In
mathematics and
physics, in particular in the theory of the
orthogonal groups (such as the
rotation or the
Lorentz groups),
spinors are
elements of a complex
vector space
introduced to expand the notion of
spatial vector. They are needed because the
full structure of the group of
rotations in
a given number of dimensions requires some extra number of
dimensions to exhibit it. Specifically, spinors are geometrical
objects constructed from a vector space endowed with a
quadratic form, such as a
Euclidean or
Minkowski space, by means of an algebraic
procedure, through
Clifford
algebras, or a
quantization procedure. A given
quadratic form may support several different types of
spinors.
Spinors in general were discovered by
Élie Cartan in 1913. Later, spinors were
adopted by
quantum mechanics in
order to study the properties of the
intrinsic angular momentum of the
electron and other
fermions. Today spinors enjoy a wide range of
physics applications. Classically,
spinors in three dimensions are
used to describe the spin of the nonrelativistic electron. Via the
Dirac equation,
Dirac spinors are required in the mathematical
description of the
quantum state of
the
relativistic electron. In
quantum field theory, spinors
describe the state of relativistic manyparticle systems. In
mathematics, particularly in
differential geometry and
global analysis, spinors have since found
broad applications to
algebraic
and
differential topology,
symplectic geometry,
gauge theory,
complex algebraic geometry,
index theory, and
special
holonomy.
The technical definitions and apparatus may make spinors seem
something imposed on geometry, but there are several reasons, now
well understood, why this is not really the case. From an algebraic
point of view, spinors are needed, because there are
representations of the infinitesimal orthogonal transformations
(the
Lie algebra) which cannot be
constructed starting from the natural rotation representation. The
existence of such "missing" representations is topological in
nature, reflecting the fact that rotation groups are not in general
simplyconnected.
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. (See
orientation entanglement.)
There are essentially two frameworks for viewing the notion of a
spinor.
One is
representation
theoretic. In this point of 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 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 the
generalized special
orthogonal group SO
^{+}(p, q,
R) on
spaces with
metric signature (p,q).
These doublecovers 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 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
(also called
geometric 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, if
V is a finitedimensional 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 = 2
g(
x,
y). It is an abstract
version of the algebra generated by the
gamma or
Pauli
matrices. The Clifford algebra
Cℓ
_{n}(
C) is
algebraically isomorphic to the algebra
Mat(2
^{k},
C) of
2
^{k} × 2
^{k} complex matrices,
if
n = dim(
V) = 2
k is even; or the
algebra
Mat(2
^{k},
C)⊕Mat(2
^{k},
C)
of two copies of the 2
^{k} ×
2
^{k} matrices, if
n = dim(
V) =
2
k+ 1 is odd. It therefore has a unique irreducible
representation (also called simple
Clifford module), commonly denoted by Δ,
whose dimension is 2
^{k}. 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. Therefore, the space Δ is also a Lie algebra
representation of
so(
V,
g) called
a
spin representation. If
n is odd, this representation is irreducible. If
n is even, it splits again into two irreducible
representations Δ = Δ
_{+} ⊕ Δ
_{−} called the
halfspin 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.
Terminology in physics
The most typical type of spinor, the
Dirac spinor, is an element of the
fundamental representation of the complexified
Clifford algebra Cℓ(p,q), into which the
spin group Spin(p,q) may be embedded. On a 2
k or
2
k+1dimensional space a Dirac spinor may be represented
as a vector of 2
^{k} complex numbers. (See
Special unitary group.) In even
dimensions, this representation is
reducible when taken as a
representation of
Spin(p,q) and may be decomposed into two: the lefthanded and
righthanded
Weyl spinor representations. In
addition, sometimes the noncomplexified version of Cℓ(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
MajoranaWeyl 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.
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
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 halfinteger linear combinations thereof.
Explicit details can be found in the
spin representation article.
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
Some important simple examples of spinors in low dimensions arise
from considering the evengraded subalgebras of the Clifford
algebra
Cℓ
_{p,q}(
R).
