# Spinor

In geometry and physics, **spinors** /spɪnɔːr/ are elements of a (complex) vector space that can be associated with Euclidean space.^{[nb 2]} Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation.^{[nb 3]} However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used: *unlike* vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors. It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.^{[1]}^{[2]} In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.^{[nb 4]}

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 famously illustrated by the belt trick puzzle (below). These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.^{[nb 5]} 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.^{[nb 6]} 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.^{[nb 7]} 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,^{[nb 8]} 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^{[nb 9]}) 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.^{[nb 10]}

## Contents

## IntroductionEdit

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) famously 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,^{[nb 12]} realizing it as a group of rotations among them,^{[nb 13]} 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. Given a suitably normalized basis of *V*, the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations, and the space of spinors is the space of column vectors with components on which those matrices act. Although the Clifford algebra can be defined abstractly in a coordinate-independent way, its particular realization as a specific algebra of matrices depends on which orthogonal axes the gamma matrices represent. So what precisely constitutes a "column vector" (or spinor) also depends on such arbitrary choices.^{[nb 14]} 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.^{[nb 15]} 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.^{[nb 16]}

## Mathematical definitionEdit

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.

### OverviewEdit

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

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

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 algebrasEdit

The language of Clifford algebras^{[3]} (sometimes called geometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras. It largely removes the need for *ad hoc* constructions.

In detail, let *V* be a finite-dimensional complex vector space with nondegenerate bilinear form *g*. The Clifford algebra Cℓ(*V*, *g*) is the algebra generated by *V* along with the anticommutation relation *xy* + *yx* = 2*g*(*x*, *y*). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If *V* = **C**^{n}, with the standard form *g*(*x*, *y*) = *x*^{t}*y* = *x*_{1}*y*_{1} + ... + *x*_{n}*y*_{n} we denote the Clifford algebra by Cℓ_{n}(**C**). Since by the choice of an orthonormal basis every complex vectorspace with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dim_{C}(*V*) = *n*. If *n* = 2*k* is even, Cℓ_{n}(**C**) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2^{k}, **C**) of 2^{k} × 2^{k} complex matrices (by the Artin-Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple). If *n* = 2*k* + 1 is odd, Cℓ_{2k+1}(**C**) is isomorphic to the algebra Mat(2^{k}, **C**) ⊕ Mat(2^{k}, **C**) of two copies of the 2^{k} × 2^{k} complex matrices. Therefore, in either case Cℓ(*V*, *g*) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2^{[n/2]}. Since the Lie algebra **so**(*V*, *g*) is embedded as a Lie subalgebra in Cℓ(*V*, *g*) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of **so**(*V*, *g*) called a spin representation. If *n* is odd, this Lie algebra representation is irreducible. If *n* is even, it splits further into two irreducible representations Δ = Δ_{+} ⊕ Δ_{−} called the Weyl or *half-spin representations*.

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

### Spin groupsEdit

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.

### Terminology in physicsEdit

The most typical type of spinor, the Dirac spinor,^{[4]} is an element of the fundamental representation of Cℓ_{p+q}(**C**), the complexification of the Clifford algebra Cℓ_{p, q}(**R**), into which the spin group Spin(*p*, *q*) may be embedded. On a 2*k*- or 2*k*+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**^{[5]} representations. In addition, sometimes the non-complexified version of Cℓ_{p,q}(**R**) has a smaller real representation, the Majorana spinor representation.^{[6]} If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana–Weyl spinor representations. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

The Dirac, Lorentz, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.^{[7]}

Massive particles, such as electrons, are described as Dirac spinors (more precisely sections of the associated Spinor bundle to account for spacetime dependence). The classical neutrino of the standard model of particle physics is 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.^{[8]} It is not known whether (spin-1/2) Weyl spinors exist in nature. In 2015, an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.^{[9]}

### Spinors in representation theoryEdit

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 understandingEdit

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".^{[10]} Stated differently:

Spinors ... provide a linear representation of the group of rotations in a space with any number of dimensions, each spinor having components where or .

^{[2]}

Several ways of illustrating everyday analogies have been formulated in terms of the plate trick, tangloids and other examples of orientation entanglement.

Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo:

No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the "square root" of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.

