These notes provide a context of algebraic representation of spinors for three and four dimensional Euclidean and pseudoeuclidean spaces. They are not even exhaustive in that respect. While they connect smoothly with the standard scattered literature on spinors, spinor fields and spin-tensor fields, as indicated in references and remarks, they search a bit more of the context of the matrix algebra of spinor concept than is customary. They attempt to make connections where the standard texts are more narrowly focused, and they also attempt to bring to light and resolve certain points of confusion and misunderstanding that are frequently the products of abuses of language, and lack of discrimination of diction.
With '†' denoting Hermitean conjugation, the matricies,
│0 1│ │0 0│
a = │ │ a† = │ │
│0 0│ │1 0│
satisfying
a a† + a† a = I(2) := σ₀ (the 2x2 identity matrix I(2))
and
a² = 0 ⇒ a†² = 0,
give the usual finite irreducible representation of the Canonical
Anticommutation Relation (CAR) of quantum theory.
Defining
σ₁ := (1/√2)(a† + a) := q
σ₂ := (i/√2)(a† - a) := p
│1 0│
iσ₃ := [q, p] = i[a, a†] = i │ │
│0 -1│
and
{q, p} = qp + pq = 0
q² = p² = ½ σ0
then
[σi, σj] = i εijk σk
with σk, k=1,2,3 representing the Pauli matrices for the lowest dimensional nontrivial representation of the Lie algebra su(2), of the covering group SU(2) of the rotation group SO(3).
In quantum theory, both CAR and the Canonical Commutation Relations (CCR) serve as kinematical algebras of physical systems. That CCR is a kinematical algebra (a rather fundamental Lie algebra within the still open general theory of nilpotent Lie algebras that is often called a Heisenberg algbera) is always understood; that CAR is also a kinematical algebra is some times obscured.
The standard parsing of the Problem of Motion in mechanics is in terms of kinematics and dynamics.
That fundamental variables (observable) satisfy the kinematical algebra before any particular dynamical assumption that defines a particular system is the kinematical constraint constraint that specifies the structure of the allowed states of the system exactly as do the fundamental Poisson brackets in classical Canonical (Hamiltonian) mechanics.
The dynamical assumption beyond this defines a system of paths of possible motions through the space of states.
A quantun theoretical kinematical algebra is characterized by the the vectors upon which it operates being the states of a system. [This is one way of looking at the state concept, at least.] Both CAR and CCR are quantum kinematical constraints on the states of physical systems to which they each apply.
The systems to which CCR applies are those that have classical cognates, while those to which CAR applies have no classical cognates, are intrinsically quantum theoretical in nature, and rather specifically are the *simplest* possible quantum theoretical entities in that by their nature they are dichotomic: when measured by some macrospcopic procedure, the eigenvalue structure of their observables has only two and exactly two possible values. Typically, in speaking of the "spin of the electron" one speaks of "spin up" and "spin down", which is misleading since the concept is has nothing to do with any specific up and down, except that which is determined by an arbitrary measuring device; the locution is merely suggestive of the fact that this spin-½ of the particle is a dichotomic in its measurable outcomes.
With the same formal algebraic structure, isotopic spin, which is really about electric charge, is also dichotomic.
Despite a few references and hints beyond those regarding the general mathematical theory of spinors, "spin of the electron", or spin-½ is the limit of what is discussed here. Most of the ideas can be generalized.
A distinction between the CCR and CAR in quantum physics is that while in CCR physics the central problem is the dynamical or Cauchy problem, that seeks to predict the future state of a system, while in CAR physics, there is ostensibly no dynamical problem contained solely within the CAR algebra. The physical quantum variables associated to spin-½, isotopic spin, spin generally, and indeed all so called "internal variable" are formally considered by present theory to be somehow exterior to any space, time or spacetime container.
That is an interesting and important distinction not often emphasized since it is rather inimcal to an underlying philosophical program of geometrization in physics due to Clifford, Riemann and Einstein. The efforts within this program are inextricably linked to the evolving generalized concept of "geometry" from Euclid, through Bolyai, Lobachewski, Riemann, Klein, Weyl, Kaehler, Niejenhuis, Alain Connes and many others.
The understanding of spinors as being attached to, and constructed from isotropic vectors in Euclidean spaces strongly suggests that a physical R³ model of space in a fundamental physical theory be replaced with a C³ that is the analytic continuation of R³. This is a matter of logical necessity more than any directly measurable empiricism. The indirect key indicator for this is the physical existence of what appear to be extremely localized physical attributed of a dichotomic nature that we model according to quantum metaprinciples using CAR.
