Spinor

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 $2\pi$ 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.

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.)

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:

${\mathbf {x}}\rightarrow X=\left({\begin{matrix}x_{3}&x_{1}-ix_{2}\\x_{1}+ix_{2}&-x_{3}\end{matrix}}\right).$

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

• det X = - (length x)².
• X^{2 = (length x)2I, where I is the identity matrix.}
• ${\frac {1}{2}}(XY+YX)=({\mathbf {x}}\cdot {\mathbf {y}})I$
• ${\frac {1}{2}}(XY-YX)=iZ$ 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 rôle, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a spinor is a column vector

$\xi =\left[{\begin{matrix}\xi _{1}\\\xi _{2}\end{matrix}}\right],$ 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.

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.,

${\mathbf {x}}\cdot {\mathbf {x}}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0.$

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

$X=2\left[{\begin{matrix}\xi _{1}\\\xi _{2}\end{matrix}}\right]\left[{\begin{matrix}-\xi _{2}&\xi _{1}\end{matrix}}\right].$

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

$\left.{\begin{matrix}\xi _{1}^{2}-\xi _{2}^{2}&=x_{1}\\i(\xi _{1}^{2}+\xi _{2}^{2})&=x_{2}\\-2\xi _{1}\xi _{2}&=x_{3}\end{matrix}}\right\}$ (1)

subject to the constraint

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0.$ (2)

This system admits the solutions

$\xi _{1}=\pm {\sqrt {\frac {x_{1}-ix_{2}}{2}}},\quad \xi _{2}=\pm {\sqrt {\frac {-x_{1}-ix_{2}}{2}}}.$ (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

$C=\left({\begin{matrix}0&1\\-1&0\end{matrix}}\right),$

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.

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

$\langle \mu |\xi \rangle ={\bar {\mu }}_{1}\xi _{1}+{\bar {\mu }}_{2}\xi _{2}$ (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:

$\left({\begin{matrix}x_{3}&x_{1}-x_{2}\\x_{1}+x_{2}&-x_{3}\end{matrix}}\right)$.

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

$\langle \mu |\xi \rangle ={\bar {\mu }}_{1}\xi _{2}-{\bar {\mu }}_{2}\xi _{1}$

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

$\langle \mu |\xi \rangle ^{2}={\hbox{length}}\left(Q({\bar {\mu }},\xi )\right)^{2}$

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 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 2 × 2 matrix K whose product with itself is the identity matrix: K^{2 = Id. The conjugate of a spinor with respect to a reality structure K is defined by}

$\xi ^{*}=K{\bar {\xi }}.$

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

${\bar {X}}=KXK$, 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, ${\bar {X}}=KXK$.
• If μ and ξ is a spinor, then the inner product

$\langle \mu |\xi \rangle =i\,^{t}\mu ^{*}C\xi$

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

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. 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

$q_{1}^{2}+\dots +q_{k}^{2}+p_{1}^{2}+\dots +p_{k}^{2}(+p_{n})^{2}$.

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

${\begin{matrix}{\mathbf {1}}=\left({\begin{matrix}1&0\\0&1\end{matrix}}\right),&{\mathbf {1}}'=\left({\begin{matrix}1&0\\0&-1\end{matrix}}\right),\\P=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right),&Q=\left({\begin{matrix}0&i\\-i&0\end{matrix}}\right)\end{matrix}}$.

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.

Even case

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

$P_{i}={\mathbf {1}}'\otimes \dots \otimes {\mathbf {1}}'\otimes P\otimes {\mathbf {1}}\otimes \dots \otimes {\mathbf {1}}$

$Q_{i}={\mathbf {1}}'\otimes \dots \otimes {\mathbf {1}}'\otimes Q\otimes {\mathbf {1}}\otimes \dots \otimes {\mathbf {1}}$

for i = 1,2,...,k (where the P or Q is considered to occupy the i-th position). The operation $\otimes$ 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:

$P_{i}^{2}=1,i=1,2,...,2k$, and $P_{i}P_{j}=-P_{j}P_{i}$ 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

$P_{i}\mapsto R(P)_{i}=\sum _{j}R_{ij}P_{j}$.

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

$R(P)_{i}=S(R)P_{i}S(R)^{-1}$ (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

$C=P\otimes Q\otimes P\otimes \dots \otimes Q.$

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,

${\hbox{ }}^{t}P_{i}\rightarrow \,^{t}S(R)^{-1}\,^{t}P_{i}\,^{t}S(R)=(CS(R)C^{-1})\,^{t}P_{i}(CS(R)C^{-1})^{-1}$

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.

Odd case

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

$P_{n}={\mathbf {1}}'\otimes \dots \otimes {\mathbf {1}}'$, (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

$R(P)_{i}=\sum _{j}R_{ij}P_{j}$

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

$R(P)_{i}=S(R)P_{i}S(R)^{-1}$

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).

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 ${\mathfrak {so}}(p,q)$ is the same as the complexification of ${\mathfrak {so}}(p+q)$, we can focus upon the latter, at least for complex representations.

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

$(\pm 1,\pm 1,0,0,\dots ,0)$

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 ${\mathfrak {so}}(2)$, which isn't semisimple.

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

$(\pm 1,\pm 1,0,0,\dots ,0)$

and the permutations of

$(\pm 1,0,0,\dots ,0)$.

For ${\mathfrak {so}}(2n)$, there is an irreducible representation whose weights are all possible combinations of

$(\pm {1 \over 2},\pm {1 \over 2},\dots ,\pm {1 \over 2})$

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

$(\pm {1 \over 2},\pm {1 \over 2},\dots ,\pm {1 \over 2})$

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 ${\mathfrak {so}}(2n+1)$. Here, there's an irreducible representation whose weights are all possible combinations of

$(\pm {1 \over 2},\pm {1 \over 2},\dots ,\pm {1 \over 2})$

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

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 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 $e^{\pm i\phi /2}$ under a rotation by angle $\phi$.

• 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 $SU(2)\cong {\mathit {Spin}}(3)$ 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 $Spin(4)\equiv SU(2)\times SU(2)$. 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.

• 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 $SL(2,C)\equiv Spin(3,1)$.

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

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:

$S_{u}=(0.8,-0.6,0.0)\cdot {\vec {\sigma }}={\begin{bmatrix}0.0&0.8+0.6i\\0.8-0.6i&0.0\end{bmatrix}}$

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 $\pm 1$ is simply$1\pm S_{u}$. That is,

$S_{u}(1\pm S_{u})=\pm 1(1\pm S_{u})$

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:

${\begin{bmatrix}1.0+(0.0)\\0.0+(0.8-0.6i)\end{bmatrix}},{\begin{bmatrix}1.0-(0.0)\\0.0-(0.8-0.6i)\end{bmatrix}}$

The trick used to find the eigenvectors is related to the concept of ideals, that is, the matrix eigenvectors $(1\pm S_{u})/2$ 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.

See also

• spinor bundle
• anyon
• twistor