Spinor

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In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of auxiliary mathematical objects introduced to expand the notion of spatial vector. They are needed because the full structure 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 an inner product by means of an algebraic^[1] or quantization^[2] 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 speaking, spinors in general were discovered by Élie Cartan, a few years ahead of their recognition as fundamental in quantum mechanics in the theory of the Dirac equation. The conventional algebraic route to their discussion is through the theory of Clifford algebras, which produce naturally the basic spin representation.

Contents 1 Overview 1.1 Basic approaches 1.2 Clifford algebras 1.3 Terminology in physics 1.4 Spinors in representation theory 1.5 History 2 Detailed discussions 2.1 Simple examples 2.1.1 Two dimensions: Cℓ02,0(R) 2.1.2 Three dimensions: Cℓ03,0(R) 2.2 Higher dimensions 2.2.1 Notation 2.2.2 Method 2.2.3 Even case 2.2.4 Weyl spinors 2.2.5 Odd case 2.3 Summary in low dimensions 2.4 Spinors of the Pauli spin matrices 2.5 Spinors of the Dirac algebra 2.5.1 Conventions 2.5.2 Construction 3 See also 4 Notes

[edit] Overview

In the classical geometry of space, a vector exhibits a certain behavior when it is 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.

Spinors can often be thought of as representing combined rotations and dilations. Using the spinor ψ and its adjoint ψ^˜ a rotation R mapping a vector v to v' can be written

One can therefore identify a spinor with a spatial vector, by considering what transformation ψ is needed to boost a unit vector e₁ to a desired vector u,

Further rotations and dilations of the vector u can then be identified with mappings of ψ to φψ, where φ is another spinor. However, spinors in general reflect much more than single points under transformation: they naturally combine to give a whole algebra, reflecting the algebra of all the transformations under further transformation. An important property of spinors is that a purely imaginary spinor, which maps ψ to iψ corresponds to a rotation of 180° (not 90°); whilst a spinor which maps ψ to -ψ corresponds to a rotation of 360°: spinors are not invariant under 360° rotation, but change sign.

[edit] Basic approaches

There are essentially two frameworks for viewing the notion of a spinor.

One is representation theoretic. In this view, one knows a priori that there are some representations of the Lie algebra of the orthogonal group which cannot be formed by the usual constructions. These missing representations are then labeled the spin representations, and their constituents spinors. In this view, a spinor must be 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 one 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.

[edit] Clifford algebras

Explicit constructions of spinors can often be useful for developing some intuition about them and their properties, as well as firmly rooting them to classical linear geometry. However, for classifying all the spin representations (especially the real representations), the explicit constructions can become cumbersome.

The language of Clifford algebras,^[3] a hybrid approach, provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations. It removes the need for ad hoc constructions, by introducing a type of geometric algebra.

Using the properties of Clifford algebras, it is 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 C(n) over the complex numbers (or, more generally, of C(p,q) over the reals). In some cases it becomes clear that the spinors split into irreducible components under the action of Spin(p,q).

[edit] Terminology in physics

The most typical type of spinor, the Dirac spinor,^[4] is an element of the fundamental representation of the complexified Clifford algebra C(p,q), into which the spin group 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^[5] representations. In addition, sometimes the non-complexified version of C(p,q) has a smaller real representation, the Majorana spinor representation.^[6] 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.

[edit] Spinors in representation theory

Main article: Spinor representation

One of the main applications 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.

[edit] History

The most general mathematical form of spinors was discovered by Élie Cartan^[7] 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^[8] in 1927, when he introduced spin matrices. The following year, Paul Dirac^[9] 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.

[edit] Detailed discussions

[edit] Simple 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

[edit] Two dimensions: Cℓ⁰_2,0(R)

The Clifford algebra Cℓ_2,0(R) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ₁ and σ₂, and one unit pseudoscalar i = σ₁σ₂. From the definitions above, it is evident that (σ₁)² = (σ₁)² = 1, and (σ₁σ₂)(σ₁σ₂) = -σ₁σ₁σ₂σ₂ = -1.

