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 similar to spatial vectors, but which change sign under a rotation of 2π radians. More precisely, spinors are geometrical objects constructed from a given vector space endowed with an inner product by means of a 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 properties of spinors are more clearly visible, but the connection with the original spatial geometry is more obscure.

Contents 1 Overview 2 Spinors in three dimensions 2.1 Isotropic vectors 2.2 Reality 3 Spinors in higher dimensions: Matrix quantization 3.1 Even case 3.1.1 Weyl spinors 3.2 Odd case 4 Clifford algebras 4.1 Properties of Clifford algebras 4.2 Examples 5 Spinors in representation theory 6 History 7 Examples in low dimensions 8 Example: Spinors of the Pauli Spin Matrices 9 See also 10 References

[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.

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.

At this point one may encounter various hybrid notions of what defines a spinor. For instance, a popular approach to spinors is by means of Clifford algebras. 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). 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.

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.)

[edit] Spinors in three dimensions

On a three dimensional euclidean space, spinors manifest themselves in the familiar algebra of the dot and cross product. This algebra admits a convenient description, due to William Rowan Hamilton, by means of quaternions. In detail, given a vector x = (x₁, x₂, x₃) of real (or complex) numbers, one can associate the matrix of complex numbers:

Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:

• det X = - (length x)².
• X² = (length x)²I, where I is the identity matrix.
• 
• where Z is the matrix associated to the cross product z = x × y.
• If u is a unit vector, then UXU is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u.
• It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u₁ followed by the plane perpendicular to u₂, then the matrix U₂U₁XU₁U₂ represents the rotation of the vector x through R.

Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a spinor is a column vector

with complex entries ξ₁ and ξ₂.

The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique. Clearly, if X → RXR^-1 is a representation of a rotation, then replacing R by -R will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always double-valued.

[edit] Isotropic vectors

Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that X is a matrix representing a vector x in complex 3-space. Suppose further that x is isotropic: i.e.,

Then, from the properties of these matrices, X²=0. Any such matrix admits a factorization as an outer product

This factorization yields an overdetermined system of equations in the coordinates of the vector x:

(1)

subject to the constraint

(2)

This system admits the solutions

(3)

Either choice of sign solves the system (1). Thus a spinor may be viewed as an isotropic vector, along with a choice of sign. Note that because of the logarithmic branching, it is impossible to choose a sign consistently so that (3) varies continuously along a full rotation among the coordinates x. In spite of this ambiguity of the representation of a rotation on a spinor, the rotations do act unambiguously on the ratio ξ₁:ξ₂ since one choice of sign in the solution (3) forces the choice of the second sign. In particular, the space of spinors is a projective representation of the orthogonal group.

As a consequence of this point of view, spinors may be regarded as a kind of "square root" of isotropic vectors. Specifically, introducing the matrix

the system (1) is equivalent to solving X = 2 ξ ^tξ C for the undetermined spinor ξ.

A fortiori, if the rôles of ξ and x are now reversed, the form Q(ξ) = x defines, for each spinor ξ, a vector x quadratically in the components of ξ. If this quadratic form is polarized, it determines a bilinear vector-valued form on spinors Q(μ,ξ). This bilinear form then transform tensorially under a reflection or a rotation.

[edit] Reality

The above considerations apply equally well whether the original euclidean space under consideration is real or complex. When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group. Suppose, for simplicity, that the inner product on 3-space has positive-definite signature:

length(x)² = x₁² + x₂² + x₃² (4).

With this convention, real vectors correspond to hermitian matrices. Furthermore, real rotations preserving the form (4) correspond (in the double-valued sense) to unitary matrices of determinant one. In modern terms, this presents the special unitary group SU(2) as a double-cover of SO(3). As a consequence, the spinor hermitian product

(5)

is preserved by all rotations, and therefore is canonical.

If, however, the signature of the inner product on 3-space is indefinite (i.e., non-degenerate, but also not positive definite), then the foregoing analysis must be adjusted to reflect this. Suppose then that the length form on 3-space is given by:

length(x)² = x₁² - x₂² + x₃² (4').

Then the construction of spinors of the preceding sections proceeds, but with x₂ replacing i x₂ in all the formulas. With this new convention, the matrix associated to a real vector (x₁,x₂,x₃) is itself real:

.

The form (5) is no longer invariant under a real rotation (or reversal), since the group stabilizing (4') is now a Lorentz group O(2,1). Instead, the antihermitian form

defines the appropriate notion of inner product for spinors in this metric signature. This form is invariant under transformations in the connected component of the identity of O(2,1).