^{[11]}

## HistoryEdit

The most general mathematical form of spinors was discovered by Élie Cartan in 1913.^{[12]} The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.^{[13]}

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.^{[14]} The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.^{[15]} By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.

Spinor spaces were represented as left ideals of a matrix algebra in 1930, by G. Juvet^{[16]} and by Fritz Sauter.^{[17]}^{[18]} More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, **C**).^{[19]}^{[20]}

In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes^{[21]}^{[22]} replaced spinor spaces by the even subalgebra Cℓ^{0}_{1,3}(**R**) of the spacetime algebra Cℓ_{1,3}(**R**).^{[18]}^{[20]} 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.

## ExamplesEdit

Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓ_{p, q}(**R**). This is an algebra built up from an orthonormal basis of *n* = *p* + *q* mutually orthogonal vectors under addition and multiplication, *p* of which have norm +1 and *q* of which have norm −1, with the product rule for the basis vectors

### Two dimensionsEdit

The Clifford algebra Cℓ_{2,0}(**R**) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, *σ*_{1} and *σ*_{2}, and one unit pseudoscalar *i* = *σ*_{1}*σ*_{2}. From the definitions above, it is evident that (*σ*_{1}*) ^{2} = (*σ

*)*

_{2}^{2}= 1, and (

*σ*

_{1}

*σ*

_{2})(

*σ*

_{1}

*σ*

_{2}) = −

*σ*

_{1}

*σ*

_{1}

*σ*

_{2}

*σ*

_{2}= −1.

The even subalgebra Cℓ^{0}_{2,0}(**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 *σ*_{1}*σ*_{2}. As a real algebra, Cℓ^{0}_{2,0}(**R**) is isomorphic to the 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

- .

which, by the Clifford relations, can be written

- .

The action of an even Clifford element *γ* ∈ Cℓ^{0}_{2,0}(**R**) on vectors, regarded as 1-graded elements of Cℓ_{2,0}(**R**), is determined by mapping a general vector *u* = *a*_{1}*σ*_{1} + *a*_{2}*σ*_{2} to the vector

- ,

where *γ*^{∗} is the conjugate of *γ*, and the product is Clifford multiplication. In this situation, a **spinor**^{[23]} is an ordinary complex number. The action of *γ* on a spinor *φ* is given by ordinary complex multiplication:

- .

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:

- .

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 *γ*^{2} = exp(*θ σ*_{1}*σ*_{2}), so that the corresponding action on spinors is via *γ* = ± exp(*θ σ*_{1}*σ*_{2}/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is 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

- corresponds to a vector rotation of 90° from
*σ*_{1}around towards*σ*_{2}, which can be checked by confirming that - It corresponds to a spinor rotation of only 45°, however:

- Similarly the even-graded element
*γ*= −*σ*_{1}*σ*_{2}corresponds to a vector rotation of 180°:

- but a spinor rotation of only 90°:

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

- but a spinor rotation of 180°.

### Three dimensionsEdit

The Clifford algebra Cℓ_{3,0}(**R**) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, *σ*_{1}, *σ*_{2} and *σ*_{3}, the three unit bivectors *σ*_{1}*σ*_{2}, *σ*_{2}*σ*_{3}, *σ*_{3}*σ*_{1} and the pseudoscalar *i* = *σ*_{1}*σ*_{2}*σ*_{3}. It is straightforward to show that (*σ*_{1})^{2} = (*σ*_{2})^{2} = (*σ*_{3})^{2} = 1, and (*σ*_{1}*σ*_{2})^{2} = (*σ*_{2}*σ*_{3})^{2} = (*σ*_{3}*σ*_{1})^{2} = (*σ*_{1}*σ*_{2}*σ*_{3})^{2} = −1.

The sub-algebra of even-graded elements is made up of scalar dilations,

and vector rotations

where

- (1)

corresponds to a vector rotation through an angle *θ* about an axis defined by a unit vector *v* = *a*_{1}*σ*_{1} + *a*_{2}*σ*_{2} + *a*_{3}*σ*_{3}.