The kinematical cognates between CAR and CCR are:
σ₁ <−> q
σ₂ <−> p
σ₃ <−> I
From here on, we will simply be considering CAR and the embedding
of the concept of spinor in its context, which is to say the context
of some simple matrix algebra.
Vectors in R³, v = (x, y, z) are in 1−1
correspondence with 2x2 Hermitean matrices, an instance of which is
│z (x - iy)│
(x, y, z) −> │ │ = M(v)
│(x + iy) -z │
from which traces, Tr(.) and determinants, Det(.) are easily
calculated:
Tr( M(v) ) = 0
Det( M(v) ) = - (x² + y² + z²) = - v∙v = - |v|²
M²(v) = (x² + y² + z²) σ0
Tr( M²(v) ) = 2 v∙v
The eigenvalues of M(v) are
r± = ±√(x² + y² + z²)
and the eigenvectors (not normalized) are,
│(x - iy)/(r± - z)│
|r±) = b±│ │
│ 1 │
with b± arbitrary.
If we take
b± = [z(r± - z)/(x - iy)]1/2
then the eigenvectors can be written as
│[(x - iy)/(r± - z)]1/2│
|r±) = √z │ │
│[(x + iy)/(r± + z)]1/2│
These are sometimes called nonsimple spinors generally.
As the eigenvalues degenerate,
r± → 0,
and the eigenvectors degenerate to the single simple spinor attached to an isotropic vector as defined below. Generalize the spinor notion to nonisotropic vectors to eigenvectors of 2x2 Hermitean matrices. Then to any vector in R³ there is associated a pair of spinors. The eigenvectors are then connected by the transformation,
│ [(r - z)/(r + z)]½ 0 │
|r₋) = -i │ │ |r₊)
│ 0 [(r + z)/(r - z)]½ │
which approaches I(2) as r→0.
If the map to an analytic dual space is simply transpose,
2z
(r±|r±) = ------- (xr± - iyz) = (r±|σ₀|r±)
x² + y²
which as r± → 0, goes to 2iy.
(r±|σ₁|r±) = 2z
(r±|σ₂|r±) = 0
2z²
(r±|σ₃|r±) = ------------ (x - iy)
(r±² - z²)
And,
-i2zx
(r±|r∓) = ------------ = (r±|σ₀|r∓)
√(r±² - z²)
2zr
(r±|σ₁|r∓) = ------------
√(r±² - z²)
±2z²
(r±|σ₂|r∓) = ------------
√(r±² - z²)
-2zy
(r±|σ₃|r∓) = ------------
√(r±² - z²)
This is consistent with the bilinears of the spinor defined below.
Also then for a dual map with complex conjugation, i.e., Hermitean
conjugate dual,
2zr±
(r±*|r±) = -----------
√(r±² - z²)
(r±*|r∓) = -2iz
And,
2zx
(r±*|σ₁|r∓) = ---------
(r² - z²)
2zy
(r±*|σ₂|r∓) = ---------
(r² - z²)
2zz
(r±*|σ₃|r∓) = ----------
√(r² - z²)
2z
(r±*|σ₁|r∓) = --------- (±zy - irx)
(r² - z²)
2z
(r±*|σ₂|r∓) = --------- (∓zx - iry)
(r² - z²)
(r±*|σ₃|r∓) = 0
If v₁ and v₂ are arbitrary vectors of R³ then
M(v₁ + v₂) = M(v₁) + M(v₂)
and
M(v₁) M(v₂) = v₁∙v₂ I(2) + i M(v₁ˆv₂)
where the caret, or circumflex, 'ˆ', is the antisymmetric exterior
or Grassmann product.
Then also,
M(v₁ˆv₂) = - M(v₂ˆv₁)
and
M(v₂) M(v₁) = v₁∙v₂ I(2) - i M(v₁ˆv₂)
Then, algebraically, one defines the inner product and the exterior
product of vectors represented by Cartan matrices by Jordan symmetric
product (anticommutator) and Lie antisymmetric product (commutator)
respectively.
v₁∙v₂ I(2) = (1/2) {M(v₂), M(v₁)}
M(v₁ˆv₂) = (i/2) [M(v₂), M(v₁)]
The Euclidean inner product may also be expressed in terms of
the invariant trace. This is obvious from the expression for
the eigenvalues.
v₁∙v₂ = (1/2) Tr( M(v₁) M(v₂) )
= (1/2) Tr( M(v₂) M(v₁) )
= (1/4) Tr( {M(v₂), M(v₁)} )
The trace expresses an inner product that implies a norm on M(2, C).
