Spinor
In geometry and physics, spinors /spɪnər/ are elements of a complex numberbased vector space that can be associated with Euclidean space.^{[b]} A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation,^{[c]} but unlike geometric vectors and tensors, 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 sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).
It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.^{[1]}^{[d]} In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron 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 the rotation group). There are two topologically distinguishable classes (homotopy classes) 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.^{[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 group act 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 doublevalued 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 basisindependent 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 an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anticommutation 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,^{[i]} and the twocomponent 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 be irreducible if the dimension is odd, or it will decompose into a pair of socalled "halfspin" or Weyl representations if the dimension is even.^{[k]}
Introduction[edit]
What characterizes spinors and distinguishes them from geometric 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 undergo the same rotation as the coordinates. More broadly, any tensor associated with the system (for instance, the 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 (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 pathdependence. It turns out that, for any final configuration of the coordinates, there are actually two ("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 puzzle (shown) 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 signreversal 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 momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the twocomponent complex column vectors 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 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,^{[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 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 space of spinors is the space of column vectors with 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 and metric signature, this realization of spinors as column vectors may be irreducible or it may decompose into a pair of socalled "halfspin" or Weyl representations.^{[p]} When the vector space V is fourdimensional, the algebra is described by the gamma matrices.
Mathematical definition[edit]
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 finitedimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finitedimensional group representation of the spin group on which the center acts nontrivially.
Overview[edit]
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[edit]
From a 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 representations, and their constituents spinors. From this view, a spinor must belong to a representation of the double cover of the rotation group SO(n,), or more generally of a double cover of the generalized special orthogonal group SO^{+}(p, q, ) on spaces with a metric signature of (p, q). These double covers are Lie groups, called the spin groups Spin(n) or 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 doublevalued projective 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 where is a vector space over or and is a homomorphism , a spinor is an element of the vector space .
Geometric point of view[edit]
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 identities, are needed.
Clifford algebras[edit]
The language of Clifford algebras^{[4]} (sometimes called geometric algebras) 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 finitedimensional complex vector space with nondegenerate symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = , with the standard form g(x, y) = x^{T}y = x_{1}y_{1} + ... + x_{n}y_{n} we denote the Clifford algebra by Cℓ_{n}(). Since by the choice of an orthonormal basis every complex vectorspace with nondegenerate form is isomorphic to this standard example, this notation is abused more generally if dim_{$\mathbb {C}$}(V) = n. If n = 2k is even, Cℓ_{n}() is isomorphic as an algebra (in a nonunique way) to the algebra Mat(2^{k}, ) of 2^{k} × 2^{k} complex matrices (by the Artin–Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple). If n = 2k + 1 is odd, Cℓ_{2k+1}() is isomorphic to the algebra Mat(2^{k}, ) ⊕ Mat(2^{k}, ) of two copies of the 2^{k} × 2^{k} complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2^{[n/2]}. Since the Lie algebra so(V, g) is embedded as a Lie subalgebra in Cℓ(V, g) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of 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]} into two irreducible representations Δ = Δ_{+} ⊕ Δ_{−} called the Weyl or halfspin 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 article for more details.
Spin groups[edit]
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 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 cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors 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 oneparameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a oneparameter 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 boosts, but otherwise the theory is substantially similar.
Spinor fields in physics[edit]
The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zerodimensional spacetime. To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4dimensional spacetime (Minkowski space). Effectively, one starts with the tangent manifold of spacetime, each point of which is a 4dimensional 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 fibre 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 waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zerodimensional) Clifford algebra/spin representation theory described above. Such planewave solutions (or other solutions) of the differential equations can then properly be called fermions; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of oneanother.
It appears that all fundamental particles in nature that are spin1/2 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 noncomplexified version of Cℓ_{2,2}(), the Majorana spinor.^{[5]} There also does not seem to be any particular prohibition to having Weyl 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.^{[6]} Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.
Weyl spinors are insufficient to describe massive particles, such as electrons, since the Weyl planewave 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.^{[q]} However, because of observed neutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors.^{[7]} 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 threedimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors 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.^{[8]}
Spinors in representation theory[edit]
One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special 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 the Atiyah–Singer index theorem, and to provide constructions in particular for discrete series representations of semisimple groups.
