Spinor

Overview

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

Clifford algebras

The language of Clifford algebras^[2] (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 symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V subject to 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, one writes Cℓ_n(ℂ) for this Clifford algebra; since every nondegenerate symmetric bilinear form on a complex vector space is equivalent to the standard one, this notation is often used whenever dim_ℂ(V) = n. If n = 2k is even, then Cℓ_n(ℂ) is isomorphic, noncanonically, to Mat(2^k, ℂ), so it has a unique irreducible module Δ of dimension 2^k. If n = 2k + 1 is odd, then Cℓ_n(ℂ) is isomorphic to Mat(2^k, ℂ) ⊕ Mat(2^k, ℂ), and therefore has two inequivalent irreducible modules, each of dimension 2^k. The Lie algebra so(V, g) embeds in the even part of the Clifford algebra, equipped with the commutator bracket, and hence acts on these modules. When n is odd, the two irreducible Clifford modules restrict to isomorphic irreducible representations of so(V, g); this representation is called the spin representation and is often denoted Δ. When n is even, the unique irreducible Clifford module Δ remains irreducible for the full Clifford algebra, but on restriction to the even Clifford algebra, or equivalently to the spin group, it splits as $${\displaystyle \Delta =\Delta _{+}\oplus \Delta _{-},}$$ where Δ₊ and Δ₋ are 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.

Spin groups

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 one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product, or Minkowski space with its Lorentz metric. In the latter case, the "rotations" include the Lorentz boosts, but otherwise the theory is substantially similar.^{[citation needed]}

Spinor fields in physics

In physics, a spinor field is a field whose values lie in a spinor representation. On Minkowski space, or more generally on a spacetime manifold that admits a spin structure, one forms a spinor bundle associated to the principal spin bundle and a chosen spin representation; spinor fields are sections of this bundle. In flat spacetime this bundle may be trivialized, so spinor fields are often written simply as spinor-valued functions on spacetime.

The most common spinor fields in relativistic physics are Dirac, Weyl, and Majorana spinor fields. A Dirac spinor is a section of the full complex spinor bundle. In even dimensions, when the spin representation splits into chiral halves, sections of the two summands are called Weyl spinors. A Majorana spinor is a spinor satisfying a reality condition, when the relevant spin representation admits one.

Spinor fields enter physics through equations such as the Dirac equation and the Weyl equation, which are first-order differential equations on the spinor bundle. These equations describe relativistic fields of spin 1⁄2 and play a central role in quantum field theory and differential geometry. For further details, see Dirac spinor, Weyl spinor, Majorana spinor, and spinor bundle.

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 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".^[3] Stated differently:

Spinors ... provide a linear representation of the group of rotations in a space with any number ${\displaystyle n}$ of dimensions, each spinor having ${\displaystyle 2^{\nu }}$ components where ${\displaystyle n=2\nu +1}$ or ${\displaystyle 2\nu }$.^[4]

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.^[5]

Two dimensions

The Clifford algebra Cℓ_2,0(${\displaystyle \mathbb {R} }$) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ₁ 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(${\displaystyle \mathbb {R} }$), spanned by even-graded basis elements of Cℓ_2,0(${\displaystyle \mathbb {R} }$), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ₁σ₂. As a real algebra, Cℓ⁰_2,0(${\displaystyle \mathbb {R} }$) is isomorphic to the field of complex numbers ${\displaystyle \mathbb {C} }$. As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by $${\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}}$$ which, by the Clifford relations, can be written $${\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}.}$$

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

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

On the other hand, in comparison with its action on spinors ${\displaystyle \gamma (\phi )=\gamma \phi }$, the action of ${\displaystyle \gamma }$ 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 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.

Three dimensions

The Clifford algebra Cℓ_3,0(${\displaystyle \mathbb {R} }$) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ₁, σ₂ 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, $${\displaystyle u'=\rho ^{\left({\frac {1}{2}}\right)}u\rho ^{\left({\frac {1}{2}}\right)}=\rho u,}$$ and vector rotations $${\displaystyle u'=\gamma u\gamma ^{*},}$$ where

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

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

$${\displaystyle \left[\cos \left({\frac {\theta }{2}}\right)-i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]\sigma _{3}\left[\cos \left({\frac {\theta }{2}}\right)+i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]=\left[\cos ^{2}\left({\frac {\theta }{2}}\right)+\sin ^{2}\left({\frac {\theta }{2}}\right)\right]\sigma _{3}=\sigma _{3}.}$$

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

$${\displaystyle {\begin{aligned}\mathbf {i} &=-\sigma _{2}\sigma _{3}=-i\sigma _{1}\\\mathbf {j} &=-\sigma _{3}\sigma _{1}=-i\sigma _{2}\\\mathbf {k} &=-\sigma _{1}\sigma _{2}=-i\sigma _{3}\end{aligned}}}$$

With the identification of the even-graded elements with the algebra ${\displaystyle \mathbb {H} }$ of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.^[f] Thus the (real^[g]) 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.