[Appendix A]
With the σk as unit basis vectors, one can project from M(v) its vector
components using this inner product:
(1/2) Tr( M(v) σ₀ ) = 0
(1/2) Tr( M(v) σ₁ ) = x
(1/2) Tr( M(v) σ₂ ) = y
(1/2) Tr( M(v) σ₃ ) = z
or more compactly, letting x₁=x, x₂=y, x₃=z, by
Tr( M(v) (1/2) σk ) = xk
The full algebra M(2, C), as a Clifford algebra, expresses more than
just vectors. A general element M(2, C) can be interpreted as an
element of a real Clifford algebra, with real coefficients s, v, a, p,
(scalar, vector, bivector or axial vector or pseudovector, volume or
pseudoscalar), recalling that for bivectors,
εijk σj σk = i σi
and for the pseudoscalar or volume unit element,
σ₁ σ₂ σ₃ = i σ₀
s σ₀ + vk σk + ak i σk + p σ₄
using an Einstein summation convention for upper, contravariant and
lower, covariant repeated indicies,
or, as an element of a complex Clifford algebra, with complex s, v, p,
s σ₀ + va σa + p σ₃
where a = 1,2.
These are the Clifford algebras for R³ and C² respectively. In the
two dimensional space, vectors coalesce with pseudovectors.
In introducing the geometrical notion of spinor in E³, one analytically continues the real vector (x, y, z) to (x, iy, z) and then to the semicomplex space (x + iy, x − iy, z). If the space is to be independent of the coordinates, the model R³ should be thought of as embedded in a C³, i.e., an R⁶; further, the linear space of 2x2 complex matrices is homeomorphic to R⁸, and the image space R⁶ is a linear subspace of this R⁸.
A real vector in C³ is mapped to an Hermitean M with vanishing trace. An imaginary vector in C³ is mapped to a skew Hermitean M with vanishing trace. Constraining the general complex matrices of M(2, C) to have vanishing trace reduces the number of free parameters from 8 to 6. Since a linear combination of matrices of trace 0 also has trace 0 by virtue of of the linarity of the trace functional, the R⁶ image subspace is linear in R⁸ (isomorphic to the C⁴ of M(2, C).
The terms of the Euclidean quadratic form expressing the square of the length of a vector can be factored. The condition for an isotropic vector,
x² + y² + z² = 0
has a solution that is not trivial, i.e., not (0, 0, 0).
For the analytically continued, or rotated components write
(x - iy)(x + iy) + z² = 0
This describes a cone in the continued space with the z-axis as its line of symmetry. The 2 spinor components are then defined by
ζ₀² = -(x - iy) x = (1/2) (ζ₀² - ζ₁²)
ζ₁² = +(x + iy) y = (-i/2)(ζ₀² + ζ₁²)
ζ₀ζ₁ = z z = ζ₀ζ₁
The mapping between isotropic vectors and the two component
spinor, using modified Dirac notation |.) [Dirac 1958] to
distinguish spinors,
│ζ₀│
|ζ) = │ │
│ζ₁│
is not 1−1, but 2−1 since by simple observation,
(x, iy, z) <→ ±|ζ)
For an isotropic vector and the associated spinor, ±r=0,
and the eigenvalue equation becomes a null one:
M(v) |ζ) = 0
which is a limit of the eigenvalue equation analyzed above, as
|v| → - |v|
For a nontrivial solution of any null eigenvalue equation it is well
known that the condition Det( M ) = 0 must be met, but then this is
exactly the isotropic condition. It is an essential part of the
assumptions in spinor black magic that the R³ model of physical
space be *physically* considered as embedded in a complex C³ mapped
to the linear subspace of M(2, C) with vanishing trace. [Re: the
question of "black magic", see Classical Geometry & Physics Redux
In the case of a nonisotropic vector with *two* associated eigenvalues,
it becomes natural to consider four component objects of the form of
direct products or direct sums,
|r₊) X |r₋)
or
|r₊) + |r₋)
one of which is associated with a nonisotropic vector.
See the section below on four-spinors.
Before going any further and misleading the reader, this brief introduction to spinors deals only with with the elementary aspects of the concept.
