Spinor

A continuous rotation can be visualized as a ribbon in space (the TNB frame of the ribbon defines a rotation continuously for each value of the arc length parameter). Two continuous rotations with different classes, one through 2π and one through 4π, are illustrated here in the belt trick puzzle. A solution of the puzzle is a (continuous) manipulation of the belt, fixing the endpoints, that untwists it. This is impossible with the 2π rotation, but possible with the 4π rotation. A solution, shown in the second animation, actually gives an explicit homotopy in the rotation group between the continuous 4π rotation and the trivial (identity) rotation.

Spinors are vector-like objects in geometry and physics that, like geometric vectors and other tensors, transform linearly when the system containing them is subjected to a continuous rotation of its frame of reference. Here a "continuous rotation" refers to a rotation that depends continuously on a single parameter. The coordinate representations of all objects in the system transform with respect to continuous rotations, but depending on the type of the objects, they will transform differently. The components of a geometrical vector undergo the same rotation as the coordinates. In particular, the transformation law for the components of a vector only depends on the final configuration of the rotation, not the details of how the continuous rotation arrived there. Spinors, like vectors, do form a vector space, but unlike geometric vectors (and other tensors), spinors undergo the same rotation as the coordinates apart from a sign which does depend on how the continuous rotation arrived at its final configuration. This sign ambiguity is called the class of the particular continuous rotation. This class actually has a topological origin, famously illustrated in the belt trick puzzle (shown), which demonstrates two different continuous rotations having the same final configurations but different classes. All tensors will have undergone the same transformation under these two topologically distinguishable rotations, but spinors will have undergone such a transformation apart from difference of sign. It turns out that there are only two possible classes for any given rotation, just as there are two different choices of sign.

In the language of group theory, spinors are ("projective") representations of the rotation group, which is to say that they are vector-like objects that transform under rotations. It is actually more accurate to say that they are representations of the spin group, which consists of continuous rotations keeping track of the class of the rotation. To say that spinors form a representation of the spin group formally means that there is a group homomorphism of the spin group into the general linear group of the space of spinors. The representation is spinorial if the transformation law under continuous rotations genuinely depends on the class of the rotation; that is, if this homomorphism is one-to-one. Spinors can be associated, as an auxiliary mathematical object, to 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. The general theory concerns Clifford algebras, in the mathematical treatment of the subject, and Pauli spin matrices, Dirac matrices and similar explicit devices in physics.

Specifically, the spin representations can be obtained as (a subspace of) the irreducible representation space of the Clifford algebra. Since the spin group can be identified concretely as a subgroup of the invertible elements in the Clifford algebra, every representation of the Clifford algebra automatically gives a representation of the Spin group and the irreducible representation is spinorial. For many applications (e.g., the Dirac equation) this Clifford algebra representation is in fact more fundamental than the representation of the spin group. Irreducible representations of Clifford algebra are called the space of Dirac spinors. For even dimensions these may split up in two irreducible representation of the even Clifford algebra called Weyl spinors. After a choice of basis, and a choice of Pauli spin or Gamma matrices such a space of Dirac spinors can be identified with the complex space on which these matrices act. Other more abstract constructions give more coordinate invariant ways to construct spinors but still depend on additional choices unless extra structure is available. The real representation theory and in particular the existence and interpretation of real spinors (Majorana spinors) is more intricate (see spin representation).

General spinors were discovered by Élie Cartan in 1913. Soon after, spinors turned out to be essential in quantum physics, and currently enjoy a wide range of applications. Spinors in three dimensions are needed to describe non-relativistic electrons and other fermions which have spin-½. Dirac spinors, spinors of the Lorentz metric in dimension 4, are required to describe the quantum state of the relativistic electron via the Dirac equation. In quantum field theory, spinors describe the state of relativistic many-particle systems. In mathematics, particularly in differential geometry and global analysis, spinors have also found broad applications in algebraic and differential topology, symplectic geometry, gauge theory, complex algebraic geometry, index theory, and special holonomy.

