In geometry and physics, spinors are
elements of a vector space that can be
associated with Euclidean space. Like
geometric vectors and more general
tensors, spinors transform linearly when
the Euclidean space is subjected to a
slight rotation. When a sequence of such
small rotations is composed to form an
overall final rotation, however, the
resulting spinor transformation depends
on which sequence of small rotations was
used, unlike for vectors and tensors. A
spinor transforms to its negative when
the space is rotated through a complete
turn from 0° to 360°, and it is this
property that characterizes spinors. It
is also possible to associate a
substantially similar notion of spinor
to Minkowski space in which case the
Lorentz transformations of special
relativity play the role of rotations.
Spinors were introduced in geometry by
Élie Cartan in 1913. In the 1920s
physicists discovered that spinors are
essential to describe the intrinsic
angular momentum, or "spin", of the
electron and other subatomic particles.
Spinors are characterized by the
specific way in which they behave under
rotations. They change in different ways
depending not just on the overall final
rotation, but the details of how that
rotation was achieved. There are two
topologically distinguishable classes of
paths through rotations that result in
the same overall rotation, as famously
illustrated by the belt trick puzzle.
These two inequivalent classes yield
spinor transformations of opposite sign.
The spin group is the group of all
rotations keeping track of the class. It
doubly covers the rotation group, since
each rotation can be obtained in two
inequivalent ways as the endpoint of a
path. The space of spinors by definition
is equipped with a linear representation
of the spin group, meaning that elements
of the spin group act as linear
transformations on the space of spinors,
in a way that genuinely depends on the
homotopy class.
Although spinors can be defined purely
as elements of a representation space of
the spin group, they are typically
defined as elements of a vector space
that carries a linear representation of
the Clifford algebra. The Clifford
algebra is an associative algebra that
can be constructed from Euclidean space
and its inner product in a basis
independent way. Both the spin group and
its Lie algebra are embedded inside the
Clifford algebra in a natural way, and
in applications the Clifford algebra is
often the easiest to work with. After
choosing an orthonormal basis of
Euclidean space, a representation of the
Clifford algebra is generated by gamma
matrices, matrices that satisfy a set of
canonical anti-commutation relations.
The spinors are the column vectors on
which these matrices act. In three
Euclidean dimensions, for instance, the
Pauli spin matrices are a set of gamma
matrices, and the two-component complex
column vectors on which these matrices
act are spinors. However, the particular
matrix representation of the Clifford
algebra, and hence what precisely
constitutes a "column vector", involves
the choice of basis and gamma matrices
in an essential way. As a representation
of the spin group, this realization of
spinors as column vectors will either be
irreducible if the dimension is odd, or
it will decompose into a pair of
so-called "half-spin" or Weyl
representations if the dimension is
even.
Introduction 
What characterizes spinors and
distinguishes them from geometric
vectors and other tensors is subtle.
Consider applying a rotation to the
coordinates of a system. No object in
the system itself has moved, only the
coordinates have, so there will always
be a compensating change in those
coordinate values when applied to any
object of the system. Geometrical
vectors, for example, have components
that will undergo the same rotation as
the coordinates. More broadly, any
tensor associated with the system also
has coordinate descriptions that adjust
to compensate for changes to the
coordinate system itself. Spinors do not
appear at this level of the description
of a physical system, when one is
concerned only with the properties of a
single isolated rotation of the
coordinates. Rather, spinors appear when
we imagine that instead of a single
rotation, the coordinate system is
gradually rotated between some initial
and final configuration. For any of the
familiar and intuitive quantities
associated with the system, the
transformation law does not depend on
the precise details of how the
coordinates arrived at their final
configuration. Spinors, on the other
hand, are constructed in such a way that
makes them sensitive to how the gradual
rotation of the coordinates arrived
there: they exhibit path-dependence. It
turns out that, for any final
configuration of the coordinates, there
are actually two inequivalent gradual
rotations of the coordinate system that
result in this same configuration. This
ambiguity is called the homotopy class
of the gradual rotation. The belt trick
puzzle famously demonstrates two
different rotations, one through an
angle of 2π and the other through an
angle of 4π, having the same final
configurations but different classes.