As "vectors" do not always represent displacements, nor are spinorial quantities connected with displacements. Even in Euclidean spaces, as one can speak of vector fields, vectors that are function of position in the space, one can also speak of spinor fields, and these are the animals that appear in physics when speaking of the spin of electron.
As a tensorial quantity is a special kind of geometric object, so is a spinorial quantity.
As the concept of tensor necessarily has to do not only with specifics of a transformation law which is linear and homogeneous, it also entails specifying the group of such transformations. The group in considering tensors and spinors is often neglected since the group is most often a tacit contextual assumption. With tensors in the context of linear spaces, the group is usually a group of the general group of basis substitutions, or a subgroup thereof. With tensor fields in the context of nonlinear spaces, manifolds the group is usually the infinite pseudo group of coordinate substitutions which induce at each point the local transformations.
For a real manifold, xk → yk(x), using Einstein summation
convention, the prototypically contravariant object transforms
dyj = ∂yj/∂xk dxk
and the protypically covariant object transforms inversely:
∂/∂yj = ∂xk/∂yj ∂/∂xk
These local transformations are then induced elements of some
GL(n, R), the group of general linear basis substitutions in
the tangent and cotangent spaces at a point.
In tensor calculus, there are both covariant and contravariant aspects of the transformation properties of general tensors, as above. Similarly, there is a similar dichotomy for spinors (spinor fields).
As there are in tensor calculus, tensor densities, so there are also spinor densities.
Finally, tensor analysis can be combined with spinor analysis, into a spinor-tensor analysis where the spinors are elements of half odd integral rank, while the tensors keep their integral rank. The spinors spoken of here become generalized to spin-tensors of rank (1/2), a kind of "square root" of a vector.
The, mixed spintensors and mixed spintensor densities with tensor and spinor indicies also exist, and are defined in the context of such analysis over differentiable Riemannian manifolds. That would be the context of General Relativity modified to allow for the existence of electron spin. Where there is an affine connection for the transport of tensorial quantities and an associated curvature tensor, there is then also a spinor connection with an associated spinor curvature. Life is always more complicated than we might like to think.
Next, it will be determined that spinors associated with the orthogonal group O(3) must have a transformation law by unimodular matrics, elements of SU(2), in fact. It is important to note that this is only true in a Euclidean space; when spinor fields on a curved manifold are considered, this will *not* be true: a spinor in this, its most general understanding is a spin-tensor density of rank (1/2).
On the other hand, what is presented in this webpage is the basic concept embodied in its simplest possible physcial representation, using three and four dimensional spaces.
Continuing now with the simpler concept of spinor: A geometry can be characterized by a group of transformations that preserve a given set of invariants. A Euclidean space Eⁿ can be characterized by the disconnected Lie group IO(n), its maximal group of isometries. In particular, E³ is characterised by the inhomogeneous group Lie group IO(3), the inhomogeneous group (including the group of translations T(3)) [3] orthogonal (rotations) group O(3) which preserves lengths and angles. That reflections (think discontinuous flips of coordinate axes) of the space are possible which have distinct geometrical properties causes such a space to be called orientable.
While a Euclidean plane E² is orientable, the two dimensionsional Möbius strip is nonorientable. For E³, starting with (+++), taking into account of the fact that the labeling of coordinate axes is, geometrically speaking, nothing but a useful fiction, one might consider the flips (+--), (-++), (---). But, the secondary geometrical invariants, bivector = pseudovector (cross product) and volume = pseudoscalar change sign only under a reflection of an odd number of coordinate axes, so all possible flips fall into two (chiral) equivalence classes [(+++), (+--)], [(++-), (---)] that arbitrarily called right handed and left handed. The transformation connecting these chiral classes is called a parity transformation or a chiral transformation.
Because these chiral transformations (for E³, there is only one, and it is idempotent) are elements of O(3), this group has two disconnected componets, say O₊(3) and O₋(3). But, only O₊(3) contains the identity element of O(3), and only it, by itself is a group.
The chiral class of a member of O(3) can be specified by its determinant of it is ±1, and these signs correspond to the chiral class. Generalizing this idea notationally, 'S' (for "special") designating prefix means that all matrices which represent the group in its defining representation have determinant +1, so as a matter of a standard notation,
SO(3) = O₊(3)
The "special orthogonal group in three dimensions", SO(3) is one of the classical Lie groups, and it is therefore simultaneously: an analytic manifold, a group and a measure space, with a left invariant Haar measure, invariant under the left action of the group on itself. Actually because it is compact as a manifold, the Haar measure is both left and right invariant, and so generally invariant. [Chevalley 1946]
As a compact (contains all its limits points) manifold, SO(3) is connected, but it is not simply connected. Although failure to be simply connected is more complicated generally, for SO(3), think of a doughnut (torus) with a hole in the middle. That hole is what is responsible for lack of simply connectedness. The complication is that failure of simply connectedness of any topological space can be described by the number of holes and their dimensions. These "connectedness properties" are topological invariants.
