\mathrm{\Lambda }}_{\mathbb{C}}^{\cdot }\overline{V}}$ (as well as its complex conjugate${\displaystyle {\mathrm{\Lambda }}_{\mathbb{C}}^{\cdot }V}$) 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 analmost Hermitian manifold and is the reason why everyalmost complex manifold (in particular everysymplectic manifold) has aSpin^c structure. Likewise, every complex vector bundle on a manifold carries a Spin^c structure.^[27]

A number ofClebsch–Gordan decompositions are possible on thetensor product of one spin representation with another.^[28] 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

• Γ_r = Λ^rV, the representation of the orthogonal group on skew tensors of rankr.

In addition, for the real orthogonal groups, there are threecharacters (one-dimensional representations)

• σ₊ : O(p, q) → {−1, +1} given byσ₊(R) = −1, ifR reverses the spatial orientation ofV, +1, ifR preserves the spatial orientation ofV. (The spatial character.)
• σ₋ : O(p, q) → {−1, +1} given byσ₋(R) = −1, ifR reverses the temporal orientation ofV, +1, ifR preserves the temporal orientation ofV. (The temporal character.)
• σ =σ₊σ₋ . (The orientation character.)

The Clebsch–Gordan decomposition allows one to define, among other things:

• An action of spinors on vectors.
• AHermitian metric on the complex representations of the real spin groups.
• ADirac operator on each spin representation.

Ifn = 2k is even, then the tensor product of Δ with thecontragredient representation decomposes as$${\displaystyle \mathrm{\Delta }\otimes {\mathrm{\Delta }}^{\ast }\cong \underset{p=0}{\overset{n}{⨁}}{\mathrm{\Gamma }}_p\cong \underset{p=0}{\overset{k-1}{⨁}}\left({\mathrm{\Gamma }}_p\oplus \sigma {\mathrm{\Gamma }}_p\right)\oplus {\mathrm{\Gamma }}_k}$$which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elementsαω ⊗ βω′. The rightmost formulation follows from the transformation properties of theHodge star operator. Note that on restriction to the even Clifford algebra, the paired summandsΓ_p ⊕σΓ_p 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:$${\displaystyle (\alpha \omega {)}^{\ast }=\omega \left({\alpha }^{\ast }\right).}$$SoΔ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose

For the complex representations of the real Clifford algebras, the associatedreality 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 conjugateΔ of the representation Δ, and the following isomorphism is seen to hold:$${\displaystyle \overline{\mathrm{\Delta }}\cong {\sigma }_{-}{\mathrm{\Delta }}^{\ast }}$$

In particular, note that the representation Δ of the orthochronous spin group is aunitary representation. In general, there are Clebsch–Gordan decompositions$${\displaystyle \mathrm{\Delta }\otimes \overline{\mathrm{\Delta }}\cong \underset{p=0}{\overset{k}{⨁}}\left({\sigma }_{-}{\mathrm{\Gamma }}_p\oplus {\sigma }_{+}{\mathrm{\Gamma }}_p\right).}$$

In metric signature(p, q), the following isomorphisms hold for the conjugate half-spin representations

• Ifq is even, then${\displaystyle {\overline{\mathrm{\Delta }}}_{+}\cong {\sigma }_{-}\otimes {\mathrm{\Delta }}_{+}^{\ast }}$ and${\displaystyle {\overline{\mathrm{\Delta }}}_{-}\cong {\sigma }_{-}\otimes {\mathrm{\Delta }}_{-}^{\ast }.}$
• Ifq is odd, then${\displaystyle {\overline{\mathrm{\Delta }}}_{+}\cong {\sigma }_{-}\otimes {\mathrm{\Delta }}_{-}^{\ast }}$ and${\displaystyle {\overline{\mathrm{\Delta }}}_{-}\cong {\sigma }_{-}\otimes {\mathrm{\Delta }}_{+}^{\ast }.}$

Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representationsΔ_± ⊗Δ_±.

Ifn = 2k + 1 is odd, then$${\displaystyle \mathrm{\Delta }\otimes {\mathrm{\Delta }}^{\ast }\cong \underset{p=0}{\overset{k}{⨁}}{\mathrm{\Gamma }}_{2p}.}$$In the real case, once again the isomorphism holds$${\displaystyle \overline{\mathrm{\Delta }}\cong {\sigma }_{-}{\mathrm{\Delta }}^{\ast }.}$$Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by$${\displaystyle \mathrm{\Delta }\otimes \overline{\mathrm{\Delta }}\cong {\sigma }_{-}{\mathrm{\Gamma }}_0\oplus {\sigma }_{+}{\mathrm{\Gamma }}_1\oplus \cdots \oplus {\sigma }_{\pm }{\mathrm{\Gamma }}_k}$$

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 spinorsϕψ as a scalar. In physical terms, a spinor should determine aprobability amplitude for thequantum state.
• A manner of regarding the productψϕ 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 asψvψ. In physical terms, this represents anelectric current of Maxwell'selectromagnetic theory, or more generally aprobability current.

• In 1 dimension (a trivial example), the single spinor representation is formally Majorana, areal 1-dimensional representation that does not transform.
• In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-componentcomplex representations, i.e. complex numbers that get multiplied bye^±iφ/2 under a rotation by angleφ.
• In 3 Euclidean dimensions, the single spinor representation is 2-dimensional andquaternionic. The existence of spinors in 3 dimensions follows from the isomorphism of thegroupsSU(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 asPauli matrices.
• In 4 Euclidean dimensions, the corresponding isomorphism isSpin(4) ≅ SU(2) × SU(2). There are two inequivalentquaternionic 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
• In 5 Euclidean dimensions, the relevant isomorphism isSpin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4-dimensional and quaternionic.
• In 6 Euclidean dimensions, the isomorphismSpin(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 ofSpin(8) calledtriality.
• Ind + 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure ind dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. SeeBott periodicity.
• In spacetimes withp spatial andq time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the(p + q)-dimensional Euclidean space, but the reality projections mimic the structure in|p − q| Euclidean dimensions. For example, in3 + 1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphismSL(2, ${\displaystyle \mathbb{C}}$) ≅ Spin(3,1).

• Brauer, Richard;Weyl, Hermann (1935). "Spinors inn dimensions".American Journal of Mathematics.57 (2). The Johns Hopkins University Press:425–449.doi:10.2307/2371218.JSTOR 2371218.
• Cartan, Élie (1913)."Les groupes projectifs qui ne laissent invariante aucune multiplicité plane"(PDF).Bull. Soc. Math. Fr.41:53–96.doi:10.24033/bsmf.916.
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• Pauli, Wolfgang (1927). "Zur Quantenmechanik des magnetischen Elektrons".Zeitschrift für Physik.43 (9–10):601–632.Bibcode:1927ZPhy...43..601P.doi:10.1007/BF01397326.S2CID 128228729.
• Tomonaga, Sin-Itiro (1998). "Lecture 7: The quantity which is neither vector nor tensor".The Story of Spin. University of Chicago Press. p. 129.ISBN 0-226-80794-0.

• Cartan, Élie (1981) [1966].The Theory of Spinors (reprint ed.). Paris, FR: Hermann (1966); Dover Publications (1981).ISBN 978-0-486-64070-9.
• Gilkey, Peter B. (1984).Invariance Theory: The heat equation, and the Atiyah–Singer index theorem. Publish or Perish.ISBN 0-914098-20-9.
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• Spinors
• Rotation in three dimensions
• Quantum mechanics
• Quantum field theory

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