Spinor

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Two continuous rotations, one through 2π and one through 4π, illustrated here in the game of tangloids. A solution of the game is a (continuous) manipulation of the two cords, fixing the endpoints, that untwists them. This is impossible with the 2π rotation, but possible with the 4π rotation. A solution, shown in the last frames of the animation, actually gives an explicit homotopy in the rotation group between the continuous 4π rotation and the trivial (identity) rotation, using the arc length parameter along each of the two curves as the homotopy parameter.

Spinors are mathematical objects in geometry and physics that, like vectors, and more generally tensors, transform in a certain way when the system containing them is subjected to a continuous rotation of its frame of reference. Here a "continuous rotation" refers to a rotation that depends continuously on a single parameter. All objects in the system transform with respect to the continuous rotation, but depending on the type of the objects, they will transform differently. However, spinors are unlike vectors (and other tensors) in the sense that, whereas a vector (or tensor) undergoes the same rotation as the frame of reference, spinors account for an additional sign ambiguity in the continuous rotation. This sign ambiguity is called the class of the particular continuous rotation. This class actually has a topological origin, famously illustrated in the game of tangloids (shown), which demonstrates two different continuous rotations having the same final configurations but different classes. All tensors will have undergone the same transformation under these two topologically distinguishable rotations, but spinors will have undergone such a transformation apart from difference of sign. It turns out that there are only two possible classes for any given rotation, just as there are two different choices of sign.

In the language of group theory, spinors are ("projective") representations of the rotation group, which is to say that they are vector-like objects that transform under rotations. It is actually more accurate to say that they are representations of the spin group, which consists of continuous rotations keeping track of the class of the rotation. Spinors form a vector space, often denoted by Δ, and to say that they are a representation of the spin group means precisely that there is a group homomorphism of the spin group into the general linear group of Δ. The representation is distinguished as a spin representation if the transformation law under continuous rotations genuinely depends on the class of the rotation; that is, if this homomorphism is one-to-one. Spinors can be associated, as an auxiliary mathematical object, to any vector space equipped with a quadratic form such as Euclidean space with its standard dot product, or Minkowski space with its Lorentz metric. The general theory concerns Clifford algebras, in the mathematical treatment of the subject, and Pauli spin matrices, Dirac matrices and similar explicit devices in physics.

General spinors were discovered by Élie Cartan in 1913.[1][2] Soon after, spinors turned out to be essential in quantum physics, and currently enjoy a wide range of applications. Spinors in three dimensions are needed to describe non-relativistic electrons and other fermions which have spin-½. Dirac spinors, spinors of the Lorentz metric in dimension 4, are required to describe the quantum state of the relativistic electron via the Dirac equation. In quantum field theory, spinors describe the state of relativistic many-particle systems. In mathematics, particularly in differential geometry and global analysis, spinors have also found broad applications in algebraic and differential topology,[3] symplectic geometry, gauge theory, complex algebraic geometry,[4] index theory,[5] and special holonomy.[6]

Overview[edit]

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

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

One is representation theoretic. In this point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors. In this view, a spinor must belong to a representation of the double cover of the rotation group SO(n, R), or more generally of double cover of the generalized special orthogonal group SO+(p, q, R) on spaces with metric signature (p, q). These double covers are Lie groups, called the spin groups Spin(n) or Spin(p, q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield projective representations of the groups themselves, which do not meet the full definition of a representation.

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

Clifford algebras[edit]

For more details on this topic, see Clifford algebra.

The language of Clifford algebras[7] (sometimes called geometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras. It largely removes the need for ad hoc constructions.

In detail, let V be a finite-dimensional complex vector space with nondegenerate bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = Cn, with the standard form g(x, y) = xty = x1y1 + ... + xnyn we denote the Clifford algebra by Cℓn(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 dimC(V) = n. If n = 2k is even, Cℓn(C) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2k, C) of 2k × 2k 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, Cℓ2k+1(C) is isomorphic to the algebra Mat(2k, C) ⊕ Mat(2k, C) of two copies of the 2k × 2k complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2[n/2]. Since the Lie algebra so(V, g) is embedded as a Lie subalgebra in Cℓ(V, g) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of so(V, g) called a spin representation. If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further into two irreducible representations Δ = Δ+ ⊕ Δ called the Weyl or half-spin representations.