A torus T² = T¹ X T¹, a topological product of two circles, is a surface of a doughnut - or a coffee cup if you're a real topologist. It has two descriptive radii, one (R) extending from its center of greatest symmetry (which ends on its centroid line), and the other (r) that extends from the centroid line to its surface. You can generate T² as a geometric Gedankenexperiment by letting the T¹ with radius r attached by its center to the end of the radius R of the other T¹ trace out, on Rotation of the radius R, the surface of T². It is explicitly a "two dimensional surface of revolution".
We have a torus T² now. Slice it along some R. Grab one side of the slice, turn it by π radians, and glue the slice together again. Now, contract the radius R, letting the torus surface pass through itself, as R→0. The surface will become "double valued" in such a way that the the final reult will be a sphere S² but with all antipodal points identified. This is a projective sphere and it is doubly connected.
Complicate things by considering a similar generation of a toroid S² X T¹, swinging the two dimensional sphere by the circle T¹, for which we will need (visually, anyhow) a four dimensional space exactly as one might more easily consider the three dimesnional sphere S³ defined from a four dimensional space by
x² + y² + z² + u² = 0
Now, we look at the manifold structure of SO(3). SO(3) is the collection of all rotations continuously connected to the identity element of the group. In E³, a rotation is specified by a direction (of the axis of rotation) and a magnitude of the rotation. That requires a total of three parameters, and indeed, SO(3) is a three parameter Lie group. (Check)
Take the direction to be a vector emanating from the origin of E³, and let the magnitude of the rotation be the vector's magnitude. For any direction, consider a growing rotation of a vector orthogonal to the axis of rotation. When the rotation reaches π, the rotated vector will have rotated into its negative, which eactly the same as would be achieved with a rotation -π. Then, the space of SO(3) looks like a three dimensional ball centered about its group identity element whose two dimensional surface has its antipodal points identitfied.
For a rotation transformation on E³, the rotations of v will preserve angles and lengths, meaning that the transformations of M(v) need to do the same, by preserving the Determinant and the trace of M(v). Preserving these is no problem since they happen both to be preserved under arbitrary similarlity transformations.
M → S M S-1
XYZZY Unitarity and specialness of S Keeping the vectors in E³ real means in the corresponding M Hermitean, then S must also be unitary, so the rotational mapping of M is written,
M → S M S†
with
S S† = I
While such a mapping of Hermitean M matricies by unitary S matricies will preserve the trace of M, the determinant will not be preserved unless Det( S ) = 1, which means that a rotation of SO(3) acting on v in E³ is mirrored in the rotation of M(v) as an element of SU(2).
Please note that the following barely touches the subject of tensor calculus amd its applications to differential geometry. Contravariant and covariant tensors are dual in the sense of being linear functionals for each other, each maps the other by contraction over their indicies to a scalar which is by definition a number, invariant under coordinate substitutions.
For example, prototypically,
dΦ = ∂Φ/∂xk dxk
where the first factor on the RHS is the covariant gradient vector
(tensor of the first rank) of the scalar function Φ. The contraction
of the repeated index 'k' appearing once covariantly and once
contravariantly is summed over by the Einstein summation convention.
If the context of the tensor calculus is a Riemannian manifold,
(or a pseudoriemannian manifold), i.e, one that possesses a metric
gkj(x) tensor field, a second rank tensor field, symmetric
in its indicies, with gkj(x) dxk dxj > 0 (or positivity not required),
then this tensor turns out to be able to convert between the two
possible variances of an index by contraction. E.g.,
gkj(x) Ak = Ak
converts a contravariant Ak to a covariant form Ak.
Similarly for spinor dual, the dual map is a Hermitean conjugate
for these spinors because S⁻¹=S†.
More generally, the *inverse* nature of this duality will dominate,
(The group of transformations matters quite substantively)
and the concept of spinor metric
(an unfortunate and misleading standard terminology) will be needed.