The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the 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 halfinteger linear combinations thereof. Explicit details can be found in the spin representation article.
Attempts at intuitive understanding[edit]
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".^{[9]} Stated differently:
Spinors ... provide a linear representation of the group of rotations in a space with any number of dimensions, each spinor having components where or .^{[2]}
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:
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.^{[10]}
History[edit]
The most general mathematical form of spinors was discovered by Élie Cartan in 1913.^{[11]} The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.^{[12]}
Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.^{[13]} The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.^{[14]} By the 1930s, Dirac, 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 ideals of a matrix algebra in 1930, by G. Juvet^{[15]} and by Fritz Sauter.^{[16]}^{[17]} More specifically, instead of representing spinors as complexvalued 2D column vectors as Pauli had done, they represented them as complexvalued 2 × 2 matrices in which only the elements of the left column are nonzero. In this manner the spinor space became a minimal left ideal in Mat(2, ).^{[r]}^{[19]}
In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes^{[20]}^{[21]} replaced spinor spaces by the even subalgebra Cℓ^{0}_{1,3}() of the spacetime algebra Cℓ_{1,3}().^{[17]}^{[19]} As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.
Examples[edit]
Some simple examples of spinors in low dimensions arise from considering the evengraded subalgebras of the Clifford algebra Cℓ_{p, q}(). This is an algebra built up from an orthonormal basis of 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
Two dimensions[edit]
The Clifford algebra Cℓ_{2,0}() is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ_{1} and σ_{2}, and one unit pseudoscalar i = σ_{1}σ_{2}. From the definitions above, it is evident that (σ_{1})^{2} = (σ_{2})^{2} = 1, and (σ_{1}σ_{2})(σ_{1}σ_{2}) = −σ_{1}σ_{1}σ_{2}σ_{2} = −1.
The even subalgebra Cℓ^{0}_{2,0}(), spanned by evengraded basis elements of Cℓ_{2,0}(), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ_{1}σ_{2}. As a real algebra, Cℓ^{0}_{2,0}() is isomorphic to the field of complex numbers . As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by
The action of an even Clifford element γ ∈ Cℓ^{0}_{2,0}() on vectors, regarded as 1graded elements of Cℓ_{2,0}(), is determined by mapping a general vector u = a_{1}σ_{1} + a_{2}σ_{2} to the vector
An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the evengraded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that evengraded elements conjugatecommute with ordinary vectors:
On the other hand, in comparison with its action on spinors , the action of 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 γ^{2} = exp(θ σ_{1}σ_{2}), so that the corresponding action on spinors is via γ = ± exp(θ σ_{1}σ_{2}/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is twovalued.
In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of evengraded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by 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 evengraded element corresponds to a vector rotation of 90° from σ_{1} around towards σ_{2}, which can be checked by confirming thatIt corresponds to a spinor rotation of only 45°, however:
 Similarly the evengraded element γ = −σ_{1}σ_{2} corresponds to a vector rotation of 180°: but a spinor rotation of only 90°:
 Continuing on further, the evengraded element γ = −1 corresponds to a vector rotation of 360°: but a spinor rotation of 180°.
Three dimensions[edit]
The Clifford algebra Cℓ_{3,0}() is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ_{1}, σ_{2} and σ_{3}, the three unit bivectors σ_{1}σ_{2}, σ_{2}σ_{3}, σ_{3}σ_{1} and the pseudoscalar i = σ_{1}σ_{2}σ_{3}. It is straightforward to show that (σ_{1})^{2} = (σ_{2})^{2} = (σ_{3})^{2} = 1, and (σ_{1}σ_{2})^{2} = (σ_{2}σ_{3})^{2} = (σ_{3}σ_{1})^{2} = (σ_{1}σ_{2}σ_{3})^{2} = −1.
The subalgebra of evengraded elements is made up of scalar dilations,

(1) 
corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a_{1}σ_{1} + a_{2}σ_{2} + a_{3}σ_{3}.