The corresponding subset algebra of spinors, Cℓ⁰_2,0(R), is made up of the even-graded elements of Cℓ_2,0(R), ie linear combinations of 1 and σ₁σ₂. This can be factorised into scalar dilations ρ^1/2, which map a general vector u = a₁σ₁ + a₂σ₂ to

and rotations ψ = exp(-σ₁σ₂ θ/2) = cos(θ/2) - σ₁σ₂ sin(θ/2), which rotate u through an angle θ in the orientation from σ₁ around towards σ₂,

where ψ^˜ is the "reverse" of ψ, ψ^˜ = exp(-σ₂σ₁ θ/2) = cos(θ/2) - σ₂σ₁ sin(θ/2)

Examples

• Thus, for example, the spinor

corresponds to a rotation of 90° from σ₁ around towards σ₂, which can be checked by confirming that

• Similarly the spinor ψ = -σ₁σ₂ corresponds to a rotation of 180°:

• Continuing on further, the spinor ψ = -1 corresponds to a rotation of 360°:

[edit] Three dimensions: Cℓ⁰_3,0(R)

Main articles Spinors in three dimensions, Quaternions and spatial rotation

The Clifford algebra Cℓ_3,0(R) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ₁, σ₂ and σ₃, the three unit bivectors σ₁σ₂, σ₂σ₃, σ₃σ₁ and the pseudoscalar i = σ₁σ₂σ₃. It is straightforward to show that (σ₁)² = (σ₂)² = (σ₃)² = 1, and (σ₁σ₂)² = (σ₂σ₃)² = (σ₃σ₁)² = (σ₁σ₂σ₃)² = -1.

The sub-algebra of spinors is made up of scalar dilations,

and rotations

where

corresponds to a rotation through an angle θ about an axis defined by a unit vector v = a₁σ₁ + a₂σ₂ + a₃σ₃

As a special case, it is easy to see that if v = σ₃ this reproduces the σ₁σ₂ rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ₃ direction invariant, since

The bivectors σ₂σ₃, σ₃σ₁ and σ₁σ₂ are in fact Hamilton's quaternions i, j and k, discovered in 1843:

The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable compactness of the corresponding spinors, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.

[edit] Higher dimensions

In the case of spinors in higher dimensions, there is a sharp distinction between the even and odd dimensions.

In either case, spinors may be constructed geometrically through a procedure of quantization due to Richard Brauer and Hermann Weyl.^[10]

[edit] Notation

Let n = 2k or 2k + 1 be the dimension, and suppose that the length in the euclidean space of dimension n on the variables (p_i, q_i) is given by

.

Define matrices 1, 1', P, and Q by

.

[edit] Method

In even or in odd dimensionality, the quantization procedure amounts to replacing the ordinary p, q coordinates with non-commutative coordinates constructed from P, Q in a suitable fashion.

P and Q correspond to the generalized "position" and "momentum" for the Weyl quantization, but this physical fact is not important for the abstract construction of the spinors.

[edit] Even case

In the case when n = 2k is even, let

for i = 1,2,...,k (where the P or Q is considered to occupy the i-th position). The operation is the tensor product of matrices. It is no longer important to distinguish between the Ps and Qs, so we shall simply refer to them all with the symbol P, and regard the index on P_i as ranging from i = 1 to i = 2k. For instance, the following properties hold:

, and P_iP_j = − P_jP_i for all unequal pairs i and j. (Clifford relations.)

Thus the algebra generated by the P_i is the Clifford algebra of euclidean n-space.

Let A denote the algebra generated by these matrices. By counting dimensions, A is a complete 2^k×2^k matrix algebra over the complex numbers. As a matrix algebra, therefore, it acts on 2^k-dimensional column vectors (with complex entries). These column vectors are the spinors.

We now turn to the action of the orthogonal group on the spinors. Consider the application of an orthogonal transformation to the coordinates, which in turn acts upon the P_i via

.

Since the P_i generate A, the action of this transformation extends to all of A and produces an automorphism of A. From elementary linear algebra, any such automorphism must be given by a change of basis. Hence there is a matrix S, depending on R, such that

R(P)_i = S(R)P_iS(R) ^{− 1} (1).

In particular, S(R) will act on column vectors (spinors). By decomposing rotations into products of reflections, one can write down a formula for S(R) in much the same way as in the case of three dimensions.

However, just as in the three-dimensional case, there will be more than one matrix S(R) which produces the action in (1). The ambiguity defines S(R) up to a nonevanescent scalar factor c. Since S(R) and cS(R) define the same transformation (1), the action of the orthogonal group on spinors is not single-valued, but instead descends to an action on the projective space associated to the space of spinors. This multiple-valued action can be sharpened by normalizing the constant c in such a way that (det S(R))² = 1. In order to do this, however, it is necessary to discuss how the space of spinors (column vectors) may be identified with its dual (row vectors).

