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

In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space.[b] Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation.[c] 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 continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).

It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.[1][d] In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.[e]

Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as illustrated by the belt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.[f] 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.[g] 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.[h] A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.[3] 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,[i] 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[j]) 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.[k]

An object attached to belts or strings can spin continuously without becoming tangled. Notice that after the cube completes a 360° rotation, the spiral is reversed from its initial configuration. The belts return to their original configuration after spinning a full 720°.
A more extreme example demonstrating that this works with any number of strings. In the limit, a piece of solid continuous space can rotate in place like this without tearing or intersecting itself

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

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

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

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.

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

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

The language of Clifford algebras[4] (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.

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.

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

Spinors form a vector space, usually over the complex numbers, equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations (see diagram). The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product, or Minkowski space with its Lorentz metric. In the latter case, the "rotations" include the Lorentz boosts, but otherwise the theory is substantially similar.

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

The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.[6] Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

Weyl spinors are insufficient to describe massive particles, such as electrons, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed. The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor.[q] However, because of observed neutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors.[7] It is not known whether Weyl spinor fundamental particles exist in nature.

The situation for condensed matter physics is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors to far more exotic materials. In 2015, an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.[8]

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.

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".[9] Stated differently:

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.[10]

The most general mathematical form of spinors was discovered by Élie Cartan in 1913.[11] The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.[12]

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.[13] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.[14] 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.

where γ is the conjugate of γ, and the product is Clifford multiplication. In this situation, a spinor[s] 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 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.

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

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

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

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.

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.

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.

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.[22] 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.

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

Note that ω2 = 0 (i.e., ω is nilpotent of order 2), and moreover, wω = 0 for all wW. The following facts can be proven easily:

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.

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

where the ai1...ip are scalars, and the Bj 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 wi:

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.

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.[23][24]

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 wW. The action of v on a spinor is given by

where i(w) is interior product with w using the nondegenerate quadratic form to identify V with V, and ε(w) denotes the exterior product. This action is sometimes called the Clifford product. It may be verified that

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[25] (the half-spin representations, or Weyl spinors) via

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

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

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.

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.[26]

A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another.[27] These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.

In addition, for the real orthogonal groups, there are three characters (one-dimensional representations)

The Clebsch–Gordan decomposition allows one to define, among other things:

If n = 2k 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 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:

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

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

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

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