As a special case, it is easy to see that, if v = σ_{3}, this reproduces the σ_{1}σ_{2} rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ_{3} direction invariant, since
The bivectors σ_{2}σ_{3}, σ_{3}σ_{1} and σ_{1}σ_{2} are in fact Hamilton's quaternions i, j, and k, discovered in 1843:
With the identification of the evengraded elements with the algebra of quaternions, as in the case of two dimensions the only representation of the algebra of evengraded elements is on itself.^{[t]} Thus the (real^{[u]}) spinors in threedimensions are quaternions, and the action of an evengraded element on a spinor is given by ordinary quaternionic multiplication.
Note that the expression (1) for a vector rotation through an angle θ, the angle appearing in γ was halved. Thus the spinor rotation γ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the spinor ψ through an angle onehalf the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is twovalued: the expression (1) with (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[edit]
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[edit]
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows. Choose an orthonormal basis e^{1} ... e^{n} for V i.e. g(e^{μ}e^{ν}) = η^{μν} where η^{μμ} = ±1 and η^{μν} = 0 for μ ≠ ν. Let k = ⌊n/2⌋. Fix a set of 2^{k} × 2^{k} matrices γ^{1} ... γ^{n} such that γ^{μ}γ^{ν} + γ^{ν}γ^{μ} = 2η^{μν}1 (i.e. fix a convention for the gamma matrices). Then the assignment e^{μ} → γ^{μ} extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2^{k}, ) by sending the monomial e^{μ1} ⋅⋅⋅ e^{μk} in the Clifford algebra to the product γ^{μ1} ⋅⋅⋅ γ^{μk} of matrices and extending linearly. The space 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 sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics. Likewise using the 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 Cℓ(V, g), the Lie algebra so(V, g), and the Spin group 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^{k} complex numbers and is denoted with spinor indices (usually α, β, γ). In the physics literature, such indices are often used to denote spinors even when an abstract spinor construction is used.
Abstract spinors[edit]
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 Cℓ(V, g) on itself. These are subspaces of the Clifford algebra of the form Cℓ(V, g)ω, admitting the evident action of Cℓ(V, g) by leftmultiplication: c : xω → cxω. There are two variations on this theme: one can either find a primitive element 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.^{[22]} 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 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 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 V is given.
As above, we let (V, g) be an ndimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification and let g denote the induced bilinear form on . Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g_{W} = 0. If n = 2k is even, then let W′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd, let W′ be a maximal isotropic subspace with W ∩ W′ = 0, and let U be the orthogonal complement of W ⊕ W′. In both the even and odddimensional cases W and W′ have dimension k. In the odddimensional case, U is onedimensional, spanned by a unit vector u.
Minimal ideals[edit]
Since W′ is isotropic, multiplication of elements of W′ inside Cℓ(V, g) is skew. Hence vectors in W′ anticommute, and Cℓ(W′, g_{W′}) = Cℓ(W′, 0) is just the exterior algebra Λ^{∗}W′. Consequently, the kfold product of W′ with itself, W′^{k}, is onedimensional. Let ω be a generator of W′^{k}. In terms of a basis w′_{1}, ..., w′_{k} of in W′, one possibility is to set
Note that ω^{2} = 0 (i.e., ω is nilpotent of order 2), and moreover, w′ω = 0 for all w′ ∈ W′. The following facts can be proven easily:
 If n = 2k, then the left ideal Δ = Cℓ(V, g)ω is a minimal left ideal. Furthermore, this splits into the two spin spaces Δ_{+} = Cℓ^{even}ω and Δ_{−} = Cℓ^{odd}ω on restriction to the action of the even Clifford algebra.
 If n = 2k + 1, then the action of the unit vector u on the left ideal 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 nonzero left ideal contained in Cℓ(V, g)ω. We shall show that I must be equal to Cℓ(V, g)ω by proving that it contains a nonzero scalar multiple of ω.
