In geometry and physics, spinors (pronounced "spinner"; ) are elements of a complex vector space that can be associated with Euclidean space. Spinors can be thought of as companion geometric objects to Euclidean space that, like Euclidean vectors, respond when the Euclidean space is subjected to a rotation. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. Spinors are therefore often described heuristically as "square roots" of (geometric) vectors, and a geometric vector can be constructed quadratically from a spinor. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. Mathematically, spinors are elements of spaces carrying representations of the spin group or of the associated Clifford algebra. After choosing a matrix realization of the Clifford algebra, spinors may be represented concretely as column vectors on which the corresponding gamma matrices act.
Introduction
What characterizes spinors and distinguishes them from geometric vectors and other tensors is a subtle difference in how they respond to rotations: briefly, spinors respond to rotations in a path-dependent way, while vectors respond without seeing the path through which a rotation was achieved. 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 (shown, in which both ends of the rotated object are physically tethered to an external reference) 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. To see how this might work in practice, we consider the set H 2 {\displaystyle H_{2}} of 2 × 2 {\displaystyle 2\times 2} Hermitian matrices, with complex entries, whose traces are zero. Any such matrix can be written as
X = [ x z z ¯ − x ] , {\displaystyle X={\begin{bmatrix}x&z\\{\bar {z}}&-x\end{bmatrix}},}
where x {\displaystyle x} is real, z = u + i v {\displaystyle z=u+iv} is complex, and z ¯ = u − i v {\displaystyle {\bar {z}}=u-iv} is the complex conjugate of z {\displaystyle z} . Then H 2 {\displaystyle H_{2}} is a three-dimensional vector space over the real field. The negative determinant of X {\displaystyle X} is − det X = x 2 + | z | 2 = x 2 + u 2 + v 2 {\displaystyle -\det X=x^{2}+|z|^{2}=x^{2}+u^{2}+v^{2}} , which is the sum of the squares of the three real coordinates ( x , u , v ) {\displaystyle (x,u,v)} . Thus, H 2 {\displaystyle H_{2}} , equipped with this form, is a real Euclidean three-space (like R 3 {\displaystyle \mathbb {R} ^{3}} equipped with its dot product). To describe rotations of this Euclidean space, consider all 2 × 2 {\displaystyle 2\times 2} complex matrices
U = [ α β − β ¯ α ¯ ] {\displaystyle U={\begin{bmatrix}\alpha &\beta \\-{\bar {\beta }}&{\bar {\alpha }}\end{bmatrix}}}
satisfying det U = | α | 2 + | β | 2 = 1. {\displaystyle \det U=|\alpha |^{2}+|\beta |^{2}=1.} One has that U ∗ U = U U ∗ = I 2 , {\displaystyle U^{*}U=UU^{*}=I_{2},} the 2 × 2 {\displaystyle 2\times 2} identity matrix, where the star denotes the Hermitian conjugate:
U ∗ = [ α ¯ − β β ¯ α ] . {\displaystyle U^{*}={\begin{bmatrix}{\bar {\alpha }}&-\beta \\{\bar {\beta }}&\alpha \end{bmatrix}}.}
The set of such matrices is the special unitary group SU ( 2 ) . {\displaystyle \operatorname {SU} (2).} Since U ∗ = U − 1 {\displaystyle U^{*}=U^{-1}} , the operation R U , {\displaystyle R_{U},} given by R U ( X ) = U X U ∗ {\displaystyle R_{U}(X)=UXU^{*}} for all X ∈ H 2 , {\displaystyle X\in H_{2},} preserves the determinant. That is, R U {\displaystyle R_{U}} is a rotation of the Euclidean space H 2 . {\displaystyle H_{2}.} Thus, rotations of H 2 {\displaystyle H_{2}} can be described by special unitary matrices U . {\displaystyle U.} But, for each U ∈ SU ( 2 ) , {\displaystyle U\in \operatorname {SU} (2),} the pair of unitary maps U {\displaystyle U} and − U {\displaystyle -U} induce the same rotation on H 2 {\displaystyle H_{2}} : R U ( X ) = U X U ∗ = ( − U ) X ( − U ) ∗ = R − U ( X ) . {\displaystyle R_{U}(X)=UXU^{*}=(-U)X(-U)^{*}=R_{-U}(X).} So X {\displaystyle X} does not feel the difference between the distinct unitary matrices U {\displaystyle U} and − U . {\displaystyle -U.}