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

Terminology in physics[edit]

The most typical type of spinor, the Dirac spinor,[8] is an element of the fundamental representation of Cℓp+q(C), the complexification of the Clifford algebra Cℓp, q(R), into which the spin group Spin(p, q) may be embedded. On a 2k- or 2k+1-dimensional space a Dirac spinor may be represented as a vector of 2k 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[9] representations. In addition, sometimes the non-complexified version of Cℓp,q(R) has a smaller real representation, the Majorana spinor representation.[10] If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana–Weyl spinor representations.

Of all these, only the Dirac representation exists in all dimensions.[clarification needed][citation needed] 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.[11]

Spinors in representation theory[edit]

Main article: Spin representation

One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah–Singer index theorem, and to provide constructions in particular for discrete series representations of semisimple groups.

The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article.

Attempts at intuitive understanding[edit]

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”.[12] Stated differently:[2]

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

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

Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo:[13]

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[edit]

The most general mathematical form of spinors was discovered by Élie Cartan in 1913.[14] The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.[15]

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.[16] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.[17] 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[18] and by Fritz Sauter.[19][20] 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).[21][22]

In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes[23][24] replaced spinor spaces by the even subalgebra Cℓ01,3(R) of the spacetime algebra Cℓ1,3(R).[20][22] 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[edit]

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

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

Two dimensions[edit]

The Clifford algebra Cℓ2,0(R) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ1 and σ2, and one unit pseudoscalar i = σ1σ2. From the definitions above, it is evident that (σ1)2 = (σ2)2 = 1, and (σ1σ2)(σ1σ2) = −σ1σ1σ2σ2 = −1.

The even subalgebra Cℓ02,0(R), spanned by even-graded basis elements of Cℓ2,0(R), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ1σ2. As a real algebra, Cℓ02,0(R) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by

(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1\,.

which, by the Clifford relations, can be written

(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2\,.

The action of an even Clifford element γ ∈ Cℓ02,0(R) on vectors, regarded as 1-graded elements of Cℓ2,0(R), is determined by mapping a general vector u = a1σ1 + a2σ2 to the vector

\gamma(u) = \gamma u \gamma^*\,,

where γ is the conjugate of γ, and the product is Clifford multiplication. In this situation, a spinor[25] is an ordinary complex number. The action of γ on a spinor φ is given by ordinary complex multiplication:

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

An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:

 \gamma(u) = \gamma u \gamma^* = \gamma^2 u\,.

On the other hand, comparing with the action on spinors γ(φ) = γφ, γ on ordinary vectors acts as the square of its action on spinors.

Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ2 = exp(θ σ1σ2), so that the corresponding action on spinors is via γ = ± exp(θ σ1σ2/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.

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

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

Three dimensions[edit]

Main articles Spinors in three dimensions, Quaternions and spatial rotation

The Clifford algebra Cℓ3,0(R) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ1, σ2 and σ3, the three unit bivectors σ1σ2, σ2σ3, σ3σ1 and the pseudoscalar i = σ1σ2σ3. It is straightforward to show that (σ1)2 = (σ2)2 = (σ3)2 = 1, and (σ1σ2)2 = (σ2σ3)2 = (σ3σ1)2 = (σ1σ2σ3)2 = −1.

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

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

and vector rotations

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

where

\left.\begin{matrix} \gamma & = & \cos(\theta/2) - \{a_1 \sigma_2\sigma_3 + a_2 \sigma_3\sigma_1 + a_3 \sigma_1\sigma_2\} \sin(\theta/2) \\
& = & \cos(\theta/2) - i \{a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3\} \sin(\theta/2) \\
& = & \cos(\theta/2) - i v \sin(\theta/2) \end{matrix}\right\} (1)

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

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

(\cos(\theta/2) - i \sigma_3 \sin(\theta/2)) \, \sigma_3 \, (\cos(\theta/2) + i \sigma_3 \sin(\theta/2))
= (\cos^2(\theta/2) + \sin^2(\theta/2)) \, \sigma_3 = \sigma_3.

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

\begin{matrix}\mathbf{i} = -\sigma_2 \sigma_3 = -i \sigma_1 \\
\mathbf{j} = -\sigma_3 \sigma_1 = -i \sigma_2 \\
\mathbf{k} = -\sigma_1 \sigma_2 = -i \sigma_3. \end{matrix}

With the identification of the even-graded elements with the algebra H of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.[26] Thus the (real[27]) 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[edit]

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

Component spinors[edit]

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 e1en for V i.e. g(eμeν) = ημν where ημμ = ±1 and ημν = 0 for μν. Let k = ⌊ n/2 ⌋. Fix a set of 2k × 2k matrices γ1γn such that γμγν + γνγμ = ημν1 (i.e. fix a convention for the gamma matrices). Then the assignment eμγμ extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2k, C) by sending the monomial eμ1eμk in the Clifford algebra to the product γμ1γμk of matrices and extending linearly. The space Δ = C2k on which the gamma matrices act is a now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics. Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic quantum field theory. In general, in order to define gamma matrices of the required kind, one can use the Weyl–Brauer matrices.