Spinors actually exhibit a sign-reversal
that genuinely depends on this homotopy
class. This distinguishes them from
vectors and other tensors, none of which
can feel the class.
Spinors can be exhibited as concrete
objects using a choice of Cartesian
coordinates. In three Euclidean
dimensions, for instance, spinors can be
constructed by making a choice of Pauli
spin matrices corresponding to the three
coordinate axes. These are 2×2 matrices
with complex entries, and the
two-component complex column vectors on
which these matrices act by matrix
multiplication are the spinors. In this
case, the spin group is isomorphic to
the group of 2×2 unitary matrices with
determinant one, which naturally sits
inside the matrix algebra. This group
acts by conjugation on the real vector
space spanned by the Pauli matrices
themselves, realizing it as a group of
rotations among them, but it also acts
on the column vectors.
More generally, a Clifford algebra can
be constructed from any vector space V
equipped with a quadratic form, such as
Euclidean space with its standard dot
product or Minkowski space with its
standard Lorentz metric. Given a
suitably normalized basis of V, the
Clifford algebra is generated by gamma
matrices, matrices that satisfy a set of
canonical anti-commutation relations,
and the space of spinors is the space of
column vectors with  components on which
those matrices act. Although the
Clifford algebra can be defined
abstractly in a coordinate-independent
way, its particular realization as a
specific algebra of matrices depends on
which orthogonal axes the gamma matrices
represent. So what precisely constitutes
a "column vector" also depends on such
arbitrary choices. The orthogonal Lie
algebra and the spin group associated to
the quadratic form are both contained in
the Clifford algebra, so every Clifford
algebra representation also defines a
representation of the Lie algebra and
the spin group. Depending on the
dimension and metric signature, this
realization of spinors as column vectors
may be irreducible or it may decompose
into a pair of so-called "half-spin" or
Weyl representations.
Overview 
There are essentially two frameworks for
viewing the notion of a spinor.
One is representation theoretic. In this
point of view, one knows beforehand that
there are some representations of the
Lie algebra of the orthogonal group that
cannot be formed by the usual tensor
constructions. These missing
representations are then labeled the
spin representations, and their
constituents spinors. In this view, a
spinor must belong to a representation
of the double cover of the rotation
group SO(n, R), or more generally of
double cover of the generalized special
orthogonal group SO+(p, q, R) on spaces
with metric signature. These double
covers are Lie groups, called the spin
groups Spin(n) or Spin(p, q). All the
properties of spinors, and their
applications and derived objects, are
manifested first in the spin group.
Representations of the double covers of
these groups yield projective
representations of the groups
themselves, which do not meet the full
definition of a representation.
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 the
advantage of providing a concrete and
elementary description of what a spinor
is. However, such a description becomes
unwieldy when complicated properties of
spinors, such as Fierz identities, are
needed.
= Clifford algebras =
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, via the
classification of Clifford algebras. It
largely removes the need for ad hoc
constructions.
In detail, let V be a finite-dimensional
complex vector space with nondegenerate
bilinear form g. The Clifford algebra
Cℓ(V, g) is the algebra generated by V
along with the anticommutation relation
xy + yx = 2g(x, y). It is an abstract
version of the algebra generated by the
gamma or Pauli matrices. If V = Cn, with
the standard form g(x, y) = xty = x1y1 +
... + xnyn we denote the Clifford
algebra by Cℓn(C). Since by the choice
of an orthonormal basis every complex
vectorspace with non-degenerate form is
isomorphic to this standard example,
this notation is abused more generally
if dimC(V) = n. If n = 2k is even,
Cℓn(C) is isomorphic as an algebra to
the algebra Mat(2k, C) of 2k × 2k
complex matrices. If n = 2k + 1 is odd,
Cℓ2k+1(C) is isomorphic to the algebra
Mat(2k, C) ⊕ Mat(2k, C) of two copies of
the 2k × 2k complex matrices. Therefore,
in either case Cℓ(V, g) has a unique
irreducible representation, commonly
denoted by Δ, of dimension 2[n/2]. Since
the Lie algebra so(V, g) is embedded as
a Lie subalgebra in Cℓ(V, g) equipped
with the Clifford algebra commutator as
Lie bracket, the space Δ is also a Lie
algebra representation of so(V, g)
called a spin representation. If n is
odd, this Lie algebra representation is
irreducible. If n is even, it splits
further into two irreducible
representations Δ = Δ+ ⊕ Δ− called the
Weyl or half-spin representations.