In spinor calculus, however, there is no such spinor metric as there is for
one of the above manifolds, and a very different kind of duality
borrowed from tensors is used. It is most frequently called
Hodge-* Duality.
The Hodge-* Dual map is effected through the completely antisymmetric ε tensors and generalizes the idea of cross product that is peculiar to three dimensions. In an n-dimensional space there are tensors and tensor densities with ranks 0-n. The dimension of the linear space of tensors of rank k is a binomial coefficient (n k) = (n n-k). The Hodge-* dual map is a 1-1 map from tensors of rank (n k) to tensors of rank (n n-k), and also from tensors of rank (n n-k) to tensors of rank (n k), which is to say it is involutive of order 2, exactly what one would expect for any "dual map".
In a three dimensional space, with εijk being the symbol completely antisymmetric in its indicies that is equal to +1 for any cyclic permutation of its indicies, equal to -1 for any other permutation of indicies, and 0 for any index equality, the cross product of two contravariant vectors might be be written:
εijk Aj Bk
which superficially seems to map a pair of contravariant vectors, a
bivector actually, to a covariant vector. Under a chiral inversion
of the coordinate system, a vector does not change sign, but the
this purported vector does, as A X B = - B X A. In specialized
3-space jargon, it is called an an axial vector; in the more general
n-space language, tensorial objects which change sign under a chiral
inversion of coordinates are denoted with prefix "pseudo" (Gr. false).
Thus, the above object is a covariant pseudovector. Fitted in with
the discussion of Clifford algebras above, it becomes clear that
Clifford algebras are subalgebras of tensor algebras. Elements of a
Clifford algebra are in 1-1 correspondence with sequences of tensors
that are completely antisymmetric in their indicies.
In 3 dimensions, there are:
Rank: 0 1 2 3
scalars vectors bivectors pseudoscalars
pseudovectors
Dimension: 1 3 3 1
A bivector is equivalent to a pseudovector.
In 4 dimensions, there are:
Rank: 0 1 2 3 4
scalars vectors bivectors pseudovectors pseudoscalars
Dimension: 1 4 6 4 1
The linear space of bivectors is Hodge-* selfdual.
In a Clifford algebra, the pseudoscalar element turns out to be a generator of a one parameter group of algebra inner automorphisms that generalizes the the Hodge-* operation, i.e.,
For k = 1, 2, ..., n, and (without loss of generality) Hermitean Γk
Γk → exp( +i θ Γ(n+1) ) Γk exp( -i θ Γ(n+1) )
is a one parameter group of automorphisms (inner by their very form)
of the complex Clifford algebra CL(n, C) generated by the Γk,
and when θ = π, the map is the Hodge-* dual map.
It turns out that all SO(n), n > 2 manifolds are doubly connected like SO(3), and that they all have universal covering groups Spin(n), with Spin(2) "accidentally" being isomorphic to SU(2); the other Spin(n) either have such isomorphisms that are not so easy or have none at all. The ease of the SU(2) isomorphism with Spin(3) is all too beguiling and misleading.
Note that the universal covering groups Spin(n), (which are simply connected) of SO(n) are not isomorphic to their SO(n). Rather, there exists a homomorphism (and a *local* isomorphism) of Spin(n) onto SO(n) where two elements of Spin(n) are mapped to one in SO(n).
The existence of local isomorphism is what causes the abstract Lie *algebras* so(n) and spin(n) to be isomorphic. But, Only half of the representations of *group* Spin(n) will be true representations of the *group* SO(n). The analysis and construction of the Spin(n) groups using Clifford algebras can be found in [Chevalley 1946].
It should also be noted that the spin representations for the SO(n) Lie groups can be extended nontrivially to spin representations of SU(n) by complexification of the parameters of SO(n). It is easy to see by first noticing that the complexification of real Lie algebra so(n) yields the su(n) algebra, and then remembering that the exp function maps a Lie algebra to a Lie group:
exp: so(n) → SO(n)
exp: su(n) → SU(n)
A real four vector u = (x, y, z, t) = (v, t) can be mapped to a 2x2
Hermitean complex matrix:
│(t + z) (x - iy)│
│ │ = M4(u)
│(x + iy) (t - z)│
treating "t" as a scalar of the Clifford algebra. Then M4(u) is Hermitean and the Minkowskian inner product can be algebraically defined by the determinant.