This is an algebra built up from an orthonormal basis of
n
=
p +
q mutually orthogonal vectors under
addition and multiplication,
p of which have norm +1 and
q of which have norm −1, with the product rule for the
basis vectors
 e_i e_j = \Bigg\{ \begin{matrix} +1 & i=j, \, i \in (1
\ldots p) \\
1 & i=j, \, i \in (p+1 \ldots n) \\
 e_j e_i & i \not = j. \end{matrix}
Two dimensions
The Clifford algebra
Cℓ
_{2,0}(
R)
is built up from a basis of one unit scalar, 1, two orthogonal unit
vectors,
σ_{1} and
σ_{2}, and one
unit
pseudoscalar i =
σ_{1}σ_{2}. From the definitions
above, it is evident that (
σ_{1})^{2}
= (σ
_{2})
^{2} = 1, and
(
σ_{1}σ_{2})(
σ_{1}σ_{2})
=

σ_{1}σ_{1}σ_{2}σ_{2}
= 1.
The even subalgebra
Cℓ
^{0}_{2,0}(
R), spanned
by
evengraded basis elements of
Cℓ
_{2,0}(
R), determines the space
of spinors via its representations. It is made up of real linear
combinations of 1 and
σ_{1}σ_{2}.
As a real algebra, Cℓ
^{0}_{2,0}(
R)
is isomorphic to field of
complex
numbers C. As a result, it admits a
conjugation operation (analogous to
complex conjugation), sometimes called the
reverse of a Clifford element, defined by
 (a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1\,.
which, by the Clifford relations, can be written
 (a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 =
ab\sigma_1\sigma_2\,.
The action of an even Clifford element γ ∈
Cℓ
^{0}_{2,0} on vectors, regarded as
1graded elements of Cℓ
_{2,0}, is determined by mapping a
general vector
u =
a_{1}σ_{1} +
a_{2}σ_{2} to the vector
 \gamma(u) = \gamma u \gamma^*\,,
where γ
^{*} is the conjugate of γ, and the product is
Clifford multiplication. In this situation, a
spinorThese are the righthanded Weyl spinors in
twodimensions. For the lefthanded Weyl spinors, the
representation is via \gamma(\phi)=\bar{\gamma}\phi. The Majorana
spinors are the common underlying real representation for the Weyl
representations. is an ordinary complex number. The action of γ on
a spinor φ is given by ordinary complex multiplication:
 \gamma(\phi) = \gamma\phi\,.
An important feature of this definition is the distinction between
ordinary vectors and spinors, manifested in how the evengraded
elements act on each of them in different ways. In general, a quick
check of the Clifford relations reveals that evengraded elements
conjugatecommute with ordinary vectors:
 \gamma(u) = \gamma u \gamma^* = \gamma^2 u\,.
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
γ
^{2} = exp(θ σ
_{1}σ
_{2}), so that the
corresponding action on spinors is via γ = ± exp(θ
σ
_{1}σ
_{2}/2). In general, because of
logarithmic branching, it is impossible to choose
a sign in a consistent way. Thus the representation of
planerotations on spinors is twovalued.
In applications of spinors in two dimensions, it is common to
exploit the fact that the algebra of evengraded elements (which is
just the ring of complex numbers) 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 (as applied, for example, to
computer graphics) they make
sense.
