Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors. However, spinors transform well under the infinitesimal orthogonal transformations (like infinitesimal rotations or infinitesimal Lorentz transformations). Under the full orthogonal group, however, they do not quite transform well, but only "up to a sign". This means that a 360 degree rotation transforms a spinor into its negative, and so it takes a rotation of 720 degrees for a spinor to be transformed into itself. Specifically, spinors are objects associated to a vector space with a quadratic form (like Euclidean space with the standard metric or Minkowski space with the Lorentz metric), and are realized as elements of representation spaces of Clifford algebras. For a given quadratic form, several different spaces of spinors with extra properties may exist.

Spinors in general were discovered by Élie Cartan in 1913.^[1] Later, spinors were adopted by quantum mechanics in order to study the properties of the intrinsic angular momentum of the electron and other fermions. Today spinors enjoy a wide range of physics applications. Classically, spinors in three dimensions are used to describe the spin of the non-relativistic electron and other spin-½ particles. Via the Dirac equation, Dirac spinors are required in the mathematical description of the quantum state of the relativistic electron. In quantum field theory, spinors describe the state of relativistic many-particle systems. In mathematics, particularly in differential geometry and global analysis, spinors have since found broad applications to algebraic and differential topology,^[2] symplectic geometry, gauge theory, complex algebraic geometry,^[3] index theory,^[4] and special holonomy.^[5]

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 a priori that there are some representations of the Lie algebra of the orthogonal group which 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 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(p,q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group.

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

The language of Clifford algebras^[6] (also 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, if V is 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. The Clifford algebra Cℓ_n(C) is algebraically isomorphic to the algebra Mat(2^k,C) of 2^k × 2^k complex matrices, if n = dim(V) = 2k is even; or the algebra Mat(2^k,C)⊕Mat(2^k,C) of two copies of the 2^k × 2^k matrices, if n = dim(V) = 2k+ 1 is odd. It therefore has a unique irreducible representation (also called simple Clifford module), commonly denoted by Δ, whose dimension is 2^k. 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. Therefore, the space Δ is also a Lie algebra representation of so(V,g) called a spin representation. If n is odd, this representation is irreducible. If n is even, it splits again into two irreducible representations Δ = Δ₊ ⊕ Δ₋ called the 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,^[7] is an element of the fundamental representation of the complexified Clifford algebra Cℓ(p,q), 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^[8] representations. In addition, sometimes the non-complexified version of Cℓ(p,q) has a smaller real representation, the Majorana spinor representation.^[9] 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.

Spinors in representation theory

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 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.

History

The most general mathematical form of spinors was discovered by Élie Cartan in 1913.^[10] The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.^[11]

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced spin matrices.^[12] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.^[13] By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute created games such as Tangloids to teach and model the calculus of spinors.

Examples

Some important 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 on vectors, regarded as 1-graded elements of Cℓ_2,0, 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^[14] 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 (which 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

$\left.\begin{matrix} \gamma & = & \cos(\theta/2) - \{a_1 \sigma_2\sigma_3 + a_2 \sigma_3\sigma_1 + a_3 \sigma_1\sigma_2\} \sin(\theta/2) \\ & = & \cos(\theta/2) - i \{a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3\} \sin(\theta/2) \\ & = & \cos(\theta/2) - i v \sin(\theta/2) \end{matrix}\right\}$ (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.^[15] Thus the (real^[16]) 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 $k = \lfloor n/2 \rfloor$ . Fix a set of 2^k by 2^k matrices $\gamma^1 \ldots \gamma^n$ such that $\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = \eta^{\mu\nu}1$ (i.e. fix a convention for the gamma matrices). Then the assignment $e^\mu \to \gamma^\mu$ extends uniquely to an algebra homomorphism Cℓ(V,g) $\to$ Mat(2^k,C) by sending the monomial $e^{\mu_1}...e^{\mu_k}$ in the Clifford algebra to the product $\gamma^{\mu_1}...\gamma^{\mu_k}$ of matrices and extending linearly. The space Δ = C^{2^k} 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 by 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 which is a nilpotent element of the Clifford algebra, or one which is an idempotent. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.^[17] 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 which 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 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′₁ of in W′, one possibility is to set

$\omega=w'_1w'_2\cdots w'_k.$

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:

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.
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 in fact 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

$\alpha = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} + \sum_j B_jw'_j$

where the a_{i₁...i_p} 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

$\Delta = Cl(W)\omega = (\wedge^* W)\omega$ .

as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that $\Delta$ 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 $\wedge^* W = \oplus_j \wedge^j W$ of the isotropic subspace W. Let $\Delta= \wedge^* W$ 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.^[18]

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 is easily 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^[19] (the half-spin representations, or Weyl spinors) via

$\Delta_+ = \wedge^{even} W,\, \Delta_- = \wedge^{odd} W$ .

