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In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.〔Spinors can always be defined over the complex numbers. However, in some signatures there exist real spinors. Details can be found in spin representation.〕 Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation.〔A formal definition of spinors at this level is that the space of spinors is a linear representation of the Lie algebra of infinitesimal rotations of a certain kind.〕 When a sequence of such small rotations is composed (integrated) to form an overall final rotation, however, the resulting spinor transformation depends on which sequence of small rotations was used, ''unlike'' for vectors and tensors. A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360° (see picture), and it is this property that characterizes spinors. It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.〔.〕〔Quote from Elie Cartan: ''The Theory of Spinors'', Hermann, Paris, 1966, first sentence of the Introduction section of the beginning of the book (before the page numbers start): "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 of dimensions, each spinor having components where or ." 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.〔More precisely, it is the fermions 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 three-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 classes) of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle (below). These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.〔Formally, the spin group is the group of relative homotopy classes 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 act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.〔More formally, the space of spinors can be defined as an (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).〕 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.〔Geometric algebra is a name for the Clifford algebra in an applied setting.〕 After choosing an 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,〔the Pauli matrices correspond to angular momenta operators about the three coordinate axes. This makes them slightly a-typical gamma matrices because in addition to their anti commutation 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, and 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〔The metric signature relevant as well if we are concerned with real spinors. See spin representation.〕) column vectors will either be 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.〔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.〕 ==Introduction== What characterizes spinors and distinguishes them from geometric vectors 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 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 puzzle (shown) famously 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. 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 momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors 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 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,〔This is the set of 2×2 complex traceless hermitian matrices.〕 realizing it as a group of rotations among them,〔Except for a kernel of corresponding to the two different elements of the spin group that go to the same rotation.〕 but it also acts on the column vectors (that is, the spinors). More generally, a Clifford algebra can be constructed from any vector space equipped with a (nondegenerate) quadratic form, such as Euclidean space with its standard dot product or Minkowski space with its standard Lorentz metric. Given a suitably normalized basis of , the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations, and the space of spinors is the space of column vectors with components on which those matrices act. Although the Clifford algebra can be defined abstractly in a coordinate-independent way, its particular realization as a specific algebra of matrices depends on which orthogonal axes the gamma matrices represent. So what precisely constitutes a "column vector" (or spinor) also depends on such arbitrary choices.〔Although there are several more intrinsic constructions, the spin representations are not functorial in the quadratic form, so they cannot be built up naturally within the tensor algebra.〕 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.〔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 or it may decompose into a pair of so-called "half-spin" or Weyl representations.〔The Clifford algebra can be given an even/odd 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.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「spinor」の詳細全文を読む スポンサード リンク
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