In this example, the spinors are by definition the complex column vectors on which the special unitary matrices U {\displaystyle U} act. These do feel the difference between the matrices U {\displaystyle U} and − U {\displaystyle -U} . In particular, if we take a path γ {\displaystyle \gamma } in SU ( 2 ) {\displaystyle \operatorname {SU} (2)} from the identity I 2 {\displaystyle I_{2}} to its negative − I 2 {\displaystyle -I_{2}} (such as γ ( t ) = [ cos ( t / 2 ) sin ( t / 2 ) − sin ( t / 2 ) cos ( t / 2 ) ] {\displaystyle \gamma (t)={\begin{bmatrix}\cos(t/2)&\sin(t/2)\\-\sin(t/2)&\cos(t/2)\end{bmatrix}}} for all t ∈ [ 0 , 2 π ] {\displaystyle t\in [0,2\pi ]} , which induces a rotation through the angle t {\displaystyle t} on H 2 {\displaystyle H_{2}} ), the associated rotations R γ ( t ) {\displaystyle R_{\gamma (t)}} start and end at the identity, and the space H 2 {\displaystyle H_{2}} feels no change, whereas a column vector will be transformed to its negative. The association of the rotation R U {\displaystyle R_{U}} to each U {\displaystyle U} is two-to-one, and the kernel is the group { ± I 2 } {\displaystyle \{\pm I_{2}\}} . This exhibits the double cover of the rotation group, and SU ( 2 ) {\displaystyle \operatorname {SU} (2)} as its spin group. In this setting, the Euclidean space (of "physical vectors") is the real vector space H 2 {\displaystyle H_{2}} , while the space of spinors is the complex two-dimensional vector space of column vectors on which SU ( 2 ) {\displaystyle \operatorname {SU} (2)} acts. Given a single spinor v {\displaystyle v} , a real vector in H 2 {\displaystyle H_{2}} can be formed as v v ∗ − 1 2 v ∗ v I 2 {\displaystyle vv^{*}-{\frac {1}{2}}v^{*}vI_{2}} . Thus "vectors" are quadratic in the spinors. Thus, in the case of three-dimensional rotations, spinors can be defined as follows: A spinor is a complex 2-component column vector v ∈ C 2 {\displaystyle v\in \mathbb {C} ^{2}} transforming under the natural action of SU ( 2 ) {\displaystyle \operatorname {SU} (2)} . The associated geometric vectors are the traceless Hermitian 2 × 2 {\displaystyle 2\times 2} matrices, transforming by conjugation X ↦ U X U ∗ {\displaystyle X\mapsto UXU^{*}} . Thus the spinors are the column vectors themselves, while the ordinary vectors are the objects derived from them that transform under the corresponding rotation action. Exhibiting spinors as concrete objects in higher dimensions, as in this three-dimensional example, generally requires constructions that depend on the dimension and on the signature of the quadratic form (for example, in Minkowski space). In general, spinors are described using the Clifford algebra or, equivalently, as representations of the spin group. After choosing a matrix realization of the Clifford algebra, the spinors may be represented concretely as column vectors on which the corresponding gamma matrices act.
Mathematical definition
A space of spinors is formally defined as an irreducible module of the Clifford algebra. Over the real or complex numbers, the Clifford algebra is a semisimple algebra, and therefore decomposes as a direct sum of full matrix algebras over a division ring, by the Artin–Wedderburn theorem. Spinor spaces are the irreducible spaces on which these components act. Thus a spinor is a "column vector" on which one of these matrix algebras acts. The space of spinors may also be defined as a spin representation of the orthogonal Lie algebra. These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional group representation of the spin group on which the center acts non-trivially.
