Editing Spinor (section)

== Introduction ==
{{multiple image
 | image1=Belt trick 2.gif
 | image2=Belt trick 1.gif
 | total_width=404
 | footer=A gradual rotation can be visualized as a ribbon in space.{{efn|The [[TNB frame]] of the ribbon defines a rotation continuously for each value of the arc length parameter.}} Two gradual rotations with different classes, one through 360° and one through 720° are illustrated here in the [[belt trick]] puzzle. A solution of the puzzle is a continuous manipulation of the belt, fixing the endpoints, that untwists it. This is impossible with the 360° rotation, but possible with the 720° rotation. A solution, shown in the second animation, gives an explicit [[homotopy]] in the rotation group between the 720° rotation and the 0° identity rotation.
}}
[[File:Belt Trick.ogv|thumb|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°.]]
[[File:Antitwister.ogv|thumb|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 vector]]s 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 [[Cauchy stress tensor|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 ([[continuous function|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 ("[[topology|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{{math|π}} and the other through an angle of 4{{math|π}}, 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 momentum operator|angular momenta]] about) the three coordinate axes. These are 2×2 matrices with [[complex number|complex]] entries, and the two-component complex [[column vector]]s 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 matrix|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,{{efn|This is the set of 2×2 complex [[Trace (linear algebra)|traceless]] [[hermitian matrix|hermitian matrices]].}} realizing it as a group of rotations among them,{{efn|Except for a [[kernel (algebra)|kernel]] of <math>\{\pm 1\}</math> corresponding to the two different elements of the spin group that go to the same rotation.<ref>For details, see {{cite journal |author-link=William Frederick Eberlein|first=W. F. |last=Eberlein |title=The Spin Model of Euclidean 3-Space |journal=The American Mathematical Monthly |volume=69 |issue=7 |pages=587–598 |doi=10.2307/2310821  |year=1962|jstor=2310821 }}</ref>}} but it also acts on the column vectors (that is, the spinors).

More generally, a Clifford algebra can be constructed from any vector space ''V'' equipped with a (nondegenerate) [[quadratic form]], such as [[Euclidean space]] with its standard dot product or [[Minkowski space]] with its standard Lorentz metric. The [[spin group|space of spinors]] is the space of column vectors with <math>2^{\lfloor\dim V/2\rfloor}</math> components. The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group.{{efn|So the ambiguity in identifying the spinors themselves persists from the point of view of the group theory, and still depends on choices.}} Depending on the dimension and [[metric signature]], this realization of spinors as column vectors may be [[irreducible representation|irreducible]] or it may decompose into a pair of so-called "half-spin" or Weyl representations.{{efn|The Clifford algebra can be given an even/odd [[graded algebra|grading]] from the parity of the degree in the gammas, and the spin group and its Lie algebra both lie in the even part. Whether here by "representation" we mean representations of the spin group or the Clifford algebra will affect the determination of their reducibility. Other structures than this splitting can also exist; precise criteria are covered at [[spin representation]] and [[Clifford algebra]].}} When the vector space ''V'' is four-dimensional, the algebra is described by the [[gamma matrices]].