Fix a basis w_{i} of W and a complementary basis w_{i}′ of W′ so that
Note that any element of I must have the form αω, by virtue of our assumption that I ⊂ Cℓ(V, g) ω. Let αω ∈ I be any such element. Using the chosen basis, we may write
Note that for n even, this computation also shows that
Exterior algebra construction[edit]
The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra Λ^{∗} W = ⊕_{j} Λ^{j} W of the isotropic subspace W. Let Δ = Λ^{∗} 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.^{[23]}^{[24]}
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 universal property of Clifford algebras. The details differ slightly according to whether the dimension of V is even or odd.
When dim(V) is even, V = W ⊕ W′ where W′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v = w + w′ with w ∈ W and w′ ∈ W′. The action of v on a spinor is given by
The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group^{[25]} (the halfspin representations, or Weyl spinors) via
When dim(V) is odd, 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 W ⊕ W′, while the Clifford action of (multiples of) u is defined by
Hermitian vector spaces and spinors[edit]
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 vector space (V, h), i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then splits in the ±i eigenspaces of J. These eigenspaces are isotropic for the complexification of g 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 (as well as its complex conjugate 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. Likewise, every complex vector bundle on a manifold carries a Spin^{c} structure.^{[26]}
Clebsch–Gordan decomposition[edit]
A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another.^{[27]} 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^{r}V, the representation of the orthogonal group on skew tensors of rank r.
In addition, for the real orthogonal groups, there are three characters (onedimensional representations)
 σ_{+} : O(p, q) → {−1, +1} given by σ_{+}(R) = −1, if R reverses the spatial orientation of V, +1, if R preserves the spatial orientation of V. (The spatial character.)
 σ_{−} : O(p, q) → {−1, +1} given by σ_{−}(R) = −1, if R reverses the temporal orientation of V, +1, if R preserves the temporal orientation of V. (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.
 A Hermitian metric on the complex representations of the real spin groups.
 A Dirac operator on each spin representation.
Even dimensions[edit]
If n = 2k is even, then the tensor product of Δ with the contragredient representation decomposes as
There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:
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 Δ of the representation Δ, and the following isomorphism is seen to hold:
In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions
In metric signature (p, q), the following isomorphisms hold for the conjugate halfspin representations
 If q is even, then and
 If q is odd, then and
Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the halfspin representations Δ_{±} ⊗ Δ_{±}.
Odd dimensions[edit]
If n = 2k + 1 is odd, then
Consequences[edit]
There are many farreaching 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 ϕψ as a scalar. In physical terms, a spinor should determine a probability amplitude for the quantum state.
 A manner of regarding the product ψϕ 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 ψvψ. In physical terms, this represents an electric current of Maxwell's electromagnetic theory, or more generally a probability current.
Summary in low dimensions[edit]
 In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1dimensional representation that does not transform.
 In 2 Euclidean dimensions, the lefthanded and the righthanded Weyl spinor are 1component complex representations, i.e. complex numbers that get multiplied by e^{±iφ/2} under a rotation by angle φ.
 In 3 Euclidean dimensions, the single spinor representation is 2dimensional and quaternionic. The existence of spinors in 3 dimensions follows from the isomorphism of the groups SU(2) ≅ Spin(3) that allows us to define the action of Spin(3) on a complex 2component column (a spinor); the generators of SU(2) can be written as Pauli matrices.
 In 4 Euclidean dimensions, the corresponding isomorphism is Spin(4) ≅ SU(2) × SU(2). There are two inequivalent quaternionic 2component Weyl spinors and each of them transforms under one of the SU(2) factors only.
 In 5 Euclidean dimensions, the relevant isomorphism is Spin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4dimensional and quaternionic.
 In 6 Euclidean dimensions, the isomorphism Spin(6) ≅ SU(4) guarantees that there are two 4dimensional complex Weyl representations that are complex conjugates of one another.
 In 7 Euclidean dimensions, the single spinor representation is 8dimensional 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 8dimensional representations that are related to the 8dimensional real vector representation by a special property of Spin(8) called triality.
 In 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 timelike 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, in 3 + 1 dimensions there are two nonequivalent Weyl complex (like in 2 dimensions) 2component (like in 4 dimensions) spinors, which follows from the isomorphism SL(2, ) ≅ Spin(3,1).