Overview There are essentially two frameworks for viewing the notion of a spinor: the representation theoretic point of view and the geometric point of view.
Representation theoretic point of view From a representation theoretic point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors. From this view, a spinor must belong to a representation of the double cover of the rotation group SO ( n , R ) {\displaystyle \operatorname {SO} (n,\mathbb {R} )} , or more generally of a double cover of the generalized special orthogonal group SO + ( p , q , R ) {\displaystyle \operatorname {SO} ^{+}(p,q,\mathbb {R} )} on spaces with a metric signature of (p, q). These double covers are Lie groups, called the spin groups Spin(n) or Spin(p, q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued projective representations of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.) In summary, given a representation specified by the data ( V , Spin ( p , q ) , ρ ) {\displaystyle (V,{\text{Spin}}(p,q),\rho )} where V {\displaystyle V} is a vector space over K = R {\displaystyle K=\mathbb {R} } or C {\displaystyle \mathbb {C} } and ρ {\displaystyle \rho } is a homomorphism ρ : Spin ( p , q ) → GL ( V ) {\displaystyle \rho :{\text{Spin}}(p,q)\rightarrow {\text{GL}}(V)} , a spinor is an element of the vector space V {\displaystyle V} .
Geometric point of view From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as Fierz identities, are needed.
Clifford algebras
The language of Clifford algebras (sometimes called geometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras. It largely removes the need for ad hoc constructions. In detail, let V be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V subject to the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = ℂn with the standard form g(x, y) = xTy = x1y1 + ... + xnyn, one writes Cℓn(ℂ) for this Clifford algebra; since every nondegenerate symmetric bilinear form on a complex vector space is equivalent to the standard one, this notation is often used whenever dimℂ(V) = n. If n = 2k is even, then Cℓn(ℂ) is isomorphic, noncanonically, to Mat(2k, ℂ), so it has a unique irreducible module Δ of dimension 2k. If n = 2k + 1 is odd, then Cℓn(ℂ) is isomorphic to Mat(2k, ℂ) ⊕ Mat(2k, ℂ), and therefore has two inequivalent irreducible modules, each of dimension 2k. The Lie algebra so(V, g) embeds in the even part of the Clifford algebra, equipped with the commutator bracket, and hence acts on these modules. When n is odd, the two irreducible Clifford modules restrict to isomorphic irreducible representations of so(V, g); this representation is called the spin representation and is often denoted Δ. When n is even, the unique irreducible Clifford module Δ remains irreducible for the full Clifford algebra, but on restriction to the even Clifford algebra, or equivalently to the spin group, it splits as
Δ = Δ + ⊕ Δ − , {\displaystyle \Delta =\Delta _{+}\oplus \Delta _{-},}
where Δ+ and Δ− are the Weyl, or half-spin, representations. Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.
Spin groups
Spinors form a vector space, usually over the complex numbers, equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations (see diagram). The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product, or Minkowski space with its Lorentz metric. In the latter case, the "rotations" include the Lorentz boosts, but otherwise the theory is substantially similar.
Spinor fields in physics In physics, a spinor field is a field whose values lie in a spinor representation. On Minkowski space, or more generally on a spacetime manifold that admits a spin structure, one forms a spinor bundle associated to the principal spin bundle and a chosen spin representation; spinor fields are sections of this bundle. In flat spacetime this bundle may be trivialized, so spinor fields are often written simply as spinor-valued functions on spacetime. The most common spinor fields in relativistic physics are Dirac, Weyl, and Majorana spinor fields. A Dirac spinor is a section of the full complex spinor bundle. In even dimensions, when the spin representation splits into chiral halves, sections of the two summands are called Weyl spinors. A Majorana spinor is a spinor satisfying a reality condition, when the relevant spin representation admits one. Spinor fields enter physics through equations such as the Dirac equation and the Weyl equation, which are first-order differential equations on the spinor bundle. These equations describe relativistic fields of spin 1⁄2 and play a central role in quantum field theory and differential geometry. For further details, see Dirac spinor, Weyl spinor, Majorana spinor, and spinor bundle.
