Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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The real case In view of proposition (I) and
C‘
1, 1
= R(2), the algebra C‘
k, l
is of the same type as
C‘
kl,0
if k > l and of the same type as C‘
0, lk
if k < l.SinceC‘
k, l
^
C‘
l, k
= C‘
kþl, kþl
,thetype
of C‘
l, k
is the inverse of the type of C‘
k, l
. The algebra
C‘
0
4, 0
!C‘
4, 0
is isomorphic to H H ! H(2): if
x = (x
1
, x
2
, x
3
, x
4
) 2 R
4
C‘
4, 0
,andq = ix
1
þ jx
2
þ
kx
3
þ x
4
2 H, then an isomorphism is obtained from
the Clifford map f,
f ðxÞ=
0 q
q 0
!
½16
In view of [13], the volume element satisfies
2
= 1.
By replacing
q with
q in [16], one shows that C‘
0, 4
is also isomorphic to H(2). The map R
4
R
kþl
!
H(2) C‘
k, l
given by (x, y) 7!f (x) 1 þ y has
the Clifford property and establishes the isomorphism
of algebras C‘
kþ4, l
= H C‘
k, l
. Since, similarly,
C‘
k, lþ4
= H C‘
k, l
, one obtains the isomorphism
C‘
kþ4;l
= C‘
k;lþ4
Therefore,
C‘
kþ8;l
= C‘
kþ4;lþ4
= C‘
k;lþ8
= C‘
k;l
Rð16Þ
and the algebras C‘
k, l
, C‘
kþ8, l
,andC‘
k, lþ8
are all of the
same type. This double periodicity of period 8 is
subsumed by saying that real Clifford algebras can be
arranged on a ‘‘spinorial chessboard.’’ The type of
C‘
0
k, l
!C‘
k, l
depends only on k l mod 8; the eight
types have the following low-dimensional algebras as
representatives: C‘
1, 0
, C‘
2, 0
, C‘
3, 0
, C‘
4, 0
= C‘
0, 4
, C‘
0, 3
,
C‘
0, 2
,andC‘
0, 1
. The Brauer–Wall group of R is Z
8
,
generated by the type of C‘
0
1, 0
!C‘
1, 0
, that is, by R !
C. Bearing in mind the isomorphism C‘
k, l
= C‘
0
kþ1, l
and abbreviating C ! R(2) to C ! R,etc.,onecan
arrange the types of real Clifford algebras in the form
of a ‘‘spinorial clock’’:
R !
7
R R !
0
R
6 "#1
CC
5 "#2
H
4
H H
3
H
½17
Proposition (K) Recipe for determining C‘
0
k, l
!
C‘
k, l
:
(i) find the integers and such that
k l = 8 þ and 0 v 7;
(ii) from the spinorial clock, read off A
0
v
!vA
v
and
compute the real dimensions, dim A
0
v
= 2
0
and
dim A
v
= 2
; and
(iii) form C‘
0
k, l
= A
0
v
(2
(1=2)(kþl1
0
)
) and C‘
k, l
=
A
v
(2
(1=2)(kþl)
).
The spinorial clock is symmetric with respect to
the reflection in the vertical line through its center;
this is a consequence of the isomorphism of algebras
C‘
k, lþ2
= C‘
l, k
R(2).
Note that the ‘‘abstract’’ algebra C‘
k, l
carries, in
general, less information than the Clifford algebra
defined in [8], which contains V as a distinguished
vector subspace with the quadratic form
v 7!v
2
= g(v, v). For example, the algebras C‘
8, 0
,
C‘
4, 4
,andC‘
0, 8
are all graded isomorphic.
Theorem on Simplicity
From general theory (Chevalley 1954) or by inspec-
tion of [14], [15], and [17], one has
Proposition (L) Let m be the dimension of the
orthogonal space (V, g) over K.
(i) If m is even (resp., odd), then the algebra
C‘(V, g) (resp., C‘
0
(V, g)) over K is central simple.
(ii) If K = C and m is odd (resp., even), then the
algebra C‘(V, g) (resp., C‘
0
(V, g))isthedirect
sum of two isomorphic complex central simple
algebras.
(iii) If K = R and m is odd (resp., even), then the
algebra C‘(V, g) (resp., C‘
0
(V, g)) when
2
= 1 is
the direct sum of two isomorphic central simple
algebras and when
2
= 1 is simple with a
center isomorphic to C.
