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. Euclid, detail from The School of Athens by Raphael. ... A Superconductor demonstrating the Meissner Effect Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... 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. ... Group 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 in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...

Overview

A spinor is a representation of the double cover of the rotation group SO(n,R), or more generally the generalized special orthogonal group, SO(p,q,R), where p+q=n for spinors in a space with a nontrivial metric signature, which is a real Lie group called the spinor group Spin(p,q), which is odd under a rotation by 2π. Group 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 mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... 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. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics the spinor group or spin group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ...

Spinors are sometimes described as "square roots of vectors" because the vector representation sometimes 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 Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ... Ettore Majorana (Catania, Sicily, 1906 - Tyrrhenian 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.) Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ... In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field 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. In theoretical physics, the Rarita-Schwinger equation is the field equation of spin-3/2 fermions. ...

Mathematical details

Let's focus on complex representations first. 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 representations. 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 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 In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ...

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 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. If G is a group and ρ is a representation of it over the vector space V, then the dual representation is defined over the dual vector space as follows: is the transpose of ρ(g-1) for all g in G. is also a representation, as you may check explicitly. ...

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... If G is a group and ρ is a representation of it over the complex vector space V, then the complex conjugate representation ρ* is defined over the conjugate vector space V* as follows: ρ*(g) is the conjugate of ρ(g) for all g in G. ρ* is also a representation, as you may... If G is a group and ρ is a representation of it over the vector space V, then the dual representation is defined over the dual vector space as follows: is the transpose of ρ(g-1) for all g in G. is also a representation, as you may check explicitly. ...

History

The most general mathematical form of spinors was discovered by Élie Cartan in 1913. The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics. Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... 1913 (MCMXIII) is a common year starting on Wednesday. ... Paul Ehrenfest Paul Ehrenfest (Vienna, January 18, 1880 – Amsterdam, September 25, 1933) was an Austrian physicist and a mathematician, who obtained Dutch citizenship on March 24, 1922. ... Fig. ...

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced spin matrices. The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group. By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute created games such as Tangloids to teach and model the calculus of spinors. Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... Wolfgang Pauli Wolfgang Ernst Pauli (April 25, 1900 – December 15, 1958) was an Austrian physicist noted for his work on the theory of spin, and in particular the discovery of the Exclusion principle, which underpins the whole of chemistry. ... 1927 (MCMXXVII) was a common year starting on Saturday (link will take you to calendar). ... The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Wolfgang Pauli. ... 1928 (MCMXXVIII) was a leap year starting on Sunday (link will take you to calendar). ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... Properties The electron is a fundamental subatomic particle that carries a negative electric charge. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... // Events and trends A public speech by Benito Mussolini, founder of the Fascist movement The 1930s were described as an abrupt shift to more radical lifestyles, as countries were struggling to find a solution to the global depression. ... 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 Wolfgang 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