In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of 2π radians. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ... In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but this... This article is about rotation as a movement of a physical body. ...

Overview

A spinor of a certain type is an element of a specific projective representation of the rotation group SO(n,R), or more generally of the group SO(p,q,R), where p + q = n for spinors in a space of nontrivial signature. This is equivalent to an ordinary (non-projective) representation of the double cover of SO(p,q,R), which is a real Lie group called the spinor group Spin(p,q). In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q). ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ...

Spinors are often described as "square roots of vectors" because the vector representation appears in the tensor product of two copies of the spinor representation. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra C(p,q), into which Spin(p,q) may be embedded. In even dimensions, this representation is reducible when taken as a representation of Spin(p,q) and may be decomposed into two: the left-handed and right-handed Weyl spinor representations. In addition, sometimes the non-complexified version of C(p,q) has a smaller real representation, the Majorana spinor representation. If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana-Weyl spinor representations. Of all these, only the Dirac representation exists in all dimensions. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations. Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, the term irreducible is used in several ways. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician and physicist, one of the first people to combine general relativity with the laws of electromagnetism. ... Ettore Majorana (Catania, Sicily, 1906 - Tirrenian Sea (supposedly), 1938) was a great Italian physicist who abruptly disappeared at the age of 32. ...

A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2ⁿ complex numbers. (See Special unitary group.) In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ...

There are also more complicated spinors like the Rarita-Schwinger spinor, which will not be covered here.

Mathematical details

Let's focus on complex reps first. So, it's convenient to work with the complexified Lie algebra. Since the complexification of is the same as the complexification of , we can focus upon the latter, at least for complex reps only. In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

Recall that the rank of is n and its roots are the permutations of See also Simple Lie group. ...

where there are n coordinates and all but two are zero and the absolute values of the nonzero coordinates are 1. This does not apply to , which isn't semisimple. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...

Recall also that the rank of is n and its roots are the permutations of

and the permutations of

.

for , there is an irrep whose weights are all possible combinations of In mathematics, the term irreducible is used in several ways. ... Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ...

with an even number of minuses and each weight has multiplicity 1. This is a Weyl spinor and it is 2^n-1 dimensional. This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...

There is also another irrep whose weights are all possible combinations of

with an odd number of minuses and each weight has multiplicity 1. This is an inequivalent Weyl spinor and it is 2^n-1 dimensional.

The direct sum of both Weyl spinors is a Dirac spinor. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Let's now go over to . Here, there's an irrep whose weights are all possible combinations of

and each weight has multiplicity 1. This is a Dirac spinor and it is 2ⁿ dimensional.

In both even and odd dimensions, the tensor product of the Dirac representation with itself contains the trivial representation, the vector representation and the adjoint representation. The first means the Dirac representation is self-dual. The second means there is a nonzero intertwiner from the tensor product of the vector representation and the Dirac representation to the dual of the Dirac representation. This is represented by the γ matrices, γⁱ. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector space V over a field K and (ii) all elements g of G act on V as the identity mapping. ... The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ...

In 4n dimensions, each Weyl representation is self-dual. In 4n+2 dimensions, both Weyl representations are duals of each other.

One thing to note, though, is these spinors are not unitary except in Euclidean space. This means complex conjugate representations and dual representations do not coincide for unless either p or q is zero. In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group...

History

Spinors were invented by Wolfgang Pauli and Paul Dirac to describe the physical properties of spin, especially the properties of fermions whose spin numerically equals one half. The word "spinor" was coined by Paul Ehrenfest. The mathematics of spinors is said to have been anticipated by Elie Cartan as early as 1913. In the early 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute created games such as Tangloids to teach and model the calculus of spinors. This article is about Austrian-Swiss physicist Wolfgang Pauli. ... Paul Adrien Maurice Dirac, (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, spin is an intrinsic angular momentum associated with microscopic particles. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... Paul Ehrenfest (January 18, 1880 – September 25, 1933) was an Austrian physicist and mathematician from Vienna. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... Events and trends Technology Jet engine invented First atom was split with a particle accelerator Disney adopts a three-color Technicolor process for cartoons The photocopier is invented Air mail service across the Atlantic Science Nuclear fission discovered by Otto Hahn, Lise Meitner and Fritz Strassmann Pluto, the ninth planet... Piet Hein (December 16, 1905 - April 18, 1996) was a scientist, mathematician, inventor, author, and poet, often writing under the Old Norse pseudonym Kumbel meaning tombstone. His short poems, gruks (or grooks), first started to appear in the daily newspaper Politiken shortly after the Nazi Occupation in April 1940 under... The Niels Bohr Institute is part of the Niels Bohr Institute for Astronomy, Physics and Geophysics of the University of Copenhagen. ... Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of Spinors. ...

Examples in low dimensions

• In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1-dimensional representation that does not transform.

• In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component complex representations, i.e. complex numbers that get multiplied by under a rotation by angle φ.

• In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and pseudoreal. The existence of spinors in 3 dimensions follows from the isomorphism of the groups which allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as Pauli matrices.

• In 4 Euclidean dimensions, the corresponding isomorphism is . There are two inequivalent pseudoreal 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.

• In 5 Euclidean dimensions, the relevant isomorphism is which implies that the single spinor representation is 4-dimensional and pseudoreal.

• In 6 Euclidean dimensions, the isomorphism guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.

• In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.

• In 8 Euclidean dimensions, there are two Weyl-Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality.

• In d + 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in d dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity.

• In spacetimes with p spatial and q time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the p + q-dimensional Euclidean space, but the reality projections mimic the structure in | p − q | Euclidean dimensions. For example, in 3+1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism .

In mathematics and theoretical physics, a real representation is a group representation that is equivalent to its complex conjugate and that also allows the matrices representing the group elements to be real — unlike a pseudoreal representation (symplectic representation). ... In mathematics, a complex representation is a group representation that is neither real nor pseudoreal. ... In mathematics and theoretical physics, a pseudoreal representation is a group representation that is equivalent to its complex conjugate, but that is not a real representation. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. ... In mathematics and theoretical physics, a pseudoreal representation is a group representation that is equivalent to its complex conjugate, but that is not a real representation. ... In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. ... Categories: Stub | Lie groups ... In mathematics, the Bott periodicity theorem is a result from homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved to be of foundational significance for much further research, in particular in K-theory. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

See also

• spinor bundle
• anyon
• twistor