 Examples
 :\gamma = \tfrac{1}{\sqrt{2}} (1  \sigma_1 \sigma_2) \,
 corresponds to a vector rotation of 90° from
σ_{1} around towards σ_{2}, which
can be checked by confirming that
 :\tfrac{1}{2} (1  \sigma_1 \sigma_2) \,
\{a_1\sigma_1+a_2\sigma_2\} \, (1  \sigma_2 \sigma_1) =
a_1\sigma_2  a_2\sigma_1 \,
 It corresponds to a spinor rotation of only 45°, however:
 :\tfrac{1}{\sqrt{2}} (1  \sigma_1 \sigma_2) \,
\{a_1+a_2\sigma_1\sigma_2\}=
\frac{a_1+a_2}{\sqrt{2}} +
\frac{a_1+a_2}{\sqrt{2}}\sigma_1\sigma_2
 Similarly the evengraded element γ =
σ_{1}σ_{2} corresponds to a
vector rotation of 180°:
 : ( \sigma_1 \sigma_2) \, \{a_1\sigma_1 + a_2\sigma_2\} \, (
\sigma_2 \sigma_1) =  a_1\sigma_1 a_2\sigma_2 \,
 but a spinor rotation of only 90°:
 :( \sigma_1 \sigma_2) \, \{a_1 + a_2\sigma_1\sigma_2\}
=a_2  a_1\sigma_1\sigma_2
 Continuing on further, the evengraded element γ = 1
corresponds to a vector rotation of 360°:
 : (1) \, \{a_1\sigma_1+a_2\sigma_2\} \, (1) =
a_1\sigma_1+a_2\sigma_2 \,
 but a spinor rotation of 180°.
Three dimensions
 Main articles Spinors in three dimensions,
Quaternions and spatial
rotation
The Clifford algebra
Cℓ
_{3,0}(
R)
is built up from a basis of one unit scalar, 1, three orthogonal
unit vectors,
σ_{1},
σ_{2} and
σ_{3}, the three unit bivectors
σ_{1}σ_{2},
σ_{2}σ_{3},
σ_{3}σ_{1} and the
pseudoscalar i =
σ_{1}σ_{2}σ_{3}.
It is straightforward to show that
(
σ_{1})
^{2} =
(
σ_{2})
^{2} =
(
σ_{3})
^{2} = 1,
and(
σ_{1}σ_{2})
^{2} =
(
σ_{2}σ_{3})
^{2} =
(
σ_{3}σ_{1})
^{2} =
(
σ_{1}σ_{2}σ_{3})
^{2}
= 1.
The subalgebra of evengraded elements is made up of scalar
dilations,
 u^{\prime} = \rho^{(1/2)} u \rho^{(1/2)} = \rho u,
and vector rotations
 u^{\prime} = \gamma \, u \, \gamma^*,
where
 \left.\begin{matrix} \gamma & = & \cos(\theta/2) 
\{a_1 \sigma_2\sigma_3 + a_2 \sigma_3\sigma_1 + a_3
\sigma_1\sigma_2\} \sin(\theta/2) \\
& = & \cos(\theta/2)  i \{a_1 \sigma_1 + a_2 \sigma_2 +
a_3 \sigma_3\} \sin(\theta/2) \\& = & \cos(\theta/2)  i v
\sin(\theta/2) \end{matrix}\right\} (1)corresponds to a vector
rotation through an angle
θ about an axis defined by a
unit vector
v =
a_{1}σ_{1} +
a_{2}σ_{2} +
a_{3}σ_{3}
As a special case, it is easy to see that if
v =
σ_{3} this reproduces the
σ_{1}σ_{2} rotation considered in
the previous section; and that such rotation leaves the
coefficients of vectors in the
σ_{3} direction
invariant, since
 (\cos(\theta/2)  i \sigma_3 \sin(\theta/2)) \, \sigma_3 \,
(\cos(\theta/2) + i \sigma_3 \sin(\theta/2))
= (\cos^2(\theta/2) + \sin^2(\theta/2)) \, \sigma_3 =
\sigma_3.