As a special case, it is easy to see that, if *v* = *σ*_{3}, this reproduces the *σ*_{1}*σ*_{2} rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the *σ*_{3} direction invariant, since

The bivectors *σ*_{2}*σ*_{3}, *σ*_{3}*σ*_{1} and *σ*_{1}*σ*_{2} are in fact Hamilton's quaternions **i**, **j** and **k**, discovered in 1843:

With the identification of the 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.^{[24]} Thus the (real^{[25]}) 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 constructionsEdit

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 spinorsEdit

Given a vector space *V* and a quadratic form *g* an explicit matrix representation of the Clifford algebra Cℓ(*V*, *g*) can be defined as follows. Choose an orthonormal basis *e*^{1} … *e*^{n} for *V* i.e. *g*(*e*^{μ}*e*^{ν}) = *η*^{μν} where *η*^{μμ} = ±1 and *η*^{μν} = 0 for *μ* ≠ *ν*. Let *k* = ⌊*n*/2⌋. Fix a set of 2^{k} × 2^{k} matrices *γ*^{1} … *γ*^{n} such that *γ*^{μ}*γ*^{ν} + *γ*^{ν}*γ*^{μ} = 2*η*^{μν}1 (i.e. fix a convention for the gamma matrices). Then the assignment *e*^{μ} → *γ*^{μ} extends uniquely to an algebra homomorphism Cℓ(*V*, *g*) → Mat(2^{k}, **C**) by sending the monomial *e*^{μ1} … *e*^{μk} in the Clifford algebra to the product *γ*^{μ1} … *γ*^{μk} of matrices and extending linearly. The space Δ = **C**^{2k} 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, abstract spinor indices are often used to denote spinors even when an abstract spinor construction is used.

### Abstract spinorsEdit

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.^{[26]} In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of *V*, and then specify the action of the Clifford algebra *externally* to that vector space.

In either approach, the fundamental notion is that of an isotropic subspace *W*. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of *V* is given.

As above, we let (*V*, *g*) be an *n*-dimensional complex vector space equipped with a nondegenerate bilinear form. If *V* is a real vector space, then we replace *V* by its complexification *V* ⊗_{R} **C** and let *g* denote the induced bilinear form on *V* ⊗_{R} **C**. Let *W* be a maximal isotropic subspace, i.e. a maximal subspace of *V* such that *g*|_{W} = 0. If *n* = 2*k* is even, then let *W*^{∗} be an isotropic subspace complementary to *W*. If *n* = 2*k* + 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 idealsEdit

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

Note that *ω*^{2} = 0 (i.e., *ω* is nilpotent of order 2), and moreover, *w*′*ω* = 0 for all *w*′ ∈ *W*′. The following facts can be proven easily:

- If
*n*= 2*k*, then the left ideal Δ = Cℓ(*V*,*g*)*ω*is a minimal left ideal. Furthermore, this splits into the two spin spaces Δ_{+}= Cℓ^{even}*ω*and Δ_{−}= Cℓ^{odd}*ω*on restriction to the action of the even Clifford algebra. - If
*n*= 2*k*+ 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*_{i}*w*_{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

where the *a*_{i1…ip} are scalars, and the *B*_{j} are auxiliary elements of the Clifford algebra. Observe now that the product

Pick any nonzero monomial *a* in the expansion of *α* with maximal homogeneous degree in the elements *w*_{i}:

- (no summation implied),

then

is a nonzero scalar multiple of *ω*, as required.

Note that for *n* even, this computation also shows that

- .

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 constructionEdit

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.^{[27]}

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

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

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

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

The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group^{[28]} (the half-spin representations, or Weyl spinors) via

- .

When dim(*V*) is odd, *V* = *W* ⊕ *U* ⊕ *W*′, where *U* is spanned by a unit vector *u* orthogonal to *W*. The Clifford action *c* is defined as before on *W* ⊕ *W*′, while the Clifford action of (multiples of) *u* is defined by

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

### Hermitian vector spaces and spinorsEdit

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

The main example is the case that the real vector space *V* is a hermitian vector space (*V*, *h*), i.e., *V* is equipped with a complex structure *J* that is an orthogonal transformation with respect to the inner product *g* on *V*. Then *V* ⊗_{R} **C** splits in the ±*i* eigenspaces of *J*. These eigenspaces are isotropic for the complexification of *g* and can be identified with the complex vector space (*V*, *J*) and its complex conjugate (*V*, −*J*). Therefore, for a hermitian vector space (*V*, *h*) the vector space Λ^{⋅}_{C}*V* (as well as its complex conjugate Λ^{⋅}_{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 Spin^{c} structure. Likewise, every complex vector bundle on a manifold carries a Spin^{c} structure.^{[29]}

## Clebsch–Gordan decompositionEdit

A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another.^{[30]} These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.