In order to identify spinors with their duals, let C be the matrix defined by

Then conjugation by C converts a P_i matrix to its transpose: ^tP_i = C P_i C^-1. Under the action of a rotation,

whence C S(R) C^-1 = α ^tS(R)^-1 for some scalar α. The scalar factor α can be made to equal one by rescaling S(R). Under these circumstances, (det S(R))² = 1, as required.

[edit] Weyl spinors

Let U be the element of the algebra A defined by

, (k factors).

Then U is preserved under rotations, so in particular its eigenspace decomposition (which necessarily corresponds to the eigenvalues +1 and -1, occurring in equal numbers) is also stabilized by rotations. As a consequence, each spinor admits a decomposition into eigenvectors under U:

ξ = ξ⁺ + ξ^-

into a right-handed Weyl spinor ξ⁺ and a left-handed Weyl spinor ξ^-. Because rotations preserve the eigenspaces of U, the rotations themselves act diagonally as matrices S(R)⁺, S(R)^- via

(S(R)ξ)⁺ = S⁺(R) ξ⁺, and

(S(R)ξ)^- = S^-(R) ξ^-.

This decomposition is not, however, stable under improper rotations (e.g., reflections in a hyperplane). A reflection in a hyperplane has the effect of interchanging the two eigenspaces. Thus there are two irreducible spin representations in even dimensions given by the left-handed and right-handed Weyl spinors, each of which has dimension 2^k-1. However, there is only one irreducible pin representation (see below) owing to the non-invariance of the above eigenspace decomposition under improper rotations, and that has dimension 2^k.

[edit] Odd case

In the quantization for an odd number 2k+1 of dimensions, the matrices Pⁱ may be introduced as above for i = 1,2,...,2k, and the following matrix may be adjoined to the system:

, (k factors),

so that the Clifford relations still hold. This adjunction has no effect on the algebra A of matrices generated by the P_i, since in either case A is still a complete matrix algebra of the same dimension. Thus A, which is a complete 2^k×2^k matrix algebra, is not the Clifford algebra, which is an algebra of dimension 2×2^k×2^k. Rather A is the quotient of the Clifford algebra by a certain ideal.

Nevertheless, one can show that if R is a proper rotation (an orthogonal transformation of determinant one), then the rotation among the coordinates

R(P)i =	∑	RijPj
	j

is again an automorphism of A, and so induces a change of basis

R(P)_i = S(R)P_iS(R) ^{− 1}

exactly as in the even dimensional case. The projective representation S(R) may again be normalized so that (det S(R))² = 1. It may further be extended to general orthogonal transformations by setting S(R) = -S(-R) in case det R = -1 (i.e., if R is a reversal).

In the case of odd dimensions it is not possible to split a spinor into a pair of Weyl spinors, and spinors form an irreducible representation of the spin group. As in the even case, it is possible to identify spinors with their duals, but for one caveat. The identification of the space of spinors with its dual space is invariant under proper rotations, and so the two spaces are spinorially equivalent. However, if improper rotations are also taken into consideration, then the spin space and its dual are not isomorphic. Thus, while there is only one spin representation in odd dimensions, there are a pair of inequivalent pin representations. This fact is not evident from the Weyl's quantization approach, however, and is more easily seen by considering the representations of the full Clifford algebra.

[edit] Summary 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 .

Metric signature	left-handed Weyl	right-handed Weyl	conjugacy	Dirac	left-handed Majorana-Weyl	right-handed Majorana-Weyl	Majorana
	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

[edit] Spinors of the Pauli spin matrices

Often, the first example of spinors that a student of physics meets encounters are the 2x1 spinors used in Pauli's theory of electron spin. The Pauli matrices are a vector of three 2x2 matrices that are used as spin operators.

Given a unit vector in 3 dimensions, for example (a,b,c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector.

The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector.

Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix:

The eigenvectors may be found by the usual methods of linear algebra, but a convenient trick is to note that the Pauli spin matrices are square roots of unity, that is, the square of the above matrix is the identity matrix. Thus a (matrix) solution to the eigenvector problem with eigenvalues of is simply. That is,

One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are:

The trick used to find the eigenvectors is related to the concept of ideals, that is, the matrix eigenvectors are projection operators or idempotents and therefore each generates an ideal in the Pauli algebra. The same trick works in any Clifford algebra, in particular the Dirac algebra that are discussed below. These projection operators are also seen in density matrix theory where they are examples of pure density matrices.

More generally, the projection operator for spin in the (a,b,c) direction is given by

and any non zero column can be taken as the projection operator. While the two columns appear different, one can use a² + b² + c² = 1 to show that they are multiples (possibly zero) of the same spinor.

