The orthogonal Lie algebra i. Mathematical definition[ edit ] The space of spinors is formally defined as the fundamental representation of the Clifford algebra. This may or may not decompose into irreducible representations. 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.
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The orthogonal Lie algebra i. Mathematical definition[ edit ] The space of spinors is formally defined as the fundamental representation of the Clifford algebra. This may or may not decompose into irreducible representations. 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[ edit ] There are essentially two frameworks for viewing the notion of a spinor. 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. These double covers are Lie groups , called the spin groups Spin n or Spin p, q. 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. 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. Further information: Clifford algebra The language of Clifford algebras  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 bilinear form g.
It is an abstract version of the algebra generated by the gamma or Pauli matrices. If n is odd, this Lie algebra representation is irreducible. 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. 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 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 cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors 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 boosts , but otherwise the theory is substantially similar. Terminology in physics[ edit ] The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional space-time.
To obtain the spinors of physics, such as the Dirac spinor , one extends the construction to obtain a spin structure on 4-dimensional space-time Minkowski space. Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO 3,1 symmetry, and then builds the spin group at each point.
The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fibre bundle , the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation , or the Weyl equation on the fiber bundle. These equations Dirac or Weyl have solutions that are plane waves , having symmetries characteristic of the fibers, i.
Such plane-wave solutions or other solutions of the differential equations can then properly be called fermions ; fermions have the algebraic qualities of spinors.
By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another. There does not seem to be any a priori reason why this would be the case. The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra. Weyl spinors are insufficient to describe massive particles, such as electrons , since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed.
The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor. The situation for condensed matter physics is different: one can can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors to far more exotic materials.
In , an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.
At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah—Singer index theorem , and to provide constructions in particular for discrete series representations of semisimple groups. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article. Attempts at intuitive understanding[ edit ] The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space".
Clifford Algebras and Spinors: Contents
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LOUNESTO CLIFFORD ALGEBRAS AND SPINORS PDF
Clifford Algebras and Spinors