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Spinor
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=== Spin groups === [[File:Spin representations do not lift.svg|thumb|The spin representation Ξ 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 [[special orthogonal group|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 covering group|double cover]]. So for every rotation there are two elements of the spin group that represent it. [[Geometric vector]]s and other [[tensor]]s 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 boost]]s, but otherwise the theory is substantially similar.{{Citation needed|date=October 2023}}
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