[edit] Spinors of the Dirac algebra

The second example of spinors that a student studying physics likely encounters are the 4x1 spinors used in Dirac's theory of electrons and positrons. The Dirac matrices are a set of four 4x4 matrices that are used as spin and charge operators.

[edit] Conventions

There are several choices of signature and representation that are in common use in the physics literature. The Dirac matrices are typically written as γ^μ where μ runs from 0 to 3. In this notation, 0 corresponds to time, and 1 through 3 correspond to x, y, and z.

The + - - - signature is sometimes called the west coast metric, while the - + + + is the east coast metric. At this time the + - - - signature is in more common use and our example will use this signature. To switch from one example to the other, multiply all γ^μ by i.

After choosing the signature, there are many ways of constructing a representation in the 4x4 matrices, and many are in common use. In order to make this example as general as possible we will not specify a representation until the final step. At that time we will substitute in the "chiral" or "Weyl" representation as used in the popular graduate textbook, An Introduction to Quantum Field Theory, by Michael E. Peskin and Daniel V. Schroeder.

[edit] Construction

First we choose a spin direction for our electron or positron. As with the example of the Pauli algebra discussed above, the spin direction is defined by a unit vector in 3 dimensions, (a,b,c). Following the convention of Peskin & Schroeder, the spin operator for spin in the (a,b,c) direction is defined as the dot product of (a,b,c) with the vector :

σ_(a,b,c) = iaγ²γ³ + ibγ³γ¹ + icγ¹γ²

Note that the above is a root of unity, that is, it squares to 1. Consequently, we can make a projection operator from it that projects out the subalgebra of the Dirac algebra that has spin oriented in the (a,b,c) direction:

Now we must choose a charge, +1 (positron) or -1 (electron). Following the conventions of Peskin & Schroeder, the operator for charge is Q = − γ⁰, that is, electron states will take an eigenvalue of -1 with respect to this operator while positron states will take an eigenvalue of +1.

Note that Q is also a square root of unity. Furthermore, Q commutes with σ_(a,b,c). They form a complete set of commuting operators for the Dirac algebra. Continuing with our example, we look for a representation of an electron with spin in the (a,b,c) direction. Turning Q into a projection operator for charge = -1, we have

The projection operator for the spinor we seek is therefore the product of the two projection operators we've found:

The above projection operator, when applied to any spinor, will give that part of the spinor that corresponds to the electron state we seek. Therefore, to write down a 4x1 spinor we take any non zero column of the above matrix. Continuing the example, we put (a,b,c) = (0,0,1) and have

and so our desired projection operator is

The 4x4 gamma matrices used in Peskin & Schroeder are

for k=1,2,3 and where σⁱ are the usual 2x2 Pauli matrices. Substituting these in for P gives

Our answer is any non zero column of the above matrix. The division by two is just a normalization. The first and third columns give the same result:

More generally, for electrons and positrons with spin oriented in the (a,b,c) direction, the projection operator is

where the upper signs are for the electron and the lower signs are for the positron. The corresponding spinor can be taken as any non zero column. Since a² + b² + c² = 1 the different columns are multiples of the same spinor.

[edit] See also

• Spinor bundle
• Pure spinor
• Anyon
• Twistor

[edit] Notes

1. ^ Now referred to Clifford algebra.
2. ^ Another approach, which at one time had its heyday, but now has waned in popularity, is to construct the Clifford algebra ex nihilo as a matrix algebra by "quantizing" the coordinates in the original vector space. From this framework, spinors are simply the column vectors on which the matrices act. One may then appeal to techniques from linear algebra directly to split the spaces of spinors into irreducible parts.
3. ^ Named for W. K. Clifford.
4. ^ Named for Paul Dirac. A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2ⁿ complex numbers. (See Special unitary group.)
5. ^ Named for Hermann Weyl.
6. ^ Named for Ettore Majorana.
7. ^ Cartan, E, "Les groupes projectifs qui ne laissent invariante aucune multiplicité plane", Bulletin de la Société Mathématique de France, 41 (1913), 53-96.
8. ^ Pauli, W. "Zur Quantenmechanik des magnetischen Elektrons", Zeitschrift für Phisik, 43 (1927) 601-632.
9. ^ Dirac, P., "The quantum theory of the electron", Proceedings of the Royal Society of London Series A, 117 (1928) 610-624.
10. ^ Brauer, R. and Weyl, H., "Spinors in n dimensions", Amer. J. Math., 57 no. 2 (1935), 425-449.