The bivectors
σ_{2}σ_{3},
σ_{3}σ_{1} and
σ_{1}σ_{2} are in fact
Hamilton's quaternions i,
j and
k, discovered in 1843:
 \begin{matrix}\mathbf{i} = \sigma_2 \sigma_3 = i \sigma_1
\\
\mathbf{j} = \sigma_3 \sigma_1 = i \sigma_2 \\\mathbf{k} =
\sigma_1 \sigma_2 = i \sigma_3. \end{matrix}
With the identification of the evengraded elements with the
algebra
H of quaternions, as in the case of
twodimensions the only representation of the algebra of
evengraded elements is on itself.Since, for a
skew field, the kernel of the representation must
be trivial. So inequivalent representations can only arise via an
automorphism of the skewfield. In this
case, there are a pair of equivalent representations:
\gamma(\phi)=\gamma\phi, and its quaternionic conjugate
\gamma(\phi)=\phi\bar{\gamma}. Thus the (realThe complex spinors
are obtained as the representations of the
tensor product
H⊗
_{R}C
= Mat
_{2}(
C). These are considered in more
detail in
spinors in three
dimensions.) spinors in threedimensions are quaternions, and
the action of an evengraded element on a spinor is given by
ordinary quaternionic multiplication.
Note that the expression (1) for a vector rotation through an angle
θ, the angle appearing in γ was halved. Thus the spinor rotation
γ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the
spinor ψ through an angle onehalf the measure of the angle of the
corresponding vector rotation. Once again, the problem of lifting a
vector rotation to a spinor rotation is twovalued: the expression
(1) with (180° + θ/2) 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. Theequivalence 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
e^{1}...
e^{n} for
V. Let
k = \lfloor n/2 \rfloor. Fix a set of 2
^{k} by
2
^{k} matrices \gamma^1 \ldots \gamma^n such that the
assignment e^\mu \to \gamma^\mu extends to a unique algebra
homomorphism
Cℓ(
V,
g)\to
Mat(2
^{k},
C) (i.e. fix a
convention for the
gamma matrices).
The
C^{2k} on which the gamma
matrices act is then 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 by 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
WeylBrauer 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 confusionover 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 2
^{k} complex numbers and is denoted with spinor
indices (usually α, β γ). 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 leftmultiplication: c :
xω →
cxω. There are two variations on this theme:
one can either find a primitive element ω which is a
nilpotent element of the Clifford algebra, or one
which 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 which can specify a basis of the
spin space, even should a preferred basis of
V already be
given.
As above, we let (
V,
g) be an
ndimensional 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 subspace of
V such that
g
_{W}=0, (i.e.,
W is a maximal
isotropic subspace). If
n = 2
k is even, then let
W′ be an isotropic subspace complementary to
W.
If
n = 2
k+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. Consequently, the
kfold
product of
W with itself,
W^{k}, is
onedimensional. Let ω be a generator of
W^{k}. In
terms of a basis of
W,
w_{1},...,
w_{k}, one possibility is to set
 \omega=w_1w_2\cdots w_k.
Note that ω
^{2} = 0 (i.e., ω is nilpotent of order 2), and
moreover,
wω = 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_{+} =
Cℓ^{even}ω and Δ_{} =
Cℓ^{odd}ω 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 (both denoted by Δ),
corresponding to the respective eigenvalues +1 and 1.
In detail, suppose for instance that
n is even. Suppose
that
I is a nonzero left ideal contained in
Cℓ(
V,
g)ω. We shall show that
I
must in fact be equal to
Cℓ(
V,
g)ω by
proving that it contains a nonzero scalar multiple of ω.
Fix a basis
w_{i} of
W and a
complementary basis
w_{i}′ of
W′ so that
 w_{i}w_{j}′
+w_{j}′ w_{i} = δ_{ij},
and
 (w_{i}′)^{2} = 0.
Note that any element of
I must have the form αω, by
virtue of our supposition that
I ⊂
Cℓ(
V,
g)ω. Let αω ∈
I be any
such element. Using the chosen basis, we may write
 \alpha = \sum_{i_1<="">math>
where the
a_{i1...ip} are
scalars, and the
B_{j} are auxiliary elements of
the Clifford algebra. Pick any monomial
a in this
expansion of α having maximal homogeneous degree among the elements
w_{i}′:
 a = a_{i_1\dots i_p}w_{i_1}^\prime\dots w_{i_p}^\prime (no
summation implied)
Observe now that the product
 w_{i_p}\cdots w_{i_1}\alpha\omega = a_{i_1\dots i_p}\omega
is a nonzero scalar multiple of ω, as required.