Metric signature  Weyl, complex  Conjugacy  Dirac, complex 
Majorana–Weyl, real  Majorana, real  

Lefthanded  Righthanded  Lefthanded  Righthanded  
(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[edit]
Notes[edit]
 ^ 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 threedimensional pseudoEuclidean 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 SL_{2}(). In Euclidean signature, the projective plane, conic and line bundle are over the complex instead, and this picture is just a real slice.
 ^ 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 formal definition of spinors at this level is that the space of spinors is a linear representation of the Lie algebra of infinitesimal rotations of a certain kind.
 ^ "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 of dimensions, each spinor having components where or ."^{[2]} The star (*) refers to Cartan (1913).
 ^ More precisely, it is the fermions of spin1/2 that are described by spinors, which is true both in the relativistic and nonrelativistic theory. The wavefunction of the nonrelativistic electron has values in 2component spinors transforming under 3dimensional infinitesimal rotations. The relativistic Dirac equation for the electron is an equation for 4component spinors transforming under infinitesimal Lorentz transformations, for which a substantially similar theory of spinors exists.
 ^ Formally, the spin group is the group of relative homotopy classes with fixed endpoints in the rotation group.
 ^ More formally, the space of spinors can be defined as an (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).
 ^ Geometric algebra is a name for the Clifford algebra in an applied setting.
 ^ The Pauli matrices correspond to 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.
 ^ The metric signature relevant as well if we are concerned with real spinors. See spin representation.
 ^ 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.
 ^ The TNB frame of the ribbon defines a rotation continuously for each value of the arc length parameter.
 ^ This is the set of 2×2 complex traceless hermitian matrices.
 ^ Except for a kernel of corresponding to the two different elements of the spin group that go to the same rotation.
 ^ So the ambiguity in identifying the spinors themselves persists from the point of view of the group theory, and still depends on choices.
 ^ The Clifford algebra can be given an even/odd 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.
 ^ More precisely, the electron starts out as two massless Weyl spinors, left and righthanded. Upon symmetry breaking, both gain a mass, and are coupled to form a Dirac spinor.
 ^ The matrices of dimension N × N in which only the elements of the left column are nonzero form a left ideal in the N × N matrix algebra Mat(N, ) – 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 nonzero. Moreover, it can be shown that it is a minimal left ideal.^{[18]}
 ^ These are the righthanded Weyl spinors in two dimensions. For the lefthanded Weyl spinors, the representation is via γ(ϕ) = γϕ. The Majorana spinors are the common underlying real representation for the Weyl representations.
 ^ Since, for a skew field, the kernel of the representation must be trivial. So inequivalent representations can only arise via an automorphism of the skewfield. In this case, there are a pair of equivalent representations: γ(ϕ) = γϕ, and its quaternionic conjugate γ(ϕ) = ϕγ.
 ^ The complex spinors are obtained as the representations of the tensor product = Mat_{2}(). These are considered in more detail in spinors in three dimensions.
References[edit]
 ^ Cartan 1913.
 ^ ^{a} ^{b} 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.
 ^ RukhsanUlHaq (December 2016). "Geometry of Spin: Clifford Algebraic Approach". Resonance. 21 (12): 1105–1117. doi:10.1007/s1204501604225. S2CID 126053475.
 ^ Named after William Kingdon Clifford,
 ^ Named after Ettore Majorana.
 ^ Francis, Matthew R.; Kosowsky, Arthur (2005) [20 March 2004]. "The construction of spinors in geometric algebra". Annals of Physics. 317 (2): 383–409. arXiv:mathph/0403040. Bibcode:2005AnPhy.317..383F. doi:10.1016/j.aop.2004.11.008. S2CID 119632876.
 ^ Wilczek, Frank (2009). "Majorana returns". Nature Physics. Macmillan Publishers. 5 (9): 614–618. Bibcode:2009NatPh...5..614W. doi:10.1038/nphys1380. ISSN 17452473.
 ^ Xu, YangSu; et al. (2015). "Discovery of a Weyl Fermion semimetal and topological Fermi arcs". Science Magazine. AAAS. 349 (6248): 613–617. arXiv:1502.03807. Bibcode:2015Sci...349..613X. doi:10.1126/science.aaa9297. ISSN 00368075. PMID 26184916. S2CID 206636457.
