Spinor Totally Explained

Everything about Spinors totally explained

In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are elements of a complex vector space introduced to expand the notion of spatial vector. They are needed because the full structure of the group of rotations in a given number of dimensions requires some extra number of dimensions to exhibit it. More formally, spinors can be defined as geometrical objects constructed from a given vector space endowed with a quadratic form by means of an algebraic or quantization procedure. The rotation group acts upon the space of spinors, but for an ambiguity in the sign of the action. Spinors thus form a projective representation of the rotation group. One can remove this sign ambiguity by regarding the space of spinors as a (linear) group representation of the spin group Spin(n). In this alternative point of view, many of the intrinsic and algebraic properties of spinors are more clearly visible, but the connection with the original spatial geometry is more obscure. On the other hand the use of complex number scalars can be kept to a minimum.
Historically, spinors in general were discovered by Élie Cartan in 1913. Later, spinors were adopted by quantum mechanics in order to study the properties of the intrinsic angular momentum of the electron and other fermions. Today spinors enjoy a wide range of physics applications. Classically, spinors in three dimensions are used to describe the spin of the non-relativistic electron. Via the Dirac equation, Dirac spinors are required in the mathematical description of the quantum state of the relativistic electron. In quantum field theory, spinors describe the state of relativistic many-particle systems.
In mathematics, particularly in differential geometry and algebraic geometry, spinors have since found broad applications to index theory, symplectic geometry, gauge theory, complex algebraic geometry, global analysis, and algebraic and differential topology.

Overview

In the classical geometry of space, a vector exhibits a certain behavior when it's acted upon by a rotation or reflected in a hyperplane. However, in a certain sense rotations and reflections contain finer geometrical information than can be expressed in terms of their actions on vectors. Spinors are objects constructed in order to encompass more fully this geometry. (See orientation entanglement.)
   There are essentially two frameworks for viewing the notion of a spinor.
   One is representation theoretic. In this point of view, one knows a priori that there are some representations of the Lie algebra of the orthogonal group which can't be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors. In this view, a spinor must belong to a representation of the double cover of the rotation group SO(n,R), or more generally of the generalized special orthogonal group SO(p, q,R) on spaces with metric signature (p,q). These double-covers are Lie groups, called the spin groups Spin(p,q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group.
   The other point of view is geometrical. 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 being able to say precisely what a spinor is, without invoking some non-constructive theorem from representation theory. Representation theory must eventually supplement the geometrical machinery once the latter becomes too unwieldy.

Clifford algebras

The language of Clifford 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 removes the need for ad hoc constructions, by introducing a type of geometric algebra.
   Using the properties of Clifford algebras, it's then possible to determine the number and type of all irreducible spaces of spinors. In this view, a spinor is an element of the fundamental representation of the Clifford algebra Cℓ_n(C) over the complex numbers (or, more generally, of Cℓ_p,q(R) over the reals). In some cases it becomes clear that the spinors split into irreducible components under the action of Spin(p,q). In detail, if V is a finite-dimensional complex vector space with nondegenerate bilinear form g, the Clifford algebra is the algebra, Cℓ(V,g), generated by V along with the anticommutation relation xy + yx = 2g(x,y). It is an abstract version of the algebra generated by the gamma matrices or Pauli matrices. The Clifford algebra Cℓ_n(C) is algebraically isomorphic to the algebra Mat(2^k,C) of 2^k × 2^k complex matrices, if n = dim(V) = 2k; or the algebra Mat(2^k,C)⊕Mat(2^k,C) of two copies of the 2^k × 2^k matrices, if n = dim(V) = 2k+ 1. It therefore has a unique irreducible representation commonly denoted by Δ of dimension 2^k. Any such irreducible representation is, by definition, a space of spinors called a spin representation.
   The subalgebra of the Clifford algebra spanned by products of an even number of vectors in V contains the Lie algebra so(V,g) of the orthogonal group as a Lie subalgebra. Consequently, Δ is a representation of so(V,g). If n is odd, this representation is irreducible. If n is even, it splits again into two irreducible representations Δ = Δ₊ ⊕ Δ_- called the half-spin representations.
   Irreducible representations 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.

Terminology in physics

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra Cℓ(p,q), into which the spin group Spin(p,q) may be embedded. On a 2k- or 2k+1-dimensional space a Dirac spinor may be represented as a vector of 2^k complex numbers. (See Special unitary group.) 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 Cℓ(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.

Spinors in representation theory

One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the index theorem, and to provide constructions in particular for discrete series representations of semisimple groups.

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.
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.

Examples

Some important simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓ_p,q(R). This is an algebra built up from an orthonormal basis of n = p + q mutually orthogonal vectors under addition and multiplication, p of which have norm +1 and q of which have norm −1, with the product rule for the basis vectors �

e_i e_j = Bigg(3)

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

Spin(4) equiv SU(2) 	imes SU(2)

. 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

Spin(5)equiv USp(4)equiv Sp(2)

which implies that the single spinor representation is 4-dimensional and pseudoreal.

In 6 Euclidean dimensions, the isomorphism

Spin(6)equiv SU(4)

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.

d+8

dimensions, the number of distinct irreducible spinor representations and their reality (whether they're 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

SL(2,C) equiv Spin(3,1)

Metric signature	eft-handed Weyl	ight-handed Weyl	onjugacy	irac	eft-handed Majorana-Weyl	ight-handed Majorana-Weyl	ajorana
	complex	complex		complex	real	real	real
(2,0)	1	1	mutual	2	-	-	2
(1,1)	1	1	self	2	1	1	2
(3,0)	-	-	-	2	-	-	-
(2,1)	-	-	-	2	-	-	2
(4,0)	2	2	self	4	-	-	-
(3,1)	2	2	mutual	4	-	-	4
(5,0)	-	-	-	4	-	-	-
(4,1)	-	-	-	4	-	-	-
(6,0)	4	4	mutual	8	-	-	8
(5,1)	4	4	self	8	-	-	-
(7,0)	-	-	-	8	-	-	8
(6,1)	-	-	-	8	-	-	-
(8,0)	8	8	self	16	8	8	16
(7,1)	8	8	mutual	16	-	-	16
(9,0)	-	-	-	16	-	-	16
(8,1)	-	-	-	16	-	-	16

Further Information

Get more info on 'Spinors'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://spinor.totallyexplained.com">Spinor Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.