Irreducible representations over the
reals in the case when V is a real
vector space are much more intricate,
and the reader is referred to the
Clifford algebra article for more
details.
= Spin groups =
Spinors form a vector space, usually
over the complex numbers, equipped with
a linear group representation of the
spin group that does not factor through
a representation of the group of
rotations. The spin group is the group
of rotations keeping track of the
homotopy class. Spinors are needed to
encode basic information about the
topology of the group of rotations
because that group is not simply
connected, but the simply connected spin
group is its double cover. So for every
rotation there are two elements of the
spin group that represent it. Geometric
vectors and other tensors cannot feel
the difference between these two
elements, but they produce opposite
signs when they affect any spinor under
the representation. Thinking of the
elements of the spin group as homotopy
classes of one-parameter families of
rotations, each rotation is represented
by two distinct homotopy classes of
paths to the identity. If a
one-parameter family of rotations is
visualized as a ribbon in space, with
the arc length parameter of that ribbon
being the parameter, then these two
distinct homotopy classes are visualized
in the two states of the belt trick
puzzle. The space of spinors is an
auxiliary vector space that can be
constructed explicitly in coordinates,
but ultimately only exists up to
isomorphism in that there is no
"natural" construction of them that does
not rely on arbitrary choices such as
coordinate systems. A notion of spinors
can be associated, as such an auxiliary
mathematical object, with any vector
space equipped with a quadratic form
such as Euclidean space with its
standard dot product, or Minkowski space
with its Lorentz metric. In the latter
case, the "rotations" include the
Lorentz boosts, but otherwise the theory
is substantially similar.
= Terminology in physics =
The most typical type of spinor, the
Dirac spinor, is an element of the
fundamental representation of Cℓp+q(C),
the complexification of the Clifford
algebra Cℓp, q(R), into which the spin
group Spin(p, q) may be embedded. On a
2k- or 2k+1-dimensional space a Dirac
spinor may be represented as a vector of
2k complex numbers. 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 Cℓp,q(R)
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.
Dirac and Weyl spinors are complex
representations while Majorana spinors
are real representations.
The Dirac, Lorentz, Weyl, and Majorana
spinors are interrelated, and their
relation can be elucidated on the basis
of real geometric algebra.
Massive particles, such as electrons,
are described as Dirac spinors. The
classical neutrino of the standard model
of particle physics is an example of a
Weyl spinor. However, because of
observed neutrino oscillation, it is now
believed that they are not Weyl spinors,
but perhaps instead Majorana spinors. It
is not known whether Weyl spinors exist
in nature. In 2015, an international
team led by Princeton University
scientists announced that they had found
a quasiparticle that behaves as a Weyl
fermion.
= Spinors in representation theory =
One major mathematical application 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 Atiyah–Singer
index theorem, and to provide
constructions in particular for discrete
series representations of semisimple
groups.
The spin representations of the special
orthogonal Lie algebras are
distinguished from the tensor
representations given by Weyl's
construction by the weights. Whereas the
weights of the tensor representations
are integer linear combinations of the
roots of the Lie algebra, those of the
spin representations are half-integer
linear combinations thereof. Explicit
details can be found in the spin
representation article.
= Attempts at intuitive understanding =
The spinor can be described, in simple
terms, as “vectors of a space the
transformations of which are related in
a particular way to rotations in
physical space”. Stated differently:
Spinors […] provide a linear
representation of the group of rotations
in a space with any number  of
dimensions, each spinor having 
components where  or .
Several ways of illustrating everyday
analogies have been formulated in terms
of the plate trick, tangloids and other
examples of orientation entanglement.
