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 2 and the other through an angle of 4, 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 of Hermitian matrices, with complex entries, whose traces are zero. Any such matrix can be written as
where is real, is complex, and is the complex conjugate of . Then is a three-dimensional vector space over the real field. The negative determinant of is , which is the sum of the squares of the three real coordinates . Thus, , equipped with this form, is a real Euclidean three-space (like equipped with its dot product). To describe rotations of this Euclidean space, consider all complex matrices
satisfying One has that the identity matrix, where the star denotes the Hermitian conjugate:
The set of such matrices is the special unitary group Since , the operation given by for all preserves the determinant. That is, is a rotation of the Euclidean space Thus, rotations of can be described by special unitary matrices But, for each the pair of unitary maps and induce the same rotation on : So does not feel the difference between the distinct unitary matrices and
In this example, the spinors are by definition the complex column vectors on which the special unitary matrices act. These do feel the difference between the matrices and . In particular, if we take a path in from the identity to its negative (such as for all , which induces a rotation through the angle on ), the associated rotations start and end at the identity, and the space feels no change, whereas a column vector will be transformed to its negative.
The association of the rotation to each is two-to-one, and the kernel is the group . This exhibits the double cover of the rotation group, and as its spin group. In this setting, the Euclidean space (of "physical vectors") is the real vector space , while the space of spinors is the complex two-dimensional vector space of column vectors on which acts. Given a single spinor , a real vector in can be formed as . 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 transforming under the natural action of . The associated geometric vectors are the traceless Hermitian matrices, transforming by conjugation . 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 , or more generally of a double cover of the generalized special orthogonal group on spaces with a metric signature of . These double covers are Lie groups, called the spin groups or . 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 where is a vector space over or and is a homomorphism , a spinor is an element of the vector space .
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 algebrasNamed after William Kingdon Clifford, (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 is the algebra generated by V subject to the anticommutation relation . It is an abstract version of the algebra generated by the gamma or Pauli matrices. If with the standard form , one writes 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 . If is even, then is isomorphic, noncanonically, to , so it has a unique irreducible module of dimension . If is odd, then is isomorphic to , and therefore has two inequivalent irreducible modules, each of dimension . The Lie algebra embeds in the even part of the Clifford algebra, equipped with the commutator bracket, and hence acts on these modules. When is odd, the two irreducible Clifford modules restrict to isomorphic irreducible representations of ; this representation is called the spin representation and is often denoted . When is even, the unique irreducible Clifford module remains irreducible for the full Clifford algebra, but on restriction to the even Clifford algebra, or equivalently to the spin group, it splits as
where and 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