For the real or complex case, the alternating representations are

- Γ
_{r}= Λ^{r}*V*, the representation of the orthogonal group on skew tensors of rank*r*.

In addition, for the real orthogonal groups, there are three characters (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 dimensionsEdit

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

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} ⊕ *σ*Γ_{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:

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

For the complex representations of the real Clifford algebras, the associated reality structure on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate Δ of the representation Δ, and the following isomorphism is seen to hold:

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

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

- If
*q*is even, then and - If
*q*is odd, then and

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

### Odd dimensionsEdit

If *n* = 2*k* + 1 is odd, then

In the real case, once again the isomorphism holds

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

### ConsequencesEdit

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 dimensionsEdit

- In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1-dimensional representation that does not transform.
- In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component complex representations, i.e. complex numbers that get multiplied by
*e*^{±iφ/2}under a rotation by angle*φ*. - In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and quaternionic. The existence of spinors in 3 dimensions follows from the isomorphism of the groups SU(2) ≅ Spin(3) that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as Pauli matrices.
- In 4 Euclidean dimensions, the corresponding isomorphism is Spin(4) ≅ SU(2) × SU(2). There are two inequivalent quaternionic 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
- In 5 Euclidean dimensions, the relevant isomorphism is Spin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4-dimensional and quaternionic.
- In 6 Euclidean dimensions, the isomorphism Spin(6) ≅ SU(4) guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
- In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.
- In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality.
- In
*d*+ 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in*d*dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity. - In spacetimes with
*p*spatial and*q*time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the (*p*+*q*)-dimensional Euclidean space, but the reality projections mimic the structure in |*p*−*q*| Euclidean dimensions. For example, in 3 + 1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism SL(2,**C**) ≅ Spin(3,1).

Metric signature | left-handed Weyl | right-handed Weyl | conjugacy | Dirac | left-handed Majorana–Weyl | right-handed Majorana–Weyl | Majorana |
---|---|---|---|---|---|---|---|