Representations
The Pauli, Cartan, Dirac, and Weyl
Representations
Odd dimensions Let (V, g) be of dimension
m = 2n þ1 over K. From propositions (A) and (L) it
follows that the central simple algebra C‘
0
(V, g) has a
unique, up to equivalence, faithful, and irreducible
representation in the complex 2
n
-dimensional vector
space S of Pauli spinors. By putting () = I it is
extended to a Pauli representation : C‘(V, g) !
End S. Given an orthonormal frame (e
)inV, Pauli
endomorphisms (matrices if S is identified with C
2
n
)
are defined as
= (e
) 2 End S. The representations
and are complex inequivalent. For K = C
none of them is faithful; their direct sum is the faithful
Cartan representation of C‘(V, g)inS S. For K = R
and (1=2)(k l 1) even, the representations and
are real equivalent and faithful. On computing
() one finds that the contragredient representation
ˇ
is equivalent to for n even and to for n odd.
Even dimensions Similarly, for (V, g) of dimension
m = 2n over K, the central simple algebra C‘(V, g)
has a unique, up to equivalence, faithful, and
Clifford Algebras and Their Representations 525
irreducible representation : C‘(V, g) ! End S in the
2
n
-dimensional complex vector space S of Dirac
spinors. The Dirac endomorphisms (matrices) are
= (e
). Put ¼ ()sothat
2
= I:thematrix
generalizes the familiar
5
. The Dirac representation
restricted to C‘
0
(V, g) decomposes into the sum
þ
of two irreducible representations in the vector spaces
S
= fs 2 S js = sg
of Weyl (chiral) spinors. The elements of S
þ
are said
to be of opposite chirality with respect to those of
S
. The transpose
defines a similar split of S
.
The representations
þ
and
are never complex-
equivalent, but they are real equivalent and
faithful for K = R and (1=2)(k l) odd.
The representations and
ˇ
are both equiva-
lent to . It is convenient to describe simultaneously
the properties of the transpositions of the Pauli and
Dirac matrices; let
be either the Pauli matrices
for V of dimension 2n þ 1 or the Dirac matrices for
V of dimension 2n. There is a complex isomorphism
B: S ! S
such that
= ð1Þ
n
B
B
1
½18
In the case of the Dirac matrices, the factor (1)
n
in
[18] implies that this equation also holds for in
place of
. The isomorphism B preserves (resp.,
changes) the chirality of Weyl spinors for n even
(resp., odd). Every matrix of the form B
1
...
p
,
where
14
1
< <
p
2n ½19
is either symmetric or antisymmetric, depending on
p and the symmetry of B. A simple argument, based
on counting the number of such products of one
symmetry, leads to the equation
B
= ð1Þ
ð1=2Þnðnþ1Þ
B
valid in dimensions 2n and 2n þ 1.
Inner products on spinor spaces Let S be the
complex vector space of Dirac or Pauli spinors
associated with (V, g) over K. The isomorphism B:
S ! S defines on S an inner product
B(s, t) = hs, B(t)i, s, t 2 S, which is orthogonal for
m 0, 1, 6, or 7 mod 8 and symplectic for m
2, 3, 4, or 5 mod 8. For m 0 mod 4, this product
restricts to an inner product on the space of Weyl
spinors that is orthogonal for m 0 mod 8 and
symplectic for m 4 mod 8. For m 2 mod 4, the
map B defines the isomorphisms B
:S
! S
.
Example One of the most used representations :
C‘
3, 1
! C(4) is given by the Dirac matrices
1
=
0
x
x
0
!
;
2
=
0
y
y
0
!
3
=
0
z
z
0
!
;
4
=
0 I
I 0
!
½20
Change Conjugation and Majorana Spinors
Throughout this section and next, one assumes
K = R so that, given a representation : C‘(V, g) !
End S,one can form the complex- (‘‘charge’’) conjugate
representation
: C‘(V, g) ! End
S defined by
(a) =
(a) and the Hermitian conjugate representa-
tion
y
:C‘(V, g) ! End
S
, where
y
(a) =
(a).
Even dimensions The representations
and are
equivalent: there is an isomorphism C: S !
S such
that
= C
C
1
½21
The automorphism
CC is in the commutant of ;it
is, therefore, proportional to I and, by a change of
scale, one can achieve
CC = I for k l 0or
6mod8 and
CC ¼I for k l 2 or 4 mod 8.
The spinor s
c
¼ C
1
s 2 S isthechargeconjugateof
s 2 S.If :V ! S is a solution of the Dirac equation
ð
ð@
iqA
ÞÞ = 0
for a particle of electric charge q, then
c
is a
solution of the same equation with the opposite
charge. Since
=
2
CC
1
charge conjugation preserves (resp., changes) the
chirality of Weyl spinors for (1=2)(k l) even (resp.,
odd).