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 = 2η^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 ${\displaystyle \Delta =\mathbb {C} ^{2^{k}}}$ on which the gamma matrices act is now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be 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

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.^[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 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 ${\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }$ and let g denote the induced bilinear form on ${\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }$. 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 W′, one possibility is to set $${\displaystyle \omega =w'_{1}w'_{2}\cdots w'_{k}.}$$

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

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 $${\displaystyle \alpha =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}+\sum _{j}B_{j}w'_{j}}$$ where the a_i1...ip are scalars, and the B_j are auxiliary elements of the Clifford algebra. Observe now that the product $${\displaystyle \alpha \omega =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}\omega .}$$ Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements w_i: $${\displaystyle a=a_{i_{1}\dots i_{\text{max}}}w_{i_{1}}\dots w_{i_{\text{max}}}}$$ (no summation implied), then $${\displaystyle w'_{i_{\text{max}}}\cdots w'_{i_{1}}\alpha \omega =a_{i_{1}\dots i_{\text{max}}}\omega }$$ is a nonzero scalar multiple of ω, as required.

Note that for n even, this computation also shows that $${\displaystyle \Delta =\mathrm {C} \ell (W)\omega =\left(\Lambda ^{*}W\right)\omega }$$ 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.^[18][19]

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 $${\displaystyle c(v)w_{1}\wedge \cdots \wedge w_{n}=\left(\epsilon (w)+i\left(w'\right)\right)\left(w_{1}\wedge \cdots \wedge w_{n}\right)}$$ where i(w′) is interior product with w′ using the nondegenerate quadratic form to identify V with V^∗, and ε(w) denotes the exterior product. This action is sometimes called the Clifford product. It may be verified that $${\displaystyle c(u)\,c(v)+c(v)\,c(u)=2\,g(u,v)\,,}$$ 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^[20] (the half-spin representations, or Weyl spinors) via $${\displaystyle \Delta _{+}=\Lambda ^{\text{even}}W,\,\Delta _{-}=\Lambda ^{\text{odd}}W.}$$

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 $${\displaystyle c(u)\alpha ={\begin{cases}\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{even}}W\\-\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{odd}}W\end{cases}}}$$ 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, g), 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 ${\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }$ 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, g) the vector space ${\displaystyle \Lambda _{\mathbb {C} }^{\cdot }{\bar {V}}}$ (as well as its complex conjugate ${\displaystyle \Lambda _{\mathbb {C} }^{\cdot }V}$) is a spinor space for the underlying real euclidean vector space.

With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of 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.^[21]

Even dimensions

If n = 2k is even, then the tensor product of Δ with the contragredient representation decomposes as $${\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{n}\Gamma _{p}\cong \bigoplus _{p=0}^{k-1}\left(\Gamma _{p}\oplus \sigma \Gamma _{p}\right)\oplus \Gamma _{k}}$$ 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: $${\displaystyle (\alpha \omega )^{*}=\omega \left(\alpha ^{*}\right).}$$ So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose $${\displaystyle {\begin{aligned}\Delta _{+}\otimes \Delta _{+}^{*}\cong \Delta _{-}\otimes \Delta _{-}^{*}&\cong \bigoplus _{p=0}^{k}\Gamma _{2p}\\\Delta _{+}\otimes \Delta _{-}^{*}\cong \Delta _{-}\otimes \Delta _{+}^{*}&\cong \bigoplus _{p=0}^{k-1}\Gamma _{2p+1}\end{aligned}}}$$

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

In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions $${\displaystyle \Delta \otimes {\bar {\Delta }}\cong \bigoplus _{p=0}^{k}\left(\sigma _{-}\Gamma _{p}\oplus \sigma _{+}\Gamma _{p}\right).}$$

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

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

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 $${\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{k}\Gamma _{2p}.}$$ In the real case, once again the isomorphism holds $${\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}.}$$ Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by $${\displaystyle \Delta \otimes {\bar {\Delta }}\cong \sigma _{-}\Gamma _{0}\oplus \sigma _{+}\Gamma _{1}\oplus \dots \oplus \sigma _{\pm }\Gamma _{k}}$$

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 electric current of Maxwell's electromagnetic theory, or more generally a probability current.

Reality type and signature

One source of confusion is that the words real, complex, and quaternionic are used in two closely related but not identical senses: they may refer to the ground field of a chosen Clifford module, or to the type (also called the Schur type) of an irreducible spin representation. For the spin group, the second notion is the useful one.

Let ${\displaystyle S}$ be an irreducible complex spin representation of ${\displaystyle \operatorname {Spin} (p,q)}$. Then ${\displaystyle S}$ is said to be

• of real type if there is a ${\displaystyle \operatorname {Spin} (p,q)}$-equivariant antilinear map ${\displaystyle J:S\to S}$ with ${\displaystyle J^{2}=+1}$;
• of quaternionic type (or pseudoreal) if there is such a map with ${\displaystyle J^{2}=-1}$;
• of complex type if no such equivariant antilinear map exists, equivalently if ${\displaystyle S}$ and its complex conjugate ${\displaystyle {\overline {S}}}$ are inequivalent.

When ${\displaystyle S}$ is of real type, the fixed-point set of ${\displaystyle J}$ gives a real form of the representation; this is the algebraic origin of Majorana conditions. When ${\displaystyle S}$ is of quaternionic type, the representation carries an invariant quaternionic structure but no invariant real structure on an irreducible complex module.^[23]

The reason for the mod-8 pattern is that spinors are naturally modules for the even Clifford algebra ${\displaystyle \mathrm {C} \ell _{p,q}^{0}}$. If ${\displaystyle S}$ is an irreducible real ${\displaystyle \mathrm {C} \ell _{p,q}^{0}}$-module, then by Schur's lemma its commuting algebra $${\displaystyle \operatorname {End} _{\mathrm {C} \ell _{p,q}^{0}}(S)}$$ is a finite-dimensional real division algebra. By the Frobenius theorem, it is therefore isomorphic to exactly one of $${\displaystyle \mathbb {R} ,\qquad \mathbb {C} ,\qquad \mathbb {H} .}$$ These three possibilities are precisely the real, complex, and quaternionic types. Using the standard identification of even Clifford algebras with Clifford algebras in one lower dimension, together with Bott periodicity, one finds that the type depends only on ${\displaystyle p-q{\pmod {8}}}$.^[24]

More information Dimension ...

Reality type of irreducible spin representations

Dimension ${\displaystyle n=p+q}$	${\displaystyle p-q{\pmod {8}}}$	Irreducible complex spin representation(s)	Type	Consequence
odd	1, 7	${\displaystyle S}$	real	${\displaystyle S}$ admits an invariant real structure
odd	3, 5	${\displaystyle S}$	quaternionic	${\displaystyle S}$ admits an invariant quaternionic structure
even	0	${\displaystyle S^{+},S^{-}}$	real	each Weyl representation admits a real structure; Majorana–Weyl spinors are possible
even	2, 6	${\displaystyle S^{+},S^{-}}$	complex	${\displaystyle S^{-}\cong {\overline {S^{+}}}}$; a single Weyl representation is neither real nor quaternionic
even	4	${\displaystyle S^{+},S^{-}}$	quaternionic	each Weyl representation admits an invariant quaternionic structure

Close

Thus, for example, spinors in 3-dimensional Euclidean space are quaternionic, Weyl spinors in 4-dimensional Euclidean space are quaternionic, Weyl spinors in Lorentzian signature ${\displaystyle (3,1)}$ are complex conjugates of one another, Weyl spinors in split signature ${\displaystyle (1,1)}$ are real, and 8-dimensional Euclidean Weyl spinors are real.

A common source of apparent disagreement between tables in the literature is that some authors classify irreducible modules for the full real Clifford algebra ${\displaystyle \mathrm {C} \ell _{p,q}}$, while others classify spin representations of ${\displaystyle \operatorname {Spin} (p,q)}$. Since the spin group sits in the even Clifford algebra, these two tables differ by a shift of one step in the mod-8 pattern.