 ^ Jean Hladik: Spinors in Physics, translated by J. M. Cole, Springer 1999, ISBN 9780387986470, p. 3
 ^ Farmelo, Graham (2009). The Strangest Man: The hidden life of Paul Dirac, quantum genius. Faber & Faber. p. 430. ISBN 9780571222865.
 ^ Cartan 1913
 ^ Tomonaga 1998, p. 129
 ^ Pauli 1927.
 ^ Dirac 1928.
 ^ Juvet, G. (1930). "Opérateurs de Dirac et équations de Maxwell". Commentarii Mathematici Helvetici (in French). 2: 225–235. doi:10.1007/BF01214461. S2CID 121226923.
 ^ Sauter, F. (1930). "Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren". Zeitschrift für Physik. 63 (11–12): 803–814. Bibcode:1930ZPhy...63..803S. doi:10.1007/BF01339277. S2CID 122940202.
 ^ ^{a} ^{b} Pertti Lounesto: Crumeyrolle's bivectors and spinors, pp. 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 0792333667, 1995, p. 151
 ^ See also: Pertti Lounesto: Clifford algebras and spinors, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, ISBN 9780521005517, p. 52
 ^ ^{a} ^{b} Pertti Lounesto: Clifford algebras and spinors, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, ISBN 9780521005517, p. 148 f. and p. 327 f.
 ^ D. Hestenes: Space–Time Algebra, Gordon and Breach, New York, 1966, 1987, 1992
 ^ Hestenes, D. (1967). "Real spinor fields". J. Math. Phys. 8 (4): 798–808. Bibcode:1967JMP.....8..798H. doi:10.1063/1.1705279. S2CID 13371668.
 ^ This construction is due to Cartan (1913). The treatment here is based on Chevalley (1954) .
 ^ One source for this subsection is Fulton & Harris (1991).
 ^ Jurgen Jost, "Riemannian Geometry and Geometric Analysis" (2002) SpringerVerlag Univeritext ISBN 3540426272. See chapter 1.
 ^ Via the evengraded Clifford algebra.
 ^ Lawson & Michelsohn 1989, Appendix D.
 ^ Brauer & Weyl 1935.
Further reading[edit]
 Brauer, Richard; Weyl, Hermann (1935). "Spinors in n dimensions". American Journal of Mathematics. The Johns Hopkins University Press. 57 (2): 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.
 Cartan, Élie (1981) [1966]. The Theory of Spinors (reprint ed.). Paris, FR: Hermann (1966); Dover Publications (1981). ISBN 9780486640709.
 Chevalley, Claude (1996) [1954]. The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). ISBN 9783540570639.
 Dirac, Paul M. (1928). "The quantum theory of the electron". Proceedings of the Royal Society of London A. 117 (778): 610–624. Bibcode:1928RSPSA.117..610D. doi:10.1098/rspa.1928.0023. JSTOR 94981.
 Fulton, William; Harris, Joe (1991). Representation Theory: A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: SpringerVerlag. doi:10.1007/9781461209799. ISBN 0387974954. MR 1153249.
 Gilkey, Peter B. (1984). Invariance Theory: The heat equation, and the Atiyah–Singer index theorem. Publish or Perish. ISBN 0914098209.
 Harvey, F. Reese (1990). Spinors and Calibrations. Academic Press. ISBN 9780123296504.
 Hitchin, Nigel J. (1974). "Harmonic spinors". Advances in Mathematics. 14: 1–55. doi:10.1016/00018708(74)900218. MR 0358873.
 Lawson, H. Blaine; Michelsohn, MarieLouise (1989). Spin Geometry. Princeton University Press. ISBN 0691085420.
 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.
 Penrose, Roger; Rindler, W. (1988). Spinor and twistor methods in spacetime geometry. Spinors and SpaceTime. Vol. 2. Cambridge University Press. ISBN 0521347866.
 Tomonaga, SinItiro (1998). "Lecture 7: The quantity which is neither vector nor tensor". The Story of Spin. University of Chicago Press. p. 129. ISBN 0226807940.