Nonetheless, the concept is generally
considered notoriously difficult to
understand, as illustrated by Michael
Atiyah's statement that is recounted by
Dirac's biographer Graham Farmelo:
No one fully understands spinors. Their
algebra is formally understood but their
general significance is mysterious. In
some sense they describe the “square
root” of geometry and, just as
understanding the square root of −1 took
centuries, the same might be true of
spinors.
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 his 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 toys such as Tangloids
to teach and model the calculus of
spinors.
Spinor spaces were represented as left
ideals of a matrix algebra in 1930, by
G. Juvet and by Fritz Sauter. More
specifically, instead of representing
spinors as complex-valued 2D column
vectors as Pauli had done, they
represented them as complex-valued 2 × 2
matrices in which only the elements of
the left column are non-zero. In this
manner the spinor space became a minimal
left ideal in Mat(2, C).
In 1947 Marcel Riesz constructed spinor
spaces as elements of a minimal left
ideal of Clifford algebras. In
1966/1967, David Hestenes replaced
spinor spaces by the even subalgebra
Cℓ01,3(R) of the spacetime algebra
Cℓ1,3(R). As of the 1980s, the
theoretical physics group at Birkbeck
College around David Bohm and Basil
Hiley has been developing algebraic
approaches to quantum theory that build
on Sauter and Riesz' identification of
spinors with minimal left ideals.
Examples 
Some simple examples of spinors in low
dimensions arise from considering the
even-graded subalgebras of the Clifford
algebra Cℓp, 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
= Two dimensions =
The Clifford algebra Cℓ2,0(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
that2 =2 = 1, and(σ1σ2) = −σ1σ1σ2σ2 =
−1.
The even subalgebra Cℓ02,0(R), spanned
by even-graded basis elements of
Cℓ2,0(R), determines the space of
spinors via its representations. It is
made up of real linear combinations of 1
and σ1σ2. As a real algebra, Cℓ02,0(R)
is isomorphic to field of complex
numbers C. As a result, it admits a
conjugation operation, sometimes called
the reverse of a Clifford element,
defined by
which, by the Clifford relations, can be
written
The action of an even Clifford element γ
∈ Cℓ02,0(R) on vectors, regarded as
1-graded elements of Cℓ2,0(R), is
determined by mapping a general vector u
= a1σ1 + a2σ2 to the vector
where γ∗ is the conjugate of γ, and the
product is Clifford multiplication. In
this situation, a spinor is an ordinary
complex number. The action of γ on a
spinor φ is given by ordinary complex
multiplication:
An important feature of this definition
is the distinction between ordinary
vectors and spinors, manifested in how
the even-graded elements act on each of
them in different ways. In general, a
quick check of the Clifford relations
reveals that even-graded elements
conjugate-commute with ordinary vectors:
On the other hand, comparing with the
action on spinors γ(φ) = γφ, γ on
ordinary vectors acts as the square of
its action on spinors.
Consider, for example, the implication
this has for plane rotations. Rotating a
vector through an angle of θ corresponds
to γ2 = exp(θ σ1σ2), so that the
corresponding action on spinors is via γ
= ± exp(θ σ1σ2/2). In general, because
of logarithmic branching, it is
impossible to choose a sign in a
consistent way. Thus the representation
of plane rotations on spinors is
two-valued.
In applications of spinors in two
dimensions, it is common to exploit the
fact that the algebra of even-graded
elements is identical to the space of
spinors. So, by abuse of language, the
two are often conflated. One may then
talk about "the action of a spinor on a
vector." In a general setting, such
statements are meaningless. But in
dimensions 2 and 3 they make sense.
Examples
The even-graded element
corresponds to a vector rotation of 90°
from σ1 around towards σ2, which can be
checked by confirming that
It corresponds to a spinor rotation of
only 45°, however:
Similarly the even-graded element
γ = −σ1σ2 corresponds to a vector
rotation of 180°:
but a spinor rotation of only 90°:
Continuing on further, the even-graded
element γ = −1 corresponds to a vector
rotation of 360°:
but a spinor rotation of 180°.
= Three dimensions =
Main articles Spinors in three
dimensions, Quaternions and spatial
rotation
The Clifford algebra Cℓ3,0(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
that2 =2 =2 = 1, and2 =2 =2 =2 = −1.