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

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

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 which 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 which is an orthogonal transformation with respect to the inner product g on V. Then $V \otimes_{\mathbb{R}} \mathbb{C}$ splits in the $\pm i$ 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 $\wedge^\cdot_{\mathbb{C}} \bar V$ (as well as its complex conjugate $\wedge^\cdot_{\mathbb{C}} V$ ) is a spinor space for the underlying real euclidean vector space.

With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold (in particular every symplectic manifold) has a SpinC structure. Likewise, every complex vector bundle on a manifold carries a SpinC structure.^[20]

Clebsch-Gordan decomposition

A number of Clebsch-Gordan decompositions are possible on the tensor product of one spin representation with another.^[21] 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 = ∧^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 $\bar{\Delta}$ 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 $\Delta_\pm\otimes\bar{\Delta}_\pm$ .

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 $\bar{\phi}\psi$ as a scalar. In physical terms, a spinor should determine a probability amplitude for the quantum state.
A manner of regarding the product $\psi\bar{\phi}$ 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 $\psi v\bar{\psi}$ ;. 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 $e^{\pm i\phi/2}$ under a rotation by angle $\phi$ .
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 $SU(2) \cong \mathit{Spin}(3)$ which 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 $\mathit{Spin}(4) \equiv SU(2) \times SU(2)$ . 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 $\mathit{Spin}(5)\equiv USp(4)\equiv Sp(2)$ which implies that the single spinor representation is 4-dimensional and quaternionic.
In 6 Euclidean dimensions, the isomorphism $\mathit{Spin}(6)\equiv SU(4)$ 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 $\mathit{SL}(2,C) \equiv \mathit{Spin}(3,1)$ .

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

Notes

↑ Cartan 1913.
↑ Hitchin 1974, Lawson & Michelsohn 1989.
↑ Hitchin 1974, Penrose & Rindler 1988.
↑ Gilkey 1984, Lawson & Michelsohn 1989.
↑ Lawson & Michelsohn 1989, Harvey 1990. These two books also provide good mathematical introductions and fairly comprehensive bibliographies on the mathematical applications of spinors as of 1989–1990.
↑ Named after William Kingdon Clifford,
↑ Named after Paul Dirac.
↑ Named after Hermann Weyl.
↑ Named after Ettore Majorana.
↑ Cartan 1913
↑ Tomonaga 1998, p. 129
↑ Pauli 1927.
↑ Dirac 1928.
↑ These are the right-handed Weyl spinors in two-dimensions. For the left-handed Weyl spinors, the representation is via $\gamma(\phi)=\bar{\gamma}\phi$ . 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 skew-field. In this case, there are a pair of equivalent representations: $\gamma(\phi)=\gamma\phi$ , and its quaternionic conjugate $\gamma(\phi)=\phi\bar{\gamma}$ .
↑ The complex spinors are obtained as the representations of the tensor product H⊗_RC = Mat₂(C). These are considered in more detail in spinors in three dimensions.
↑ This construction is due to Cartan. The treatment here is based on Chevalley (1954).
↑ One source for this subsection is Fulton & Harris (1991).
↑ Via the even-graded Clifford algebra.
↑ Lawson & Michelsohn 1989, Appendix D.
↑ Brauer & Weyl 1935.

References

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, http://jstor.org/stable/2371218 .
Cartan, Élie (1913), "Les groupes projectifs qui ne laissent invariante aucune multiplicité plane", Bul. Soc. Math. France 41: 53–96, http://archive.numdam.org/article/BSMF_1913__41__53_1.pdf .
Cartan, Élie (1966), The theory of spinors, Paris, Hermann (reprinted 1981, Dover Publications), ISBN 978-0486640709
Chevalley, Claude (1954), The algebraic theory of spinors and Clifford algebras, Columbia University Press (reprinted 1996, Springer), ISBN 978-3540570639 .
Dirac, Paul M. (1928), "The quantum theory of the electron", Proceedings of the Royal Society of London A117: 610–624 .
Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR 1153249, ISBN 0-387-97527-6, ISBN 0-387-97495-4 .
Gilkey, Peter B. (1984), Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, ISBN 0-914098-20-9, http://www.emis.de/monographs/gilkey/index.html .
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Hitchin, Nigel J. (1974), "Harmonic spinors", Advances in Mathematics 14: 1–55, doi:10.1016/0001-8708(74)90021-8, MR 50_#11332 .
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Pauli, Wolfgang (1927), "Zur Quantenmechanik des magnetischen Elektrons", Zeitschrift für Phisik 43: 601–632, 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