Overview

In the classical geometry of space, a vector exhibits a certain behavior when it is acted upon by a rotation or reflected in a hyperplane. However, in a certain sense rotations and reflections contain finer geometrical information than can be expressed in terms of their actions on vectors. Spinors are objects constructed in order to encompass more fully this geometry. (See orientation entanglement.)

There are essentially two frameworks for viewing the notion of a spinor.

One is representation theoretic. In this 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. In this view, a spinor must belong to a representation of the double cover of the rotation group SO(n, R), or more generally of double cover of the generalized special orthogonal group SO⁺(p, q, R) on spaces with metric signature (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 projective representations of the groups themselves, which do not meet the full definition of a representation.

The other point of view is geometrical. 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 spinors, such as Fierz identities, are needed.

Clifford algebras

Further information: Clifford algebra

The language of Clifford algebras (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 finite-dimensional complex vector space with nondegenerate 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 = Cⁿ, with the standard form g(x, y) = x^ty = x₁y₁ + ... + x_ny_n we denote the Clifford algebra by Cℓ_n(C). Since by the choice of an orthonormal basis every complex vectorspace with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dim_C(V) = n. If n = 2k is even, Cℓ_n(C) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2^k, C) 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(C) is isomorphic to the algebra Mat(2^k, C) ⊕ Mat(2^k, C) 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 into two irreducible representations Δ = Δ₊ ⊕ Δ₋ called the Weyl or half-spin representations.

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

Terminology in physics

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of Cℓ_p+q(C), the complexification of the Clifford algebra Cℓ_p, q(R), into which the spin group Spin(p, q) may be embedded. On a 2k- or 2k+1-dimensional space a Dirac spinor may be represented as a vector of 2^k complex numbers. (See Special unitary group.) In even dimensions, this representation is reducible when taken as a representation of Spin(p, q) and may be decomposed into two: the left-handed and right-handed Weyl spinor representations. In addition, sometimes the non-complexified version of Cℓ_p,q(R) has a smaller real representation, the Majorana spinor representation. If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana–Weyl spinor representations.

Of all these, only the Dirac representation exists in all dimensions. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

The Dirac, Lorentz, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.

Spinors in representation theory

Main article: Spin representation

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 half-integer linear combinations thereof. Explicit details can be found in the spin representation article.

Attempts at intuitive understanding

The spinor can be described, in simple terms, as “vectors of a space the transformations of which are related in a particular way to rotations in physical space”. Stated differently:

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

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.

History

The most general mathematical form of spinors was discovered by Élie Cartan in 1913. The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices. The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group. 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 and by Fritz Sauter. More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, C).

In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes replaced spinor spaces by the even subalgebra Cℓ⁰_1,3(R) of the spacetime algebra Cℓ_1,3(R). 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

Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓ_p, q(R). 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

$e_{i}e_{j}={\Bigg \{}{\begin{matrix}+1&i=j,\,i\in (1\ldots p)\\-1&i=j,\,i\in (p+1\ldots n)\\-e_{j}e_{i}&i\not =j.\end{matrix}}$

Two dimensions

The Clifford algebra Cℓ_2,0(R) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ₁ and σ₂, and one unit pseudoscalar i = 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(R), spanned by even-graded basis elements of Cℓ_2,0(R), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ₁σ₂. As a real algebra, Cℓ⁰_2,0(R) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by

$(a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}\,$.

which, by the Clifford relations, can be written

$(a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}\,$.

The action of an even Clifford element γ ∈ Cℓ⁰_2,0(R) on vectors, regarded as 1-graded elements of Cℓ_2,0(R), is determined by mapping a general vector u = a₁σ₁ + a₂σ₂ to the vector

$\gamma (u)=\gamma u\gamma ^{*}\,$,

where γ^∗ is the conjugate of γ, and the product is Clifford multiplication. In this situation, a spinor is an ordinary complex number. The action of γ on a spinor φ is given by ordinary complex multiplication:

$\gamma (\phi )=\gamma \phi$.

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:

$\gamma (u)=\gamma u\gamma ^{*}=\gamma ^{2}u\,$.