The sub-algebra of even-graded elements
is made up of scalar dilations,
and vector rotations
where
corresponds to a vector 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:
With the identification of the
even-graded elements with the algebra H
of quaternions, as in the case of two
dimensions the only representation of
the algebra of even-graded elements is
on itself. Thus the spinors in
three-dimensions are quaternions, and
the action of an even-graded element on
a spinor is given by ordinary
quaternionic multiplication.
Note that the expression for a vector
rotation through an angle θ, the angle
appearing in γ was halved. Thus the
spinor rotation γ(ψ) = γψ will rotate
the spinor ψ through an angle one-half
the measure of the angle of the
corresponding vector rotation. Once
again, the problem of lifting a vector
rotation to a spinor rotation is
two-valued: the expression with in place
of θ/2 will produce the same vector
rotation, but the negative of the spinor
rotation.
The spinor/quaternion representation of
rotations in 3D is becoming increasingly
prevalent in computer geometry and other
applications, because of the notable
brevity of the corresponding spin
matrix, and the simplicity with which
they can be multiplied together to
calculate the combined effect of
successive rotations about different
axes.
Explicit constructions 
A space of spinors can be constructed
explicitly with concrete and abstract
constructions. The equivalence of these
constructions are a consequence of the
uniqueness of the spinor representation
of the complex Clifford algebra. For a
complete example in dimension 3, see
spinors in three dimensions.
= Component spinors =
Given a vector space V and a quadratic
form g an explicit matrix representation
of the Clifford algebra Cℓ(V, g) can be
defined as follows. Choose an
orthonormal basis e1 … en for V i.e.
g(eμeν) = ημν where ημμ = ±1 and ημν = 0
for μ ≠ ν. Let k = ⌊ n/2 ⌋. Fix a set of
2k × 2k matrices γ1 … γn such that γμγν
+ γνγμ = 2ημν1. Then the assignment eμ →
γμ extends uniquely to an algebra
homomorphism Cℓ(V, g) → Mat(2k, C) by
sending the monomial eμ1 … eμk in the
Clifford algebra to the product γμ1 …
γμk of matrices and extending linearly.
The space Δ = C2k on which the gamma
matrices act is a now a space of
spinors. One needs to construct such
matrices explicitly, however. In
dimension 3, defining the gamma matrices
to be the Pauli sigma matrices gives
rise to the familiar two component
spinors used in non relativistic quantum
mechanics. Likewise using the 4 × 4
Dirac gamma matrices gives rise to the 4
component Dirac spinors used in 3+1
dimensional relativistic quantum field
theory. In general, in order to define
gamma matrices of the required kind, one
can use the Weyl–Brauer matrices.
In this construction the representation
of the Clifford algebra Cℓ(V, g), the
Lie algebra so(V, g), and the Spin group
Spin(V, g), all depend on the choice of
the orthonormal basis and the choice of
the gamma matrices. This can cause
confusion over conventions, but
invariants like traces are independent
of choices. In particular, all
physically observable quantities must be
independent of such choices. In this
construction a spinor can be represented
as a vector of 2k complex numbers and is
denoted with spinor indices. In the
physics literature, abstract spinor
indices are often used to denote spinors
even when an abstract spinor
construction is used.
= Abstract spinors =
There are at least two different, but
essentially equivalent, ways to define
spinors abstractly. One approach seeks
to identify the minimal ideals for the
left action of Cℓ(V, g) on itself. These
are subspaces of the Clifford algebra of
the form Cℓ(V, g)ω, admitting the
evident action of Cℓ(V, g) by
left-multiplication: c : xω → cxω. There
are two variations on this theme: one
can either find a primitive element ω
that is a nilpotent element of the
Clifford algebra, or one that is an
idempotent. The construction via
nilpotent elements is more fundamental
in the sense that an idempotent may then
be produced from it. In this way, the
spinor representations are identified
with certain subspaces of the Clifford
algebra itself. The second approach is
to construct a vector space using a
distinguished subspace of V, and then
specify the action of the Clifford
algebra externally to that vector space.