If
CC = I, then
Re S = fs 2 S js
c
= sg
is a real vector space of dimension 2
n
, the space of
Dirac–Majorana spinors. The representation is
real: restricted to Re S and expressed with respect to
a frame in this space, it is given by real 2
n
2
n
matrices. For k l 0 mod 8 the representations
þ
and
are both real: in this case there are
Weyl–Majorana spinors.
Odd dimensions On computing
() one finds that
the conjugate representation is equivalent to
526 Clifford Algebras and Their Representations
(resp., )if
2
= 1 (resp.,
2
= 1). There is an
isomorphism C:S !
S such that
= ð1Þ
ð1=2Þðklþ1Þ
C
C
1
½22
and
CC = I (resp.,
CC = I)fork l 1or7mod8
(resp., k l 3 or 5 mod 8). For k l 1 mod 8, the
restriction of the Pauli representation to C‘
0
k, l
is real
and the Pauli matrices are pure imaginary; for k l
7 mod 8, the Pauli representations of C‘
k, l
are both real
and so are the Pauli matrices. In both these cases there
are Pauli–Majorana spinors.
Hermitian Scalar Products and Multivectors
For m = k þ l odd and C as in [22], the map
A =
BC: S !
S
intertwines the representations
y
and (resp., ) for k even (resp., odd),
y
= ð1Þ
k
A
A
1
By rescaling of B, the map A can be made
Hermitian. The corresponding Hermitian form
s 7!A(s, s) is definite if and only if k or l = 0;
otherwise, it is neutral.
For m = k þ l even, the representations
y
and
are equivalent and one can define a Hermitian
isomorphism A:S !
S
so that
y
= A
A
1
½23
The isomorphism A
0
= A intertwines the represen-
tations
y
and ; it can also be made Hermitian
by rescaling. The Hermitian form A(s, s) is definite
for k = 0 and A
0
(s, s) is definite for l = 0; otherwise,
these forms are neutral. For example, in the familiar
representation [20], one has A =
4
, a neutral form.
For p = 0, 1, ..., m = 2n, two spinors s and t 2 S
define the p-vector with components
A
1
...
p
ðs; tÞ= hs; A
1
...
p
ti½24
where the indices are as in [19]. The Hermiticity of
A and [23] imply
A
1
...
p
ðs; tÞ= ð1Þ
ð1=2Þpðp1Þ
A
1
...
p
ðt; sÞ
In view of
y
= (1)
k
AA
1
, the map A defines,
for k even, a nondegenerate Hermitian scalar
product on the spaces S
whereas A(s, t) = 0ifs
and t are Weyl spinors of opposite chiralities. For k
odd, A changes the chirality.
The Radon–Hurwitz Numbers
Proposition (M) For every integer m > 0, the
algebra C‘
m,0
has an irreducible real representation
of dimension 2
(m)
, where (m) is the mth Radon–
Hurwitz number given by
m = 12345678
ðmÞ= 12233334
and (m þ 8) = (m) þ 4. The matrices
2 R(2
(m)
),
= 1, ..., m, defining these representations satisfy
v
þ
v
= 2
v
I
and can be chosen so as to be antisymmetric. In all
dimensions other than m 3 mod 4 the representa-
tions are faithful.
For m 2 and 4mod8 (resp., m 1, 3, and
5mod8)the representations are the realifications of
the corresponding Dirac (resp., Pauli) representations.
In dimensions m 0 and 6mod8 (resp.,
m 7mod8) the Dirac (resp., Pauli) representations
themselves are real.
Inductive Construction
of Representations
An inductive construction of the Pauli
representations
: C‘
n1;n
! Rð2
n1
Þ; n = 1; 2; ...
and of the Dirac representations
: C‘
n;n
! Rð2
n
Þ; n = 1; 2; ...
is as follows.
1. In dimension 1, put
1
= 1.
2. Given
2 R(2
n1
), = 1, ...,2n 1, define
=
0
0
!
for = 1; ...; 2n 1
and
2n
=
0 I
I 0
!
3. Given
2 R(2
n
), = 1, ...,2n, define
=
for = 1, ...,2n, and
2nþ1
=
1
2n
.
All entries of these matrices are either 0, 1, or 1;
therefore, they can be used to construct representa-
tions of Clifford algebras of orthogonal spaces over
any commutative field of characteristic 6¼ 2.