On the other hand, comparing with the action on spinors γ(φ) = γφ, γ on ordinary vectors acts 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 γ² = exp(θ σ₁σ₂), so that the corresponding action on spinors is via γ = ± exp(θ σ₁σ₂/2). 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 two-valued.

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

Examples

• The even-graded element

$\gamma ={\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})\,$

corresponds to a vector rotation of 90° from σ₁ around towards σ₂, which can be checked by confirming that

${\tfrac {1}{2}}(1-\sigma _{1}\sigma _{2})\,\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(1-\sigma _{2}\sigma _{1})=a_{1}\sigma _{2}-a_{2}\sigma _{1}\,$

It corresponds to a spinor rotation of only 45°, however:

${\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})\,\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}={\frac {a_{1}+a_{2}}{\sqrt {2}}}+{\frac {-a_{1}+a_{2}}{\sqrt {2}}}\sigma _{1}\sigma _{2}$

• Similarly the even-graded element γ = −σ₁σ₂ corresponds to a vector rotation of 180°:

$(-\sigma _{1}\sigma _{2})\,\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(-\sigma _{2}\sigma _{1})=-a_{1}\sigma _{1}-a_{2}\sigma _{2}\,$

but a spinor rotation of only 90°:

$(-\sigma _{1}\sigma _{2})\,\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}=a_{2}-a_{1}\sigma _{1}\sigma _{2}$

• Continuing on further, the even-graded element γ = −1 corresponds to a vector rotation of 360°:

$(-1)\,\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(-1)=a_{1}\sigma _{1}+a_{2}\sigma _{2}\,$

but a spinor rotation of 180°.

Three dimensions

Main articles Spinors in three dimensions, Quaternions and spatial rotation

The Clifford algebra Cℓ_3,0(R) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ₁, σ₂ and σ₃, the three unit bivectors σ₁σ₂, σ₂σ₃, σ₃σ₁ and the pseudoscalar i = 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,

$u^{\prime }=\rho ^{(1/2)}u\rho ^{(1/2)}=\rho u,$

and vector rotations

$u^{\prime }=\gamma \,u\,\gamma ^{*},$

where

$\left.{\begin{matrix}\gamma &=&\cos(\theta /2)-\{a_{1}\sigma _{2}\sigma _{3}+a_{2}\sigma _{3}\sigma _{1}+a_{3}\sigma _{1}\sigma _{2}\}\sin(\theta /2)\\&=&\cos(\theta /2)-i\{a_{1}\sigma _{1}+a_{2}\sigma _{2}+a_{3}\sigma _{3}\}\sin(\theta /2)\\&=&\cos(\theta /2)-iv\sin(\theta /2)\end{matrix}}\right\}$ (1)

corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a₁σ₁ + a₂σ₂ + a₃σ₃.

As a special case, it is easy to see that, if v = σ₃, this reproduces the σ₁σ₂ rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ₃ direction invariant, since

$(\cos(\theta /2)-i\sigma _{3}\sin(\theta /2))\,\sigma _{3}\,(\cos(\theta /2)+i\sigma _{3}\sin(\theta /2))=(\cos ^{2}(\theta /2)+\sin ^{2}(\theta /2))\,\sigma _{3}=\sigma _{3}.$

The bivectors σ₂σ₃, σ₃σ₁ and σ₁σ₂ are in fact Hamilton's quaternions i, j and k, discovered in 1843:

${\begin{matrix}\mathbf {i} =-\sigma _{2}\sigma _{3}=-i\sigma _{1}\\\mathbf {j} =-\sigma _{3}\sigma _{1}=-i\sigma _{2}\\\mathbf {k} =-\sigma _{1}\sigma _{2}=-i\sigma _{3}.\end{matrix}}$

With the identification of the even-graded elements with the algebra H of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself. Thus the (real) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

Note that the expression (1) for a vector rotation through an angle 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 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 of θ/2 will produce the same vector rotation, but the negative of the spinor rotation.

