Editing Spinor (section)

{{short description|Non-tensorial representation of the spin group}}
[[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.{{efn|Spinors in three dimensions are points of a [[line bundle]] over a [[conic]] in the [[projective plane]]. In this picture, which is associated to spinors of a three-dimensional [[pseudo-Euclidean space]] of signature (1,2), the conic is an ordinary real conic (here the circle), the line bundle is the Möbius bundle, and the spin group is {{math|SL<sub>2</sub>(<math>\Reals</math>)}}. In Euclidean signature, the projective plane, conic and line bundle are over the complex instead, and this picture is just a real slice.}}]]

In [[geometry]] and [[physics]], '''spinors''' (pronounced "spinner" IPA {{IPAc-en|s|p|ɪ|n|ɚ}}) are elements of a [[complex numbers|complex]] [[vector space]] that can be associated with [[Euclidean space]].{{efn|Spinors can always be defined over the complex numbers. However, in some signatures there exist real spinors. Details can be found in [[spin representation]].}} A spinor transforms linearly when the Euclidean space is subjected to a slight ([[infinitesimal transformation|infinitesimal]]) rotation,{{efn|A formal definition of spinors at this level is that the space of spinors is a [[linear representation]] of the [[Lie algebra]] of [[infinitesimal rotation]]s of a [[spin representation|certain kind]].}} but unlike [[Euclidean vector|geometric vectors]] and [[tensor]]s, 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. 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 [[Section (fiber bundle)|sections]] of [[vector bundle]]s – in the case of the [[exterior algebra]] bundle of the [[cotangent bundle]], they thus become "square roots" of [[differential form]]s).

It is also possible to associate a substantially similar notion of spinor to [[Minkowski space]], in which case the [[Lorentz transformation]]s of [[special relativity]] play the role of rotations. Spinors were introduced in geometry by [[Élie Cartan]] in 1913.<ref>{{Harvnb|Cartan|1913}}.</ref>{{efn|"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 <math>n</math> of dimensions, each spinor having <math>2^\nu</math> components where <math>n = 2\nu + 1</math> or <math>2\nu</math>."<ref name="cartan-1966-quote">Quote from Elie Cartan: ''The Theory of Spinors'', Hermann, Paris, 1966, first sentence of the Introduction section at the beginning of the book, before page numbers start.</ref> The star (*) refers to Cartan (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.{{efn|More precisely, it is the [[fermion]]s of [[spin-1/2]] that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2-component spinors transforming under 3-dimensional infinitesimal rotations. The relativistic [[Dirac equation]] for the electron is an equation for 4-component spinors transforming under infinitesimal Lorentz transformations, for which a substantially similar theory of spinors exists.}}

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 class]]es) 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.{{efn|Formally, the spin group is the group of [[relative homotopy class]]es with fixed endpoints in the rotation group.}} 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 [[Group action (mathematics)|act]] as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.{{efn|More formally, the space of spinors can be defined as an ([[irreducible representation|irreducible]]) representation of the spin group that does not factor through a representation of the rotation group (in general, the connected component of the identity of the [[orthogonal group]]).}} 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.{{efn|[[Geometric algebra]] is a name for the Clifford algebra in an applied setting.}} A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.<ref>{{cite journal |author=Rukhsan-Ul-Haq |url=https://www.ias.ac.in/article/fulltext/reso/021/12/1105-1117 |title=Geometry of Spin: Clifford Algebraic Approach |journal=Resonance |date=December 2016 |volume=21 |issue=12 |pages=1105–1117|doi=10.1007/s12045-016-0422-5 |s2cid=126053475 |url-access=subscription }}</ref> After choosing an [[Orthonormality|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,{{efn|The Pauli matrices correspond to [[angular momentum operator|angular momenta]] operators about the three coordinate axes. This makes them slightly atypical gamma matrices because in addition to their anticommutation relation they also satisfy commutation relations.}} 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{{efn|The [[metric signature]] relevant as well if we are concerned with real spinors. See [[spin representation]].}}) column vectors will either be [[irreducible representation|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.{{efn|Whether the representation decomposes depends on whether they are regarded as representations of the spin group (or its Lie algebra), in which case it decomposes in even but not odd dimensions, or the Clifford algebra when it is the other way around. Other structures than this decomposition can also exist; precise criteria are covered at [[spin representation]] and [[Clifford algebra]].}}