By induction, one has
= (1)
þ1
. Therefore,
the isomorphisms appearing in [18] are
B =
2
4
2n
for both m = 2n and 2n þ 1.
By multiplying some of the matrices
or
by the
imaginary unit, one obtains complex representations
of the Clifford algebras associated with the quadratic
Clifford Algebras and Their Representations 527
forms of other signatures. For example, in dimension
3, (
1
,i
2
,
3
) are the Pauli matrices. In dimension 4,
multiplying
2
by i one obtains the Dirac matrices for g
of signature (1, 3), in the ‘‘chiral representation’’:
1
¼
0
x
x
0
;
2
¼
0
y
y
0
3
¼
0
z
z
0
;
4
¼
0 I
I 0
½25
To obtain the real Majorana representation one uses
the following fact:
Proposition (N) If the matrix C 2 R(2
n
) is such
that C
2
= I and [21] holds, then the matrices
(I þ iC)
(I þ iC)
1
, = 1, ...,2n, {\it are real}.
For the matrices [25], one can take C =
1
3
4
to
obtain
0
1
=
0
x
x
0
!
;
0
2
=
I 0
0 I
!
0
3
=
0
z
z
0
!
;
0
4
=
0 I
I 0
!
The real representations described in proposition
(M) can be obtained by the following direct inductive
construction. Consider the following seven real anti-
symmetric and anticommuting 8 8 matrices:
1
¼
z
I ";
2
¼
z
"
x
3
¼
z
"
z
;
4
¼
x
" I
5
¼
x
x
";
6
¼
x
z
"
7
¼ " I I
½26
For m = 4, 5, 6, and 7 the matrices
1
, ...,
m
gener-
ate the representations of C‘
m,0
in R
8
. The eight
matrices
=
x
, = 1, ..., 7, and
8
= " I
I I give the required representation of C‘
8, 0
in
R
16
. By dropping the first factor in
1
,
2
,
3
, one
obtains the matrices generating a representation of
C‘
3, 0
in R
4
, etc. The symmetric matrix
=
1
8
=
z
I I I anticommutes with all
the s and
2
= I. If the matrices
2 R(2
(m)
)
correspond to a representation of C‘
m,0
, then the
m þ 8 matrices
1
, ...,
m
,
1
I, ...,
8
I
generate the required representation of C‘
mþ8, 0
.
Vector Fields on Spheres
and Division Algebras
It is known that even-dimensional spheres have no
nowhere-vanishing tangent vector fields. All such
fields on odd-dimensional spheres can be constructed
with the help of the representation described in
proposition (M). Given a positive even integer N,let
m be the largest integer such that N = 2
(m)
p, where
p is an odd integer. Consider the unit sphere
S
N1
= fx 2 R
N
jjjxjj= 1g of dimension N 1. For
v 2 R
m
,put
0
(v) = (v) I,whereI 2 R(p)isthe
unit matrix. Since (v)isantisymmetric,soisthe
matrix
0
(v) 2 R(N). Therefore, for every x 2 S
N1
,
the vector
0
(v)x is orthogonal to x. The map
x 7!
0
(v)x defines a vector field on S
N1
that
vanishes nowhere unless v = 0 : the (N1)-sphere
admits a set of m tangent vector fields which are
linearly independent at every point. Using methods of
algebraic topology, it has been shown that this
method gives the maximum number of linearly
independent tangent vector fields on spheres.
If m = 1, 3, or 7, then m þ 1 = 2
(m)
and, for these
values of m, the sphere S
m
is parallelizable. More-
over, one can then introduce in R
mþ1
the structure
of an algebra A
m
as follows. Put
0
= I.Ife
0
2 R
mþ1
is a unit vector and e
=
(e
0
), then (e
0
, e
1
, ..., e
m
)
is an orthonormal frame in R
mþ1
. The product of
x =
P
m
= 0
x
e
and y =
P
m
= 0
y
e
is defined to be
x y =
X
m
;v = 0
x
y
v
ðe
v
Þ
so that e
0
is the unit element for this product.
Defining Rex=x
0
e
0
,Imx=x Rex,
x=Rex Imx,
one has
x x=e
0
jjxjj
2
and
x (x y)=(
x x) y, so that
x y=0 implies x=0ory=0: A
m
is a normed
algebra without zero divisors. The algebras A
1
and
A
3
are isomorphic to C and H, respectively, and A
7
is, by definition, the algebra O of octonions
discovered by Graves and Cayley. The algebra O is
nonassociative; its multiplication