Det( M4(u) ) = t² - x² - y² - z²
and
(1/2) Tr( M₄(u) ) = t
M₄²(u) = (x² + y² + z² + t²) I(2) + 2t M(v)
(1/2) Tr( M₄²(u) ) = x² + y² + z² + t²
The elements S of SL(2, C) preserve the determinant in the map
M ──> S† M S
A four vector u = (x, y, z, t) = (v, t) can also be mapped
to a 2x2 complex matrix:
│(it + z) (x - iy)│
│ │ = M₄(u)
│(x + iy) (it - z)│
treating "t" as a pseudoscalar of the Clifford algebra CL(2, C).
M₄²(u) = (x² + y² + z² - t²) I(2) + i2t M(v)
and
(1/2) Tr( M4²(u) ) = x² + y² + z² - t²
Det( M4(u) ) = - (x² + y² + z² + t²)
The determinant is now the negative of a Euclidean norm. The trace is invariant under similarity transformations by GL(2, C). Algebraically extracting the spatial part of the vector,
M(v) = (i/4t) ( M₄†²(u) - M₄²(u) )
where
it = (1/2) Tr( M₄(u) )
Also
-t² = (1/2) Tr²( M₄(u) )
and repeating that
(1/2) Tr( M₄²(u) ) = x² + y² + z² - t²
(1/2)( Tr( M₄²(u) ) - Tr²( M₄(u) ) ) = x² + y² + z²
(LHS side of the last expression has the form of an uncertainty or standard deviation in the value of ’it’.) we can define inner and outer products for R³ in a manner similar to the algebraic definitions above. For the Minkowskian inner product,
let
u₁ = (v₁, t₁)
and
u₂ = (v₂, t₂)
be arbitrary four vectors. Then,
M₄(u₁ + u₂) = M₄(u₁) + M₄(u₂)
[M₄(u₁), M₄(u₂)] = [M(v₁), M(v₂)]
and
{M₄(u₁), M₄(u₂)} = {M(v₁), M(v₂)}
+ i2 (t₁ M(v₂) + t₂ M(v₁))
- 2 t₁ t₂ I(2)
So,
(1/2) Tr( {M₄(u₁), M₄(u₂)} ) = v₁∙v₂ - t₁t₂
a Minkowskian "pseudoinnerproduct". [Appendix A]
The inner product so defined by a trace is invariant under the group GL(2, C). The determinant invariant under SL(2, C) is not invariant under GL(2, C), but is multiplied by Det( g ) for g in GL(2, C). The exterior product of two vectors in a four dimensional space has six components that make up a bivector and is, of course, not dual to a vector since in four dimensions, bivectors are dual to bivectors. To express Minkowskian bivectors properly, the Dirac Algebra must be used.
Since,
M₄(u) = it + M(v)
The eigenvalues of M₄(u) are then
ρ± = it ± r
where r is the positive eigenvalue as above for M(v). Furthermore, the eigenvectors of M(v) are then also eigenvectors for M₄(u).
M₄(u) |r±) = ρ± |ρ±)
The notion of mapping 3−vectors into 2x2 Hermitean matrices, generalizes in the spinor formalism of [Cartan 1946] for an even dimensional space n = 2ν to mapping an n dimensional vector to a 2ν x 2ν matrix( Hermitean if the signature of the underlying space is Euclidean). In particular for a four dimensional space with Lorentzian signature, and vector u = (x, y, z, t):
│ 0(2) M(u) │
│ │
u → │ │ := MC(u)
│ │
│ M*(u) 0(2) │
where 0(2) is a null matrix,
│ (x + iy) (z + t) │
│ │
M(u) = │ │
│ │
│ (z - t) (-x + iy) │
and M* is the complex conjugate matrix.
with
M*(u) M(u) = (x² + y² + z² - t²)
[M*(u), M(u)] = 0
In terms of direct products of Pauli matrices.