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

Explicit constructions

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

Component spinors

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows. Choose an orthonormal basis e¹ … eⁿ for V i.e. g(e^μe^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 γ¹ … γⁿ such that γ^μγ^ν + γ^νγ^μ = η^μν1 (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, C) by sending the monomial e^μ₁ … e^μ_k in the Clifford algebra to the product γ^μ₁ … γ^μ_k of matrices and extending linearly. The space Δ = C^2k on which the gamma matrices act is a 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, abstract spinor indices are often used to denote spinors even when an abstract spinor construction is used.

Abstract spinors

There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of 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 left-multiplication: 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. 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 n-dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification V ⊗_R C and let g denote the induced bilinear form on V ⊗_R C. 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 odd-dimensional cases W and W* have dimension k. In the odd-dimensional case, U is one-dimensional, spanned by a unit vector u.

Minimal ideals

Since W′ is isotropic, multiplication of elements of W′ inside Cℓ(V, g) is skew. Hence vectors in W′ anti-commute, and Cℓ(W′, g|_W′) = Cℓ(W′, 0) is just the exterior algebra Λ^∗W′. Consequently, the k-fold product of W′ with itself, W′^k, is one-dimensional. 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

$\omega =w'_{1}w'_{2}\cdots w'_{k}.$

Note that ω² = 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:

1. 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^evenω and Δ₋ = Cℓ^oddω on restriction to the action of the even Clifford algebra.
2. 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 non-zero 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

w_iw_j′ +w_j′ w_i = δ_ij, and

(w_i)² = 0, (w_i′)² = 0.

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

$\alpha =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}+\sum _{j}B_{j}w'_{j}$

where the a_i1…ip are scalars, and the B_j are auxiliary elements of the Clifford algebra. Observe now that the product

$\alpha \omega =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}\omega .$

Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements w_i:

$a=a_{i_{1}\dots i_{max}}w_{i_{1}}\dots w_{i_{max}}$ (no summation implied),

then

$w_{i_{max}}\cdots w_{i_{1}}\alpha \omega =a_{i_{1}\dots i_{max}}\omega$

is a nonzero scalar multiple of ω, as required.

Note that for n even, this computation also shows that

$\Delta =\mathrm {C} \ell (W)\omega =(\Lambda ^{*}W)\omega$.

as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a Fock space by creating spinors using anti-commuting creation operators in W acting on a vacuum ω.

Exterior algebra construction

The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra 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.

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

$c(v)w_{1}\wedge \cdots \wedge w_{n}=(\epsilon (w)+i(w'))\left(w_{1}\wedge \cdots \wedge w_{n}\right)$

where i(w′) is interior product with w′ using the non degenerate quadratic form to identify V with V^∗, and ε(w) denotes the exterior product. It may be verified that

c(u)c(v) + c(v)c(u) = 2 g(u,v),

and so c respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).

The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group (the half-spin representations, or Weyl spinors) via

$\Delta _{+}=\Lambda ^{even}W,\,\Delta _{-}=\Lambda ^{odd}W$.

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

$c(u)\alpha =\left\{{\begin{matrix}\alpha &{\hbox{if }}\alpha \in \Lambda ^{even}W\\-\alpha &{\hbox{if }}\alpha \in \Lambda ^{odd}W\end{matrix}}\right.$

As before, one verifies that c respects the Clifford relations, and so induces a homomorphism.

Hermitian vector spaces and spinors

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

The main example is the case that the real vector space V is a hermitian 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 V ⊗_R C 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 Λ^⋅
_CV (as well as its complex conjugate Λ^⋅
_CV) 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.

Clebsch–Gordan decomposition

A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another. 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 rank r.