Spinors in representation theory
One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah–Singer index theorem, and to provide constructions in particular for discrete series representations of semisimple groups. The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article.
Attempts at intuitive understanding The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space". 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.
History The most general mathematical form of spinors was discovered by Élie Cartan in 1913. The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics. Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices. The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group. By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors. Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett and by Fritz Sauter. More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, ℂ). In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes replaced spinor spaces by the even subalgebra Cℓ01,3( R {\displaystyle \mathbb {R} } ) of the spacetime algebra Cℓ1,3( R {\displaystyle \mathbb {R} } ). As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.
Examples Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓp, q( R {\displaystyle \mathbb {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 = { + 1 i = j , i ∈ ( 1 , … , p ) − 1 i = j , i ∈ ( p + 1 , … , n ) − e j e i i ≠ j . {\displaystyle e_{i}e_{j}={\begin{cases}+1&i=j,\,i\in (1,\ldots ,p)\\-1&i=j,\,i\in (p+1,\ldots ,n)\\-e_{j}e_{i}&i\neq j.\end{cases}}}
Two dimensions The Clifford algebra Cℓ2,0( R {\displaystyle \mathbb {R} } ) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ1 and σ2, and one unit pseudoscalar i = σ1σ2. From the definitions above, it is evident that (σ1)2 = (σ2)2 = 1, and (σ1σ2)(σ1σ2) = −σ1σ1σ2σ2 = −1. The even subalgebra Cℓ02,0( R {\displaystyle \mathbb {R} } ), spanned by even-graded basis elements of Cℓ2,0( R {\displaystyle \mathbb {R} } ), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ1σ2. As a real algebra, Cℓ02,0( R {\displaystyle \mathbb {R} } ) is isomorphic to the field of complex numbers C {\displaystyle \mathbb {C} } . As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by
( a + b σ 1 σ 2 ) ∗ = a + b σ 2 σ 1 {\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}}
which, by the Clifford relations, can be written
( a + b σ 1 σ 2 ) ∗ = a + b σ 2 σ 1 = a − b σ 1 σ 2 . {\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}.}
The action of an even Clifford element γ ∈ Cℓ02,0( R {\displaystyle \mathbb {R} } ) on vectors, regarded as 1-graded elements of Cℓ2,0( R {\displaystyle \mathbb {R} } ), is determined by mapping a general vector u = a1σ1 + a2σ2 to the vector
γ ( u ) = γ u γ ∗ , {\displaystyle \gamma (u)=\gamma u\gamma ^{*},}
where γ ∗ {\displaystyle \gamma ^{*}} is the conjugate of γ {\displaystyle \gamma } , and the product is Clifford multiplication. In this situation, a spinor is an ordinary complex number. The action of γ {\displaystyle \gamma } on a spinor ϕ {\displaystyle \phi } is given by ordinary complex multiplication:
γ ( ϕ ) = γ ϕ . {\displaystyle \gamma (\phi )=\gamma \phi .}
An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:
γ ( u ) = γ u γ ∗ = γ 2 u . {\displaystyle \gamma (u)=\gamma u\gamma ^{*}=\gamma ^{2}u.}
On the other hand, in comparison with its action on spinors γ ( ϕ ) = γ ϕ {\displaystyle \gamma (\phi )=\gamma \phi } , the action of γ {\displaystyle \gamma } on ordinary vectors appears as the square of its action on spinors. Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ2 = exp(θ σ1σ2), so that the corresponding action on spinors is via γ = ± exp(θ σ1σ2/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued. In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by abuse of language, the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to computer graphics) they make sense.
Examples The even-graded element γ = 1 2 ( 1 − σ 1 σ 2 ) {\displaystyle \gamma ={\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})} corresponds to a vector rotation of 90° from σ1 around towards σ2, whi