In this construction the representation of the Clifford algebra Cℓ(V, g), the Lie algebra so(V, g), and the Spin group Spin(V, g), all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2k complex numbers and is denoted with spinor indices (usually α, β, γ). In the physics literature, abstract spinor indices are often used to denote spinors even when an abstract spinor construction is used.

Abstract spinors[edit]

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 : 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.[28] 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 ⊗RC and let g denote the induced bilinear form on V ⊗RC. 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[edit]

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 ΛW′. Consequently, the k-fold product of W′ with itself, Wk, is one-dimensional. Let ω be a generator of Wk. In terms of a basis w1,..., wk of in W′, one possibility is to set

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

Note that ω2 = 0 (i.e., ω is nilpotent of order 2), and moreover, wω = 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+ = Cℓevenω and Δ = Cℓoddω 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 wi of W and a complementary basis wi′ of W′ so that

wiwj′ +wjwi = δij, and
(wi)2 = 0, (wi′)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 αωI be any such element. Using the chosen basis, we may write

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

where the ai1…ip are scalars, and the Bj are auxiliary elements of the Clifford algebra. Observe now that the product

\alpha\omega = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} \omega.

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

a = a_{i_1\dots i_{max}}w_{i_1}\dots w_{i_{max}} (no summation implied),

then

w_{i_{max}}\cdots w_{i_1}\alpha\omega = a_{i_1\dots i_{max}}\omega

is a nonzero scalar multiple of ω, as required.

Note that for n even, this computation also shows that

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

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

Exterior algebra construction[edit]

The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra Λ W = ⊕j Λj 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.[29]

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 = WW where W′ is the chosen isotropic complement. Hence any vV decomposes uniquely as v = w + w with wW and w′ ∈ W. The action of v on a spinor is given by

 c(v) w_1 \wedge\cdots\wedge w_n = (\epsilon(w) + i(w'))\left(w_1 \wedge\cdots\wedge w_n\right)

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

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

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

The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group[30] (the half-spin representations, or Weyl spinors) via

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

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

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

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

Hermitian vector spaces and spinors[edit]

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 ⊗RC 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 Spinc structure. Likewise, every complex vector bundle on a manifold carries a Spinc structure.[31]

Clebsch–Gordan decomposition[edit]

A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another.[32] 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

  • Γr = ΛrV, 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[edit]

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

\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^n \Gamma_p \cong \bigoplus_{p=0}^{k-1} \left(\Gamma_p\oplus\sigma\Gamma_p\right)\, \oplus \Gamma_k

which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements αω ⊗ βω. The rightmost formulation follows from the transformation properties of the Hodge star operator. Note that on restriction to the even Clifford algebra, the paired summands ΓpσΓp are isomorphic, but under the full Clifford algebra they are not.

There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:

(\alpha\omega)^*=\omega(\alpha^*).

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

\begin{matrix}
\Delta_+\otimes\Delta^*_+ \cong \Delta_-\otimes\Delta^*_- &\cong& \bigoplus_{p=0}^k \Gamma_{2p}\\
\Delta_+\otimes\Delta^*_- \cong \Delta_-\otimes\Delta^*_+ &\cong& \bigoplus_{p=0}^{k-1} \Gamma_{2p+1}
\end{matrix}

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

\bar{\Delta} \cong \sigma_-\Delta^*

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

\Delta\otimes\bar{\Delta} \cong \bigoplus_{p=0}^k\left(\sigma_-\Gamma_p\oplus \sigma_+\Gamma_p\right).

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

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

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

Odd dimensions[edit]

If n = 2k + 1 is odd, then

\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^k \Gamma_{2p}.

In the real case, once again the isomorphism holds

\bar{\Delta} \cong \sigma_-\Delta^*.