In addition, for the real orthogonal groups, there are three characters (one-dimensional 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

If n = 2k is even, then the tensor product of Δ with the contragredient representation decomposes as

$\Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{n}\Gamma _{p}\cong \bigoplus _{p=0}^{k-1}\left(\Gamma _{p}\oplus \sigma \Gamma _{p}\right)\,\oplus \Gamma _{k}$

which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements αω ⊗ βω′. The rightmost formulation follows from the transformation properties of the Hodge star operator. Note that on restriction to the even Clifford algebra, the paired summands Γ_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:

$(\alpha \omega )^{*}=\omega (\alpha ^{*}).$

So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose

${\begin{matrix}\Delta _{+}\otimes \Delta _{+}^{*}\cong \Delta _{-}\otimes \Delta _{-}^{*}&\cong &\bigoplus _{p=0}^{k}\Gamma _{2p}\\\Delta _{+}\otimes \Delta _{-}^{*}\cong \Delta _{-}\otimes \Delta _{+}^{*}&\cong &\bigoplus _{p=0}^{k-1}\Gamma _{2p+1}\end{matrix}}$

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:

${\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}$

In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions

$\Delta \otimes {\bar {\Delta }}\cong \bigoplus _{p=0}^{k}\left(\sigma _{-}\Gamma _{p}\oplus \sigma _{+}\Gamma _{p}\right).$

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

• If q is even, then ${\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{+}^{*}$ and ${\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{-}^{*}.$
• If q is odd, then ${\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{-}^{*}$ and ${\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{+}^{*}.$

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

Odd dimensions

If n = 2k + 1 is odd, then

$\Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{k}\Gamma _{2p}.$

In the real case, once again the isomorphism holds

${\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}.$

Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by

$\Delta \otimes {\bar {\Delta }}\cong \sigma _{-}\Gamma _{0}\oplus \sigma _{+}\Gamma _{1}\oplus \dots \oplus \sigma _{\pm }\Gamma _{k}$

Consequences

There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are

• A manner of regarding the product of two spinors 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 electrical current of Maxwell's electromagnetic theory, or more generally a probability current.

Summary in low dimensions

• In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1-dimensional representation that does not transform.
• In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component 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 2-dimensional 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 2-component 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 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
• In 5 Euclidean dimensions, the relevant isomorphism is Spin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4-dimensional and quaternionic.
• In 6 Euclidean dimensions, the isomorphism Spin(6) ≅ SU(4) guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
• In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.
• In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality.
• In d + 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in d dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity.
• In spacetimes with p spatial and q time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the (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 non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism SL(2, C) ≅ Spin(3,1).

Metric signature	left-handed Weyl	right-handed Weyl	conjugacy	Dirac	left-handed Majorana–Weyl	right-handed Majorana–Weyl	Majorana
	complex	complex		complex	real	real	real
(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

• Anyon
• Dirac equation in the algebra of physical space
• Einstein–Cartan theory
• Pure spinor
• Spin-½
• Spinor bundle
• Supercharge
• Twistor theory

Further reading

• Brauer, Richard; Weyl, Hermann (1935), "Spinors in n 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), Bul. Soc. Math. France, 41: 53–96.
• Cartan, Élie (1966), The theory of spinors, Paris, Hermann (reprinted 1981, Dover Publications), ISBN 978-0-486-64070-9
• Chevalley, Claude (1954), The algebraic theory of spinors and Clifford algebras, Columbia University Press (reprinted 1996, Springer), ISBN 978-3-540-57063-9.
• Dirac, Paul M. (1928), "The quantum theory of the electron", Proceedings of the Royal Society of London, A117: 610–624, JSTOR 94981.
• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, vol. 129, New York: Springer-Verlag, ISBN 0-387-97495-4, MR 1153249.
• Gilkey, Peter B. (1984), Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Publish or Perish, ISBN 0-914098-20-9.
• Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0-12-329650-4.
• "Spinor", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Hitchin, Nigel J. (1974), "Harmonic spinors", Advances in Mathematics, 14: 1–55, doi:10.1016/0001-8708(74)90021-8, MR 0358873.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton University Press, ISBN 0-691-08542-0.
• Pauli, Wolfgang (1927), "Zur Quantenmechanik des magnetischen Elektrons", Zeitschrift für Physik, 43 (9–10): 601–632, Bibcode:1927ZPhy...43..601P, doi:10.1007/BF01397326.
• Penrose, Roger; Rindler, W. (1988), Spinors and Space–Time: Volume 2, Spinor and Twistor Methods in Space–Time Geometry, Cambridge University Press, ISBN 0-521-34786-6.
• 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