Physics:Spinor

Short description: Non-tensorial representation of the spin group; represents fermions in physics

A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.

In geometry and physics, spinors /spɪnər/ are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).

It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.

Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as illustrated by the belt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class. It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO(3).

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. A Clifford space operates on a spinor space, and the elements of a spinor space are spinors. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

Introduction

A gradual rotation can be visualized as a ribbon in space. Two gradual rotations with different classes, one through 360° and one through 720° 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 360° rotation, but possible with the 720° rotation. A solution, shown in the second animation, gives an explicit homotopy in the rotation group between the 720° rotation and the 0° identity rotation.

File:Belt Trick.ogv File:Antitwister.ogv

What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system (for instance, the stress of some medium) also has coordinate descriptions that adjust to compensate for changes to the coordinate system itself.

Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually (continuously) rotated between some initial and final configuration. For any of the familiar and intuitive ("tensorial") quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration. Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: They exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are actually two ("topologically") inequivalent gradual (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called the homotopy class of the gradual rotation. The belt trick puzzle (shown) demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors actually exhibit a sign-reversal that genuinely depends on this homotopy class. This distinguishes them from vectors and other tensors, none of which can feel the class.

Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors).

More generally, a Clifford algebra can be constructed from any vector space V equipped with a (nondegenerate) quadratic form, such as Euclidean space with its standard dot product or Minkowski space with its standard Lorentz metric. The space of spinors is the space of column vectors with [math]\displaystyle{ 2^{\lfloor\dim V/2\rfloor} }[/math] components. The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group. Depending on the dimension and metric signature, this realization of spinors as column vectors may be irreducible or it may decompose into a pair of so-called "half-spin" or Weyl representations. When the vector space V is four-dimensional, the algebra is described by the gamma matrices.

Mathematical definition

The space of spinors is formally defined as the fundamental representation of the Clifford algebra. (This may or may not decompose into irreducible representations.) The space of spinors may also be defined as a spin representation of the orthogonal Lie algebra. These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional group representation of the spin group on which the center acts non-trivially.

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.

Representation theoretic point of view

From a representation theoretic point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors. From this view, a spinor must belong to a representation of the double cover of the rotation group SO(n,[math]\displaystyle{ \Reals }[/math]), or more generally of a double cover of the generalized special orthogonal group SO⁺(p, q, [math]\displaystyle{ \Reals }[/math]) on spaces with a metric signature of (p, q). These double covers are Lie groups, called the spin groups Spin(n) or Spin(p, q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued projective representations of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.)

In summary, given a representation specified by the data [math]\displaystyle{ (V,\text{Spin}(p,q), \rho) }[/math] where [math]\displaystyle{ V }[/math] is a vector space over [math]\displaystyle{ K = \mathbb{R} }[/math] or [math]\displaystyle{ \mathbb{C} }[/math] and [math]\displaystyle{ \rho }[/math] is a homomorphism [math]\displaystyle{ \rho:\text{Spin}(p,q)\rightarrow \text{GL}(V) }[/math], a spinor is an element of the vector space [math]\displaystyle{ V }[/math].

Geometric point of view

From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as Fierz identities, are needed.

Clifford algebras

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 symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = [math]\displaystyle{ \Complex^n }[/math], with the standard form g(x, y) = x^Ty = x₁y₁ + ... + x_ny_n we denote the Clifford algebra by Cℓ_n([math]\displaystyle{ \Complex }[/math]). 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_{[math]\displaystyle{ \Complex }[/math]}(V) = n. If n = 2k is even, Cℓ_n([math]\displaystyle{ \Complex }[/math]) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2^k, [math]\displaystyle{ \Complex }[/math]) 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([math]\displaystyle{ \Complex }[/math]) is isomorphic to the algebra Mat(2^k, [math]\displaystyle{ \Complex }[/math]) ⊕ Mat(2^k, [math]\displaystyle{ \Complex }[/math]) 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.

Spin groups

The spin representation Δ is a vector space equipped with a representation of the spin group that does not factor through a representation of the (special) orthogonal group. The vertical arrows depict a short exact sequence.

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.

Spinor fields in physics

The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional space-time. To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4-dimensional space-time (Minkowski space). Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the spin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fiber bundle, the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation, or the Weyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that are plane waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called fermions; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another.

It appears that all fundamental particles in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the neutrino. There does not seem to be any a priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of Cℓ_2,2([math]\displaystyle{ \Reals }[/math]), the Majorana spinor. There also does not seem to be any particular prohibition to having Weyl spinors appear in nature as fundamental particles.