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

\Delta\otimes\bar{\Delta} \cong \sigma_-\Gamma_0\oplus\sigma_+\Gamma_1\oplus\dots\oplus\sigma_\pm\Gamma_k

Consequences[edit]

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

  • A manner of regarding the product of two spinors ϕψ as a scalar. In physical terms, a spinor should determine a probability amplitude for the quantum state.
  • A manner of regarding the product ψϕ as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space.
  • A manner of regarding a spinor as acting upon a vector, by an expression such as ψvψ. In physical terms, this represents an electrical current of Maxwell's electromagnetic theory, or more generally a probability current.

Summary in low dimensions[edit]

  • 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±/2 under a rotation by angle φ.
  • In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and quaternionic. The existence of spinors in 3 dimensions follows from the isomorphism of the groups SU(2) ≅ Spin(3) that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as Pauli matrices.
  • In 4 Euclidean dimensions, the corresponding isomorphism is Spin(4) ≅ SU(2) × SU(2). There are two inequivalent quaternionic 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
  • In 5 Euclidean dimensions, the relevant isomorphism is Spin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4-dimensional and quaternionic.
  • In 6 Euclidean dimensions, the isomorphism Spin(6) ≅ SU(4) guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
  • In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.
  • In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality.
  • In d + 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in d dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity.
  • In spacetimes with p spatial and q time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the (p + q)-dimensional Euclidean space, but the reality projections mimic the structure in |p − q| Euclidean dimensions. For example, in 3 + 1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism SL(2, C) ≅ Spin(3,1).
Metric signature left-handed Weyl right-handed Weyl conjugacy Dirac left-handed Majorana–Weyl right-handed Majorana–Weyl Majorana
complex complex complex real real real
(2,0) 1 1 mutual 2 2
(1,1) 1 1 self 2 1 1 2
(3,0) 2
(2,1) 2 2
(4,0) 2 2 self 4
(3,1) 2 2 mutual 4 4
(5,0) 4
(4,1) 4
(6,0) 4 4 mutual 8 8
(5,1) 4 4 self 8
(7,0) 8 8
(6,1) 8
(8,0) 8 8 self 16 8 8 16
(7,1) 8 8 mutual 16 16
(9,0) 16 16
(8,1) 16 16

See also[edit]

References[edit]

  1. ^ Cartan 1913.
  2. ^ 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 n of dimensions, each spinor having 2^\nu components where n = 2\nu+1 or 2\nu." The star (*) refers to Cartan 1913.
  3. ^ Hitchin 1974, Lawson & Michelsohn 1989.
  4. ^ Hitchin 1974, Penrose & Rindler 1988.
  5. ^ Gilkey 1984, Lawson & Michelsohn 1989.
  6. ^ Lawson & Michelsohn 1989, Harvey 1990. These two books also provide good mathematical introductions and fairly comprehensive bibliographies on the mathematical applications of spinors as of 1989–1990.
  7. ^ Named after William Kingdon Clifford,
  8. ^ Named after Paul Dirac.
  9. ^ Named after Hermann Weyl.
  10. ^ Named after Ettore Majorana.
  11. ^ 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
  12. ^ Jean Hladik: Spinors in Physics, translated by J. M. Cole, Springer 1999, ISBN 978-0-387-98647-0, p. 3
  13. ^ Graham Farmelo: The Strangest Man. The Hidden Life of Paul Dirac, Quantum Genius, Faber & Faber, 2009, ISBN 978-0-571-22286-5, p. 430
  14. ^ Cartan 1913
  15. ^ Tomonaga 1998, p. 129
  16. ^ Pauli 1927.
  17. ^ Dirac 1928.
  18. ^ 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)
  19. ^ F. Sauter: Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren, Zeitschrift für Physik, Volume 63, Numbers 11–12, 803–814, doi:10.1007/BF01339277 (abstract in German language)
  20. ^ 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
  21. ^ 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
  22. ^ 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.
  23. ^ D. Hestenes: Space–Time Algebra, Gordon and Breach, New York, 1966, 1987, 1992
  24. ^ D. Hestenes: Real spinor fields, J. Math. Phys. 8 (1967), pp. 798–808
  25. ^ 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.
  26. ^ 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.
  27. ^ The complex spinors are obtained as the representations of the tensor product HR C = Mat2(C). These are considered in more detail in spinors in three dimensions.
  28. ^ This construction is due to Cartan. The treatment here is based on Chevalley (1954).
  29. ^ One source for this subsection is Fulton & Harris (1991).
  30. ^ Via the even-graded Clifford algebra.
  31. ^ Lawson & Michelsohn 1989, Appendix D.
  32. ^ Brauer & Weyl 1935.

Further reading[edit]