In either approach, the fundamental
notion is that of an isotropic subspace
W. Each construction depends on an
initial freedom in choosing this
subspace. In physical terms, this
corresponds to the fact that there is no
measurement protocol that can specify a
basis of the spin space, even if a
preferred basis of V is given.
As above, we let be an n-dimensional
complex vector space equipped with a
nondegenerate bilinear form. If V is a
real vector space, then we replace V by
its complexification V ⊗R C and let g
denote the induced bilinear form on
V ⊗R C. Let W be a maximal isotropic
subspace, i.e. a maximal subspace of V
such that g|W = 0. If n =  2k is even,
then let W∗ be an isotropic subspace
complementary to W. If n =  2k + 1 is
odd, let W∗ be a maximal isotropic
subspace with W ∩ W∗ = 0, and let U be
the orthogonal complement of W ⊕ W∗. In
both the even- and odd-dimensional cases
W and W∗ have dimension k. In the
odd-dimensional case, U is
one-dimensional, spanned by a unit
vector u.
= Minimal ideals =
Since W′ is isotropic, multiplication of
elements of W′ inside Cℓ(V, g) is skew.
Hence vectors in W′ anti-commute, and
Cℓ(W′, g|W′) = Cℓ(W′, 0) is just the
exterior algebra Λ∗W′. Consequently, the
k-fold product of W′ with itself, W′k,
is one-dimensional. Let ω be a generator
of W′k. In terms of a basis w′1,..., w′k
of in W′, one possibility is to set
Note that ω2 = 0, and moreover, w′ω = 0
for all w′ ∈ W′. The following facts can
be proven easily:
If n = 2k, then the left ideal Δ =
Cℓ(V, g)ω is a minimal left ideal.
Furthermore, this splits into the two
spin spaces Δ+ = Cℓevenω and Δ− = Cℓoddω
on restriction to the action of the even
Clifford algebra.
If n = 2k + 1, then the action of the
unit vector u on the left ideal
Cℓ(V, g)ω decomposes the space into a
pair of isomorphic irreducible
eigenspaces, corresponding to the
respective eigenvalues +1 and −1.
In detail, suppose for instance that n
is even. Suppose that I is a non-zero
left ideal contained in Cℓ(V, g)ω. We
shall show that I must be equal to
Cℓ(V, g)ω by proving that it contains a
nonzero scalar multiple of ω.
Fix a basis wi of W and a complementary
basis wi′ of W′ so that
wiwj′ +wj′ wi = δij, and
(wi)2 = 0,2 = 0.
Note that any element of I must have the
form αω, by virtue of our assumption
that I ⊂ Cℓ(V, g) ω. Let αω ∈ I be any
such element. Using the chosen basis, we
may write
where the ai1…ip are scalars, and the Bj
are auxiliary elements of the Clifford
algebra. Observe now that the product
Pick any nonzero monomial a in the
expansion of α with maximal homogeneous
degree in the elements wi:
then
is a nonzero scalar multiple of ω, as
required.
Note that for n even, this computation
also shows that
as a vector space. In the last equality
we again used that W is isotropic. In
physics terms, this shows that Δ is
built up like a Fock space by creating
spinors using anti-commuting creation
operators in W acting on a vacuum ω.
= Exterior algebra construction =
The computations with the minimal ideal
construction suggest that a spinor
representation can also be defined
directly using the exterior algebra Λ∗ W
= ⊕j Λj W of the isotropic subspace W.
Let Δ = Λ∗ W denote the exterior algebra
of W considered as vector space only.
This will be the spin representation,
and its elements will be referred to as
spinors.
The action of the Clifford algebra on Δ
is defined first by giving the action of
an element of V on Δ, and then showing
that this action respects the Clifford
relation and so extends to a
homomorphism of the full Clifford
algebra into the endomorphism ring
End(Δ) by the universal property of
Clifford algebras. The details differ
slightly according to whether the
dimension of V is even or odd.
When dim(V) is even, V = W ⊕ W′ where W′
is the chosen isotropic complement.
Hence any v ∈ V decomposes uniquely as v
= w + w′ with w ∈ W and w′ ∈ W′. The
action of v on a spinor is given by
where i(w′) is interior product with w′
using the non degenerate quadratic form
to identify V with V∗, and ε(w) denotes
the exterior product. It may be verified
that
c(u)c(v) + c(v)c(u) = 2 g(u,v),
and so c respects the Clifford relations
and extends to a homomorphism from the
Clifford algebra to End(Δ).
The spin representation Δ further
decomposes into a pair of irreducible
complex representations of the Spin
group via
When dim(V) is odd, V = W ⊕ U ⊕ W′,
where U is spanned by a unit vector u
orthogonal to W. The Clifford action c
is defined as before on W ⊕ W′, while
the Clifford action of u is defined by
As before, one verifies that c respects
the Clifford relations, and so induces a
homomorphism.
= Hermitian vector spaces and spinors =
If the vector space V has extra
structure that provides a decomposition
of its complexification into two maximal
isotropic subspaces, then the definition
of spinors becomes natural.
The main example is the case that the
real vector space V is a hermitian
vector space, i.e., V is equipped with a
complex structure J that is an
orthogonal transformation with respect
to the inner product g on V. Then V ⊗R C
splits in the ±i eigenspaces of J. These
eigenspaces are isotropic for the
complexification of g and can be
identified with the complex vector space
and its complex conjugate. Therefore,
for a hermitian vector space the vector
space Λ⋅CV is a spinor space for the
underlying real euclidean vector space.
With the Clifford action as above but
with contraction using the hermitian
form, this construction gives a spinor
space at every point of an almost
Hermitian manifold and is the reason why
every almost complex manifold has a
Spinc structure. Likewise, every complex
vector bundle on a manifold carries a
Spinc structure.
Clebsch–Gordan decomposition 
A number of Clebsch–Gordan
decompositions are possible on the
tensor product of one spin
representation with another. These
decompositions express the tensor
product in terms of the alternating
representations of the orthogonal group.
For the real or complex case, the
alternating representations are
Γr = ΛrV, the representation of the
orthogonal group on skew tensors of rank
r.
In addition, for the real orthogonal
groups, there are three characters
σ+ : O(p, q) → {−1, +1} given by σ+(R) =
−1, if R reverses the spatial
orientation of V, +1, if R preserves the
spatial orientation of V.
σ− : O(p, q) → {−1, +1} given by σ−(R) =
−1, if R reverses the temporal
orientation of V, +1, if R preserves the
temporal orientation of V.
σ = σ+σ− .
The Clebsch–Gordan decomposition allows
one to define, among other things:
An action of spinors on vectors.
A Hermitian metric on the complex
representations of the real spin groups.
A Dirac operator on each spin
representation.
= Even dimensions =
If n = 2k is even, then the tensor
product of Δ with the contragredient
representation decomposes as
which can be seen explicitly by
considering the action of the Clifford
algebra on decomposable elements
αω ⊗ βω′. The rightmost formulation
follows from the transformation
properties of the Hodge star operator.
Note that on restriction to the even
Clifford algebra, the paired summands Γp
⊕ σΓp are isomorphic, but under the full
Clifford algebra they are not.
There is a natural identification of Δ
with its contragredient representation
via the conjugation in the Clifford
algebra:
So Δ ⊗ Δ also decomposes in the above
manner. Furthermore, under the even
Clifford algebra, the half-spin
representations decompose
For the complex representations of the
real Clifford algebras, the associated
reality structure on the complex
Clifford algebra descends to the space
of spinors. In this way, we obtain the
complex conjugate Δ of the
representation Δ, and the following
isomorphism is seen to hold:
In particular, note that the
representation Δ of the orthochronous
spin group is a unitary representation.
In general, there are Clebsch–Gordan
decompositions
In metric signature, the following
isomorphisms hold for the conjugate
half-spin representations
If q is even, then  and 
If q is odd, then  and 
Using these isomorphisms, one can deduce
analogous decompositions for the tensor
products of the half-spin
representations Δ± ⊗ Δ±.
= Odd dimensions =
If n = 2k + 1 is odd, then
In the real case, once again the
isomorphism holds
Hence there is a Clebsch–Gordan
decomposition given by
= Consequences =
There are many far-reaching consequences
of the Clebsch–Gordan decompositions of
the spinor spaces. The most fundamental
of these pertain to Dirac's theory of
the electron, among whose basic
requirements are
A manner of regarding the product of two
spinors ϕψ as a scalar. In physical
terms, a spinor should determine a
probability amplitude for the quantum
state.
A manner of regarding the product ψϕ as
a vector. This is an essential feature
of Dirac's theory, which ties the spinor
formalism to the geometry of physical
space.
A manner of regarding a spinor as acting
upon a vector, by an expression such as
ψvψ. In physical terms, this represents
an electric current of Maxwell's
electromagnetic theory, or more
generally a probability current.
Summary in low dimensions 
In 1 dimension, 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±iφ/2 under a
rotation by angle φ.
In 3 Euclidean dimensions, the single
spinor representation is 2-dimensional
and quaternionic. The existence of
spinors in 3 dimensions follows from the
isomorphism of the groups SU(2) ≅
Spin(3) that allows us to define the
action of Spin(3) on a complex
2-component column; the generators of
SU(2) can be written as Pauli matrices.
In 4 Euclidean dimensions, the
corresponding isomorphism is Spin(4) ≅
SU(2) × SU(2). There are two
inequivalent quaternionic 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) ≅ USp(4) ≅ Sp(2)
that implies that the single spinor
representation is 4-dimensional and
quaternionic.
In 6 Euclidean dimensions, the
isomorphism Spin(6) ≅ 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 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 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-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
2-component spinors, which follows from
the isomorphism SL(2, C) ≅ Spin(3,1).
See also 
Anyon
Dirac equation in the algebra of
physical space
Einstein–Cartan theory
Pure spinor
Spin-½
Spinor bundle
Supercharge
Twistor theory
Notes 
References 
Further reading 
Brauer, Richard; Weyl, Hermann, "Spinors
in n dimensions", American Journal of
Mathematics 57: 425–449,
doi:10.2307/2371218, JSTOR 2371218 .
Cartan, Élie, "Les groupes projectifs
qui ne laissent invariante aucune
multiplicité plane", Bul. Soc. Math.
France 41: 53–96 .
Cartan, Élie, The theory of spinors,
Paris, Hermann, ISBN 978-0-486-64070-9 
Chevalley, Claude, The algebraic theory
of spinors and Clifford algebras,
Columbia University Press, ISBN
978-3-540-57063-9 .
Dirac, Paul M., "The quantum theory of
the electron", Proceedings of the Royal
Society of London A117: 610–624, JSTOR
94981 .
Fulton, William; Harris, Joe,
Representation theory. A first course,
Graduate Texts in Mathematics, Readings
in Mathematics 129, New York:
Springer-Verlag, ISBN 0-387-97495-4, MR
1153249 .
Gilkey, Peter B., Invariance Theory, the
Heat Equation, and the Atiyah–Singer
Index Theorem, Publish or Perish, ISBN
0-914098-20-9 .
Harvey, F. Reese, Spinors and
Calibrations, Academic Press, ISBN
978-0-12-329650-4 .
Hazewinkel, Michiel, ed., "Spinor",
Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4 
Hitchin, Nigel J., "Harmonic spinors",
Advances in Mathematics 14: 1–55,
doi:10.1016/0001-8708(74)90021-8, MR
358873 .
Lawson, H. Blaine; Michelsohn,
Marie-Louise, Spin Geometry, Princeton
University Press, ISBN 0-691-08542-0 .
Pauli, Wolfgang, "Zur Quantenmechanik
des magnetischen Elektrons", Zeitschrift
für Physik 43: 601–632,
Bibcode:1927ZPhy...43..601P,
doi:10.1007/BF01397326 .
Penrose, Roger; Rindler, W., Spinors and
Space–Time: Volume 2, Spinor and Twistor
Methods in Space–Time Geometry,
Cambridge University Press, ISBN
0-521-34786-6 .
Tomonaga, Sin-Itiro, "Lecture 7: The
Quantity Which Is Neither Vector nor
Tensor", The story of spin, University
of Chicago Press, p. 129, ISBN
0-226-80794-0