complex | complex | complex | real | real | real | ||

(2,0) | 1 | 1 | mutual | 2 | – | – | 2 |

(1,1) | 1 | 1 | self | 2 | 1 | 1 | 2 |

(3,0) | – | – | – | 2 | – | – | – |

(2,1) | – | – | – | 2 | – | – | 2 |

(4,0) | 2 | 2 | self | 4 | – | – | – |

(3,1) | 2 | 2 | mutual | 4 | – | – | 4 |

(5,0) | – | – | – | 4 | – | – | – |

(4,1) | – | – | – | 4 | – | – | – |

(6,0) | 4 | 4 | mutual | 8 | – | – | 8 |

(5,1) | 4 | 4 | self | 8 | – | – | – |

(7,0) | – | – | – | 8 | – | – | 8 |

(6,1) | – | – | – | 8 | – | – | – |

(8,0) | 8 | 8 | self | 16 | 8 | 8 | 16 |

(7,1) | 8 | 8 | mutual | 16 | – | – | 16 |

(9,0) | – | – | – | 16 | – | – | 16 |

(8,1) | – | – | – | 16 | – | – | 16 |

## See alsoEdit

## NotesEdit

**^**Spinors in three dimensions are points of a line bundle over a conic in the projective plane. In this picture, which is associated to spinors of a three-dimensional pseudo-Euclidean space of signature (1,2), the conic is an ordinary real conic (here the circle), the line bundle is the Möbius bundle, and the spin group is SL_{2}(**R**). In Euclidean signature, the projective plane, conic and line bundle are over the complex instead, and this picture is just a real slice.**^**Spinors can always be defined over the complex numbers. However, in some signatures there exist real spinors. Details can be found in spin representation.**^**A formal definition of spinors at this level is that the space of spinors is a linear representation of the Lie algebra of infinitesimal rotations of a certain kind.**^**More precisely, it is the fermions of spin-1/2 that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic Dirac equation for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.**^**Formally, the spin group is the group of relative homotopy classes with fixed endpoints in the rotation group.**^**More formally, the space of spinors can be defined as an (irreducible) representation of the spin group that does not factor through a representation of the rotation group (in general, the connected component of the identity of the orthogonal group).**^**Geometric algebra is a name for the Clifford algebra in an applied setting.**^**the Pauli matrices correspond to angular momenta operators about the three coordinate axes. This makes them slightly a-typical gamma matrices because in addition to their anti commutation relation they also satisfy commutation relations**^**The metric signature relevant as well if we are concerned with real spinors. See spin representation.**^**Whether the representation decomposes depends on whether they are regarded as representations of the spin group (or its Lie algebra), in which case it decomposes in even but not odd dimensions, or the Clifford algebra when it is the other way around. Other structures than this decomposition can also exist; precise criteria are covered at spin representation and Clifford algebra.**^**The TNB frame of the ribbon defines a rotation continuously for each value of the arc length parameter.**^**This is the set of 2×2 complex traceless hermitian matrices.**^**Except for a kernel of corresponding to the two different elements of the spin group that go to the same rotation.**^**Although there are several more intrinsic constructions, the spin representations are not functorial in the quadratic form, so they cannot be built up naturally within the tensor algebra.**^**So the ambiguity in identifying the spinors themselves persists from the point of view of the group theory, and still depends on choices.**^**The Clifford algebra can be given an even/odd grading from the parity of the degree in the gammas, and the spin group and its Lie algebra both lie in the even part. Whether here by "representation" we mean representations of the spin group or the Clifford algebra will affect the determination of their reducibility. Other structures than this splitting can also exist; precise criteria are covered at spin representation and Clifford algebra.

## ReferencesEdit

**^**Cartan 1913.- ^
^{a}^{b}Quote from Elie Cartan:*The Theory of Spinors*, Hermann, Paris, 1966, first sentence of the Introduction section of the beginning of the book (before the page numbers start): "Spinors were first used under that name, by physicists, in the field of Quantum Mechanics. In their most general form, spinors were discovered in 1913 by the author of this work, in his investigations on the linear representations of simple groups*; they provide a linear representation of the group of rotations in a space with any number of dimensions, each spinor having components where or ." The star (*) refers to Cartan 1913. **^**Named after William Kingdon Clifford,**^**Named after Paul Dirac.**^**Named after Hermann Weyl.**^**Named after Ettore Majorana.**^**Matthew R. Francis, Arthur Kosowsky:*The Construction of Spinors in Geometric Algebra*, submitted 20 March 2004, version of 18 October 2004 arXiv:math-ph/0403040**^***Wilczek, Frank (2009). "Majorana returns".*Nature Physics*. Macmillan Publishers.**5**(9): 614–618. Bibcode:2009NatPh...5..614W. doi:10.1038/nphys1380. ISSN 1745-2473. (Subscription required (help)). Cite uses deprecated parameter`|subscription=`

(help)**^**Xu, Yang-Su; et al. (2015). "Discovery of a Weyl Fermion semimetal and topological Fermi arcs".*Science Magazine*. AAAS.**349**: 613–617. arXiv:1502.03807. Bibcode:2015Sci...349..613X. doi:10.1126/science.aaa9297. ISSN 0036-8075. (Subscription required (help)). Cite uses deprecated parameter`|subscription=`