In either case, the quartic form

is fully invariant under O(3) (or O(2,1), respectively), where Q is the vector-valued bilinear form described in the previous section. The fact that this is a quartic invariant, rather than quadratic, has an important consequence. If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals. In the language of representation theory, this implies that there is only one irreducible spin representation of SO(3) (or SO(2,1)) up to isomorphism. If, however, reversals (e.g., reflections in a plane) are also allowed, then it is no longer possible to identify spinors with their duals owing to a change of sign on the application of a reflection. Thus there are two irreducible spin representations of O(3) (or O(2,1)), sometimes called the pin representations.

The differences between these two signatures can be codified by the notion of a reality structure on the space of spinors. Informally, this is a prescription for taking a complex conjugate of a spinor, but in such a way that this may not correspond to the usual conjugate per the components of a spinor. Specifically, a reality structure is specified by a hermitian 2 × 2 matrix K whose product with itself is the identity matrix: K² = Id. The conjugate of a spinor with respect to a reality structure K is defined by

The particular form of the inner product on vectors (e.g., (4) or (4')) determines a reality structure (up to a factor of -1) by requiring

, whenever X is a matrix associated to a real vector.

Thus K = i C is the reality structure in euclidean signature (4), and K = Id is that for signature (4'). With a reality structure in hand, one has the following results:

• X is the matrix associated to a real vector if, and only if, .
• If μ and ξ is a spinor, then the inner product

determines a hermitian form which is invariant under proper orthogonal transformations.

[edit] Spinors in higher dimensions: Matrix quantization

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^[1]. First, we fix some 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

.

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

[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] 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 provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations.

[edit] Properties of Clifford algebras

The following identities hold for the real Clifford algebras Cl_p,q in signature (p,q) for p+q ≤ 2:

1. Cl_1,0 = C, Cl_0,1 = R ⊕ R
2. Cl_1,1 = Cl_0,2 = R^{2× 2}
3. Cl_2,0 = H, the algebra of quaternions

Using these three identities, the following isomorphisms allow one to describe any real Clifford algebra explicitly as a matrix algebra:

1. Cl_n,0 ⊗ Cl_0,2 = Cl_0,n+2
2. Cl_0,n ⊗ Cl_2,0 = Cl_n+2,0
3. Cl_p,q ⊗ Cl_1,1 = Cl_p+1,q+1

To pass from the real Clifford algebra Cl_p,q to the complex Clifford algebra Cl_n, where n = p + q, one only needs to take a tensor product: Cl_n ≈ Cl_r,s ⊗ C. In light of this, the following are useful for constructing complex Clifford algebras up to isomorphism:

1. C ⊗_R C ≈ C ⊕ C
2. C ⊗_R H ≈ C^2×2
3. H ⊗_R H ≈ R^4×4

To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pin_p,q is the set of invertible elements in Cl_p,q which can be written as a product of unit vectors:

Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group O(p,q). The Spin group consists of those elements of Pin_p,q which are products of an even number of unit vectors. Thus Spin is a cover of the group of proper rotations SO(p,q).

Let α : Cl → Cl be the automorphism which is given by -Id acting on pure vectors. Then, for instance, Spin_p,q is the subgroup of Pin_p,q whose elements are fixed by α. Let

.

(These are precisely the elements of even degree in Cl_p,q.) Then the spin group lies within Cl_p,q⁰.

[edit] Examples

Using the properties of Clifford algebras, it is then possible to determine the number and type of all irreducible spaces of spinors.

[edit] Spinors in representation theory

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.

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

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.

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

and the permutations of

.

For , there is an irreducible representation whose weights are all possible combinations of

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

There is also another irreducible representation 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.

Let's now go over to . Here, there's an irreducible representation 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 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.

Spinor representations of are modules of the Clifford algebra Cl_p,q, which are generated as a ring by a vector space C^p+q. Weyl spinors exist when p+q is even. Weyl spinors which are annihilated by a maximum dimension ((p+q)/2) subspace of this vector space are called pure spinors.

[edit] History

The most general mathematical form of spinors was discovered by Élie Cartan^[2] 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^[3] in 1927, when he introduced spin matrices. The following year, Paul Dirac^[4] 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] 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 .

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] Example: Spinors of the Pauli Spin Matrices

The first example of spinors that a student studying physics likely 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 provide the second set of spinors the student of physics is likely to run into. These projection operators are also seen in density matrix theory where they are examples of pure density matrices.

[edit] See also

• spinor bundle
• anyon
• twistor

[edit] References

1. ^ Brauer, R. and Weyl, H., "Spinors in n dimensions", Amer. J. Math., 57 no. 2 (1935), 425-449.
2. ^ Cartan, E, "Les groupes projectifs qui ne laissent invariante aucune multiplicité plane", Bulletin de la Société Mathématique de France, 41 (1913), 53-96.
3. ^ Pauli, W. "Zur Quantenmechanik des magnetischen Elektrons", Zeitschrift für Phisik, 43 (1927) 601-632.
4. ^ Dirac, P., "The quantum theory of the electron", Proceedings of the Royal Society of London Series A, 117 (1928) 610-624.