MC(u) = x(σ₁ X σ₄) + y(-σ₂ X σ₀) + z(σ₁ X σ₁) + t(-iσ₁ X σ₂)
If Γk are define by direct products of the σk as
defined above,
Γ₁ := ( σ₁ X σ₄)
Γ₂ := ( -σ₂ X σ₀)
Γ₄ := ( σ₁ X σ₁)
Γ₄ := (-iσ₁ X σ₂)
then,
{Γk, Γj} = 2 ηkj
where ηkj is the Lorentz metric Diag[+1 +1 +1 -1], so that the matrix MC(u) expresses the vector as a linear combination of the vector subspace basis of a Clifford algebra for a four dimensional space with Lorentzian signature. Again considering the eigenvalue problem for MC(u), the four eigenvalues are easily seen to be
+t ± r and -t ± r (which embrace the 4 possible 1,3 signatures
of an inner product),
with r = √(x² + y² + z²)
An eigenvector for +t + r:
│ (r - t) │
│ │
│ 0 │
│ │
│ (x-iy)/(r²-t²) │
│ │
│ (z-t)/(r²-t²) │
An eigenvector for +t - r:
│ (r + t) │
│ │
│ 0 │
│ │
│ (x - iy) │
│ │
│ (z - t) │
An eigenvector for -t - r:
│ (t - r) │
│ │
│ 0 │
│ │
│ (x-iy)/(r²-t²) │
│ │
│ (z-t)/(r²-t²) │
An eigenvector for -t + r:
│ -(t + r) │
│ │
│ 0 │
│ │
│ (x-iy)/(r²-t²) │
│ │
│ (z-t)/(r²-t²) │
No matter what representation of the Clifford algebra we use, if |φ) is a spinor, that transforms according to the double valued representation of the group preserving ηkj in its defining or vector representation, i.e. SO(n) or one of its noncompact forms SO(p, q), then the bilinears
(φ|Γk|φ)
transform as the components of a vector, since
S⁻¹(g) Γk S(g) = Λkj(g) Γj
[Brauer 1935]
with index summation and where S(g) is the spin representation of the group and Λkj(g) is a matrix of the defining representation of the group. Similarly, other sets of bilinears transform as p−th order tensors. If the above representation of the Γk are used, and |ξ) is a general 4−spinor with components
ξ0, ξ1, ξ2, ξ3,
then the components of the associated vector as functions of the spinor components are:
(ξ| Γ1 |ξ) = ξ0* ξ2 - ξ1* ξ3 + ξ2* ξ0 - ξ3* ξ1 (ξ| Γ₂ |ξ) = i(ξ0* ξ2 + ξ1* ξ3 - ξ2* ξ0 - ξ3* ξ1) (ξ| Γ₃ |ξ) = ξ0* ξ3 + ξ1* ξ2 + ξ2* ξ1 - ξ3* ξ0 (ξ| Γ₄ |ξ) = ξ0* ξ2 - ξ1* ξ3 - ξ2* ξ0 + ξ3* ξ1
A spinor concept exists for any n dimensional Euclidean or pseudoeuclidean space. Another way of saying this is that the special orthogonal Lie groups SO(n) are always doubly connected. The above discussion extends conceptually and computationally to higher dimensions. As there Spin(n) groups that cover the SO(n) representations doubly, there are also SpinU(n) representations of the SU(n) Lie groups even though they are singly connected; the SU(n) contain SO(n) as Lie subgroups.
This brief essay gives no particular answers to the "mystery" of spinors, but perhaps puts some of them in relief. It should be fairly clear that their existence has something to do with the deep structure of Euclidean spaces, but also that this deep structure also somehow involves complex numbers and the analytic continuation of an Rⁿ to Cⁿ. They arise by factorizing quadratic forms, as in the method of Dirac strarting with the Klein-Gordon equation. They arise as 2-sided ideals in Clifford algebras. The "belt trick" indicates a connection with knot theory, noting that knots only exist in three dimensional spaces: for n < 3, you cannot make them, and for n > 3, all knots "untie" and are topologically equivalent to the "unknot".
Useful computational aspects of spinors in n dimensional spaces can be found in [Cartan 1946].
TOC
1. Note that this page is written using UTF-8 unicode encoding.
You may need to set something on your browser, or download
appropriate unicode fonts.
2. I am going to deal rather heavily here with mathematical and
physical concepts here and not the mathematical mechanics that
are available in many textbooks in mathematics and physics that
deal with spinors. One small note on language and English spelling:
Properly, as a few early British authors wrote the word, "spinor"
should be spelled according to both its construction and meaning
as "spinnor", else by it's current spelling it should be pronounced
spine-or, which it never is. The world seems now to have adopted
the perfectly ingorant US spelling. Undoubtedly, some Harvard MBA,
more concerned with some bottom line regarding expenditure of ink
than with language, or anything else for that matter, made the
determination for the posterity of the eternally stupid. For the
sake of search engines and ingrained wrongness, I adopt the standard
wrong spelling.
3. The inhomgeneous orthogonal group IO(n) is, in fact, a semidirect
product of its orthogonal subgroup O(n) with the translation
subgroup of translations T(n).
4.
Top of Page (TOC)
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