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

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

The situation for condensed matter physics is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors to far more exotic materials. In 2015, an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.

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". Stated differently:

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

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 Gustave Juvett 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, [math]\displaystyle{ \Complex }[/math]).

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([math]\displaystyle{ \Reals }[/math]) of the spacetime algebra Cℓ_1,3([math]\displaystyle{ \Reals }[/math]). 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([math]\displaystyle{ \Reals }[/math]). 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 [math]\displaystyle{ e_ie_j = \begin{cases} +1 & i=j, \, i \in (1, \ldots, p) \\ -1 & i=j, \, i \in (p+1, \ldots, n) \\ -e_j e_i & i \neq j. \end{cases} }[/math]

Two dimensions

The Clifford algebra Cℓ_2,0([math]\displaystyle{ \Reals }[/math]) 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([math]\displaystyle{ \Reals }[/math]), spanned by even-graded basis elements of Cℓ_2,0([math]\displaystyle{ \Reals }[/math]), 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([math]\displaystyle{ \Reals }[/math]) is isomorphic to the field of complex numbers  [math]\displaystyle{ \Complex }[/math]. As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by [math]\displaystyle{ (a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1. }[/math] which, by the Clifford relations, can be written [math]\displaystyle{ (a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2. }[/math]

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

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: [math]\displaystyle{ \gamma(u) = \gamma u \gamma^* = \gamma^2 u. }[/math]

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

Examples

• The even-graded element [math]\displaystyle{ \gamma = \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) }[/math] corresponds to a vector rotation of 90° from σ₁ around towards σ₂, which can be checked by confirming that [math]\displaystyle{ \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 }[/math] It corresponds to a spinor rotation of only 45°, however: [math]\displaystyle{ \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 }[/math]
• Similarly the even-graded element γ = −σ₁σ₂ corresponds to a vector rotation of 180°: [math]\displaystyle{ (- \sigma_1 \sigma_2)\{a_1\sigma_1 + a_2\sigma_2\} (- \sigma_2 \sigma_1) = - a_1\sigma_1 -a_2\sigma_2 }[/math] but a spinor rotation of only 90°:[math]\displaystyle{ (- \sigma_1 \sigma_2) \{a_1 + a_2\sigma_1\sigma_2\} = a_2 - a_1\sigma_1\sigma_2 }[/math]
• Continuing on further, the even-graded element γ = −1 corresponds to a vector rotation of 360°: [math]\displaystyle{ (-1) \{a_1\sigma_1+a_2\sigma_2\} \, (-1) = a_1\sigma_1+a_2\sigma_2 }[/math] but a spinor rotation of 180°.

Three dimensions

The Clifford algebra Cℓ_3,0([math]\displaystyle{ \Reals }[/math]) 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, [math]\displaystyle{ u' = \rho^{\left(\frac{1}{2}\right)} u \rho^{\left(\frac{1}{2}\right)} = \rho u, }[/math] and vector rotations [math]\displaystyle{ u' = \gamma u\gamma^*, }[/math] where

[math]\displaystyle{ \left.\begin{align} \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{align}\right\} }[/math]

(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

[math]\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. }[/math]

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

[math]\displaystyle{ \begin{align} \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{align} }[/math]

With the identification of the even-graded elements with the algebra [math]\displaystyle{ \mathbb{H} }[/math] 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 is a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see spinors in three dimensions.

Component spinors

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, [math]\displaystyle{ \Complex }[/math]) 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 [math]\displaystyle{ \Delta = \Complex^{2^k} }[/math] 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. 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 [math]\displaystyle{ V \otimes_\Reals \Complex }[/math] and let g denote the induced bilinear form on [math]\displaystyle{ V \otimes_\Reals \Complex }[/math]. 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 [math]\displaystyle{ \omega = w'_1 w'_2 \cdots w'_k. }[/math]

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 [math]\displaystyle{ \alpha = \sum_{i_1\lt i_2\lt \cdots\lt i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} + \sum_j B_j w'_j }[/math] where the a_i1...ip are scalars, and the B_j are auxiliary elements of the Clifford algebra. Observe now that the product [math]\displaystyle{ \alpha\omega = \sum_{i_1\lt i_2\lt \cdots\lt i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p}\omega. }[/math] Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements w_i: [math]\displaystyle{ a = a_{i_1\dots i_\text{max}}w_{i_1}\dots w_{i_\text{max}} }[/math] (no summation implied), then [math]\displaystyle{ w'_{i_\text{max}}\cdots w'_{i_1}\alpha\omega = a_{i_1\dots i_\text{max}}\omega }[/math] is a nonzero scalar multiple of ω, as required.