(help)**^**Jean Hladik:*Spinors in Physics*, translated by J. M. Cole, Springer 1999, ISBN 978-0-387-98647-0, p. 3**^**Graham Farmelo.*The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius*, Faber & Faber, 2009, ISBN 978-0-571-22286-5, p. 430.**^**Cartan 1913**^**Tomonaga 1998, p. 129**^**Pauli 1927.**^**Dirac 1928.**^**G. Juvet:*Opérateurs de Dirac et équations de Maxwell*, Commentarii Mathematici Helvelvetici, 2 (1930), pp. 225–235, doi:10.1007/BF01214461 (abstract in French language)**^**Sauter, F. (1930). "Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren".*Zeitschrift für Physik*.**63**(11–12): 803–814. Bibcode:1930ZPhy...63..803S. doi:10.1007/BF01339277.- ^
^{a}^{b}Pertti Lounesto:*Crumeyrolle's bivectors and spinors*, pp. 137–166, In: Rafał Abłamowicz, Pertti Lounesto (eds.):*Clifford algebras and spinor structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992)*, ISBN 0-7923-3366-7, 1995, p. 151 **^**The matrices of dimension*N*×*N*in which only the elements of the left column are non-zero form a*left ideal*in the*N*×*N*matrix algebra Mat(*N*,**C**) – multiplying such a matrix*M*from the left with any*N*×*N*matrix*A*gives the result*AM*that is again an*N*×*N*matrix in which only the elements of the left column are non-zero. Moreover, it can be shown that it is a*minimal left ideal*. See also: Pertti Lounesto:*Clifford algebras and spinors*, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, DOI 978-0-521-00551-7, p. 52- ^
^{a}^{b}Pertti Lounesto:*Clifford algebras and spinors*, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, DOI 978-0-521-00551-7, p. 148 f. and p. 327 f. **^**D. Hestenes:*Space–Time Algebra*, Gordon and Breach, New York, 1966, 1987, 1992**^**Hestenes, D. (1967). "Real spinor fields".*J. Math. Phys.***8**(4): 798–808. Bibcode:1967JMP.....8..798H. doi:10.1063/1.1705279.**^**These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via*γ*(*ϕ*) =*γϕ*. The Majorana spinors are the common underlying real representation for the Weyl representations.**^**Since, for a skew field, the kernel of the representation must be trivial. So inequivalent representations can only arise via an automorphism of the skew-field. In this case, there are a pair of equivalent representations:*γ*(*ϕ*) =*γϕ*, and its quaternionic conjugate*γ*(*ϕ*) =*ϕγ*.**^**The complex spinors are obtained as the representations of the tensor product**H**⊗_{R}**C**= Mat_{2}(**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.

## Further readingEdit

- Brauer, Richard; Weyl, Hermann (1935), "Spinors in n dimensions",
*American Journal of Mathematics*, The Johns Hopkins University Press,**57**(2): 425–449, doi:10.2307/2371218, JSTOR 2371218. - Cartan, Élie (1913), "Les groupes projectifs qui ne laissent invariante aucune multiplicité plane" (PDF),
*Bul. Soc. Math. France*,**41**: 53–96. - Cartan, Élie (1966),
*The theory of spinors*, Paris, Hermann (reprinted 1981, Dover Publications), ISBN 978-0-486-64070-9 - Chevalley, Claude (1954),
*The algebraic theory of spinors and Clifford algebras*, Columbia University Press (reprinted 1996, Springer), ISBN 978-3-540-57063-9. - Dirac, Paul M. (1928), "The quantum theory of the electron",
*Proceedings of the Royal Society of London*,**A117**: 610–624, JSTOR 94981. - Fulton, William; Harris, Joe (1991),
*Representation theory. A first course*, Graduate Texts in Mathematics, Readings in Mathematics,**129**, New York: Springer-Verlag, doi:10.1007/978-1-4612-0979-9, ISBN 0-387-97495-4, MR 1153249. - Gilkey, Peter B. (1984),
*Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem*, Publish or Perish, ISBN 0-914098-20-9. - Harvey, F. Reese (1990),
*Spinors and Calibrations*, Academic Press, ISBN 978-0-12-329650-4. - Hazewinkel, Michiel, ed. (2001) [1994], "Spinor",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Hitchin, Nigel J. (1974), "Harmonic spinors",
*Advances in Mathematics*,**14**: 1–55, doi:10.1016/0001-8708(74)90021-8, MR 0358873. - Lawson, H. Blaine; Michelsohn, Marie-Louise (1989),
*Spin Geometry*, Princeton University Press, ISBN 0-691-08542-0. - Pauli, Wolfgang (1927), "Zur Quantenmechanik des magnetischen Elektrons",
*Zeitschrift für Physik*,**43**(9–10): 601–632, Bibcode:1927ZPhy...43..601P, doi:10.1007/BF01397326. - Penrose, Roger; Rindler, W. (1988),
*Spinors and Space–Time: Volume 2, Spinor and Twistor Methods in Space–Time Geometry*, Cambridge University Press, ISBN 0-521-34786-6. - Tomonaga, Sin-Itiro (1998), "Lecture 7: The Quantity Which Is Neither Vector nor Tensor",
*The story of spin*, University of Chicago Press, p. 129, ISBN 0-226-80794-0