Exterior algebra construction
Let \Delta= \wedge^\cdot W = \oplus_j \wedge^j W denote the
exterior algebra of
W
considered as vector space. 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
 c(v) w_1 \wedge\cdots\wedge w_n = (\epsilon(w) +
i(w'))\left(w_1 \wedge\cdots\wedge w_n\right)
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 is easily 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 (the
halfspin representations, or Weyl spinors) via
 \Delta_+ = \wedge^{even} W,\, \Delta_ = \wedge^{odd} W.
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 (multiples of)
u is defined by
 c(u) \alpha = \left\{\begin{matrix}
\alpha&\hbox{if } \alpha\in \wedge^{even}
W\\\alpha&\hbox{if } \alpha\in \wedge^{odd}
W\end{matrix}\right.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 which provides a
decomposition of its complexification into two maximal isotropic
subspaces, then the definition of spinors (by either method)
becomes natural.
The main example is the case that the real vector space
V
is a
hermitian vector space
(
V,
h), i.e.,
V is equipped with a
complex structure J which
is an
orthogonal
transformation with respect to the inner product
g on
V. Then V \otimes_{\mathbb{R}} \mathbb{C} splits in the
\pm i eigenspaces of
J. These eigenspaces are isotropic
for the complexification of
g and can be identified with
the complex vector space (
V,
J) and its complex
conjugate (
V, 
J). Therefore for a hermitian
vector space (V, h) the vector space \wedge^\cdot_{\mathbb{C}} V 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 (in
particular every
symplectic
manifold) has a
SpinC structure. Likewise,
every complex vector bundle on a manifold carries a SpinC
structure.
ClebschGordan decomposition
A number of
ClebschGordan
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
 Γ_{r} = ∧^{r}V, the representation of
the orthogonal group on skew tensors of rank r.
In addition, for the real orthogonal groups, there are three
characters (onedimensional
representations)
 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. (The spatial character.)
 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. (The temporal character.)
 σ = σ_{+}σ_{}. (The orientation
character.)
The ClebschGordan 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 = 2
k is even, then the tensor product of Δ
with the
contragredient
representation decomposes as
 \Delta\otimes\Delta^* \cong \bigoplus_{p=0}^n \Gamma_p \cong
\bigoplus_{p=0}^{k1} \left(\Gamma_p\oplus\sigma\Gamma_p\right)\,
\oplus \Gamma_k
which can be seen explicitly by considering (in the Explicit
construction) 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
Γ
_{p} ⊕ σΓ
_{p} 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:
 (\alpha\omega)^*=\omega(\alpha^*).
So Δ⊗Δ also decomposes in the above manner. Furthermore, under the
even Clifford algebra, the halfspin representations decompose
 \begin{matrix}
\Delta_+\otimes\Delta^*_+ \cong \Delta_\otimes\Delta^*_
&\cong& \bigoplus_{p=0}^k
\Gamma_{2p}\\\Delta_+\otimes\Delta^*_ \cong
\Delta_\otimes\Delta^*_+ &\cong& \bigoplus_{p=0}^{k1}
\Gamma_{2p+1}\end{matrix}
For the complex representations of the real Clifford algebras, the
associated
reality structure on
the complex Clifford algebra descends to the space of spinors (via
the explicit construction in terms of minimal ideals, for
instance). In this way, we obtain the complex conjugate
\bar{\Delta} of the representation Δ, and the following isomorphism
is seen to hold:
 \bar{\Delta} \cong \sigma_\Delta^*
In particular, note that the representation Δ of the orthochronous
spin group is a
unitary
representation. In general, there are ClebschGordan
decompositions
 \Delta\otimes\bar{\Delta} \cong
\bigoplus_{p=0}^k\left(\sigma_\Gamma_p\oplus
\sigma_+\Gamma_p\right).
In metric signature (
p,
q), the following
isomorphisms hold for the conjugate halfspin representations
 If q is even, then \bar{\Delta}_+ \cong
\sigma_\otimes \Delta_+^* and \bar{\Delta}_ \cong \sigma_\otimes
\Delta_^*.
