Spinor

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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 <math>2\pi<math> radians.

Contents

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).

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.

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.

A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2n complex numbers. (See Special unitary group.)

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 <math>\mathfrak{so}(p,q)<math> is the same as the complexification of <math>\mathfrak{so}(p+q)<math>, we can focus upon the latter, at least for complex reps only.

Recall that the rank of <math>\mathfrak{so}(2n)<math> is n and its roots are the permutations of

<math>(\pm 1,\pm 1, 0, 0, \dots, 0)<math>

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 <math>\mathfrak{so}(2)<math>, which isn't semisimple.

Recall also that the rank of <math>\mathfrak{so}(2n+1)<math> is n and its roots are the permutations of

<math>(\pm 1, \pm 1, 0, 0, \dots, 0)<math>

and the permutations of

<math>(\pm 1, 0, 0, \dots, 0)<math>.

for <math>\mathfrak{so}(2n)<math>, there is an irrep whose weights are all possible combinations of

<math>(\pm {1\over 2},\pm {1\over 2}, \dots, \pm{1\over 2})<math>

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

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

<math>(\pm{1\over 2},\pm{1\over 2},\dots,\pm{1\over 2})<math>

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

The direct sum of both Weyl spinors is a Dirac spinor.

Let's now go over to <math>\mathfrak{so}(2n+1)<math>. Here, there's an irrep whose weights are all possible combinations of

<math>(\pm {1\over 2},\pm {1\over 2},\dots,\pm{1\over 2})<math>

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 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 <math>\mathfrak{so}(p,q)<math> unless either p or q is zero.

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.

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 <math>e^{\pm i\phi/2}<math> under a rotation by angle <math>\phi<math>.
  • 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 <math>SU(2) \cong \mathit{Spin}(3)<math> which allows us to define the action of <math>Spin(3)<math> on a complex 2-component column (a spinor); the generators of <math>SU(2)<math> can be written as Pauli matrices.
  • In 4 Euclidean dimensions, the corresponding isomorphism is <math>Spin(4) \equiv SU(2) \times SU(2)<math>. There are two inequivalent pseudoreal 2-component Weyl spinors and each of them transforms under one of the <math>SU(2)<math> factors only.
  • In 5 Euclidean dimensions, the relevant isomorphism is <math>Spin(5)\equiv USp(4)\equiv Sp(2)<math> which implies that the single spinor representation is 4-dimensional and pseudoreal.
  • In 6 Euclidean dimensions, the isomorphism <math>Spin(6)\equiv SU(4)<math> 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 <math>d+8<math> dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in <math>d<math> dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity.
  • In spacetimes with <math>p<math> spatial and <math>q<math> time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the <math>p+q<math>-dimensional Euclidean space, but the reality projections mimic the structure in <math>|p-q|<math> 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 <math>SL(2,C) \equiv Spin(3,1)<math>.
Metric signatureleft handed Weylright handed WeylconjugacyDiracleft handed Majorana-Weylright handed Majorana-WeylMajorana
complexcomplex complexrealrealreal
(2,0)11mutual2--2
(1,1)11self2112
(3,0)---2---
(2,1)---2--2
(4,0)22self4---
(3,1)22mutual4--4
(5,0)---4---
(4,1)---4---
(6,0)44mutual8--8
(5,1)44self8---
(7,0)---8--8
(6,1)---8---
(8,0)88self168816
(7,1)88mutual16--16
(9,0)---16--16
(8,1)---16--16

See also

de:Spinor

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