Note that for n even, this computation also shows that [math]\displaystyle{ \Delta = \mathrm{C}\ell(W)\omega = \left(\Lambda^* W\right)\omega }[/math] 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 [math]\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) }[/math] 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 [math]\displaystyle{ c(u)\,c(v) + c(v)\,c(u) = 2\,g(u,v)\,, }[/math] 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 [math]\displaystyle{ \Delta_+ = \Lambda^\text{even} W,\, \Delta_- = \Lambda^\text{odd} W. }[/math]

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 [math]\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} }[/math] 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 [math]\displaystyle{ V \otimes_\Reals \Complex }[/math] 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 [math]\displaystyle{ \Lambda^\cdot_\Complex \bar V }[/math] (as well as its complex conjugate [math]\displaystyle{ \Lambda^\cdot_\Complex V }[/math] 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 [math]\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 }[/math] 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: [math]\displaystyle{ (\alpha\omega)^* = \omega\left(\alpha^*\right). }[/math] So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose [math]\displaystyle{ \begin{align} \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{align} }[/math]

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: [math]\displaystyle{ \bar{\Delta} \cong \sigma_-\Delta^* }[/math]

In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions [math]\displaystyle{ \Delta \otimes\bar{\Delta} \cong \bigoplus_{p=0}^k\left(\sigma_-\Gamma_p \oplus \sigma_+\Gamma_p\right). }[/math]

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

• If q is even, then [math]\displaystyle{ \bar{\Delta}_+ \cong \sigma_- \otimes \Delta_+^* }[/math] and [math]\displaystyle{ \bar{\Delta}_- \cong \sigma_- \otimes \Delta_-^*. }[/math]
• If q is odd, then [math]\displaystyle{ \bar{\Delta}_+ \cong \sigma_- \otimes \Delta_-^* }[/math] and [math]\displaystyle{ \bar{\Delta}_- \cong \sigma_- \otimes \Delta_+^*. }[/math]

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 [math]\displaystyle{ \Delta\otimes\Delta^* \cong \bigoplus_{p=0}^k \Gamma_{2p}. }[/math] In the real case, once again the isomorphism holds [math]\displaystyle{ \bar{\Delta} \cong \sigma_-\Delta^*. }[/math] Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by [math]\displaystyle{ \Delta \otimes \bar{\Delta} \cong \sigma_-\Gamma_0\oplus\sigma_+\Gamma_1\oplus\dots\oplus\sigma_\pm\Gamma_k }[/math]

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.

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, [math]\displaystyle{ \Complex }[/math]) ≅ Spin(3,1).

Metric signature	Weyl, complex		Conjugacy	Dirac, complex	Majorana–Weyl, real		Majorana, real
	Left-handed	Right-handed			Left-handed	Right-handed
(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
• Eigenspinor
• Einstein–Cartan theory
• Projective representation
• Pure spinor
• Spin-1/2
• Spinor bundle
• Supercharge
• Twistor theory

Further reading

• 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. 
• Cartan, Élie (1913). "Les groupes projectifs qui ne laissent invariante aucune multiplicité plane". Bull. Soc. Math. Fr. 41: 53–96. doi:10.24033/bsmf.916. http://archive.numdam.org/article/BSMF_1913__41__53_1.pdf. 
• Cartan, Élie (1981). The Theory of Spinors (reprint ed.). Paris, FR: Hermann (1966); Dover Publications (1981). ISBN 978-0-486-64070-9. 
• Chevalley, Claude (1996). The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). ISBN 978-3-540-57063-9. 
• Dirac, Paul M. (1928). "The quantum theory of the electron". Proceedings of the Royal Society of London A 117 (778): 610–624. doi:10.1098/rspa.1928.0023. Bibcode: 1928RSPSA.117..610D. 
• Fulton, William; Harris, Joe (1991). Representation Theory: A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. 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. 
• Harvey, F. Reese (1990). Spinors and Calibrations. Academic Press. ISBN 978-0-12-329650-4. 
• Hitchin, Nigel J. (1974). "Harmonic spinors". Advances in Mathematics 14: 1–55. doi:10.1016/0001-8708(74)90021-8. 
• 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. doi:10.1007/BF01397326. Bibcode: 1927ZPhy...43..601P. 
• Penrose, Roger; Rindler, W. (1988). Spinor and twistor methods in space-time geometry. Spinors and Space-Time. 2. 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.