 If q is odd, then \bar{\Delta}_+ \cong \sigma_\otimes
\Delta_^* and \bar{\Delta}_ \cong \sigma_\otimes
\Delta_+^*.
Using these isomorphisms, one can deduce analogous decompositions
for the tensor products of the halfspin representations
\Delta_\pm\otimes\bar{\Delta}_\pm.
Odd dimensions
If
n = 2
k+1 is odd, then
 \Delta\otimes\Delta^* \cong \bigoplus_{p=0}^k \Gamma_{2p}.
In the real case, once again the isomorphism holds
 \bar{\Delta} \cong \sigma_\Delta^*.
Hence there is a ClebschGordan decomposition (again using the
Hodge star to dualize) given by
 \Delta\otimes\bar{\Delta} \cong
\sigma_\Gamma_0\oplus\sigma_+\Gamma_1\oplus\dots\oplus\sigma_\pm\Gamma_k
Consequences
There are many farreaching consequences of the ClebschGordan
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 \bar{\phi}\psi
as a scalar. In physical terms, a spinor should determine a
probability amplitude for the
quantum state.
 A manner of regarding the product \bar{\phi}\psi 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 \psi v\bar{\psi};. In physical terms, this is
represents an electrical current
of Maxwell's electromagnetic
theory, or more generally a probability current.
Summary in low dimensions
 In 1 dimension (a trivial example), the single spinor
representation is formally Majorana, a real 1dimensional representation that
does not transform.
 In 2 Euclidean dimensions, the lefthanded and the righthanded
Weyl spinor are 1component 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
2dimensional and quaternionic. The existence of
spinors in 3 dimensions follows from the isomorphism of the
group SU(2) \cong
\mathit{Spin}(3) which allows us to define the action of Spin(3) on
a complex 2component column (a spinor); the generators of SU(2)
can be written as Pauli
matrices.
 In 4 Euclidean dimensions, the corresponding isomorphism is
\mathit{Spin}(4) \equiv SU(2) \times SU(2). There are two
inequivalent quaternionic 2component Weyl
spinors and each of them transforms under one of the SU(2) factors
only.
 In 5 Euclidean dimensions, the relevant isomorphism is
\mathit{Spin}(5)\equiv USp(4)\equiv Sp(2) which implies that the
single spinor representation is 4dimensional and
quaternionic.
 In 6 Euclidean dimensions, the isomorphism
\mathit{Spin}(6)\equiv SU(4) guarantees that there are two
4dimensional complex Weyl representations that are complex
conjugates of one another.
 In 7 Euclidean dimensions, the single spinor representation is
8dimensional 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 WeylMajorana real
8dimensional representations that are related to the 8dimensional
real vector representation by a special property of Spin 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 timelike directions, the
dimensions viewed as dimensions over the complex numbers coincide
with the case of the p+qdimensional Euclidean space, but the
reality projections mimic the structure in pq Euclidean
dimensions. For example, in 3+1 dimensions there are two
nonequivalent Weyl complex (like in 2 dimensions) 2component
(like in 4 dimensions) spinors, which follows from the isomorphism
\mathit{SL}(2,C) \equiv \mathit{Spin}(3,1).
Metric signature 
lefthanded Weyl 
righthanded Weyl 
conjugacy 
Dirac 
lefthanded MajoranaWeyl 
righthanded MajoranaWeyl 
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 
See also
Notes
 .
 , .
 , .
 , .
 , . These two books also provide good mathematical
introductions and fairly comprehensive bibliographies on the
mathematical applications of spinors as of 1989–1990.
 Named after William Kingdon Clifford,
 Named after Paul
Dirac.
 Named after Hermann Weyl.
 Named after Ettore Majorana.
 .
 .
 This construction is due to Cartan. The treatment here is based
on .
 One source for this subsection is .
 Via the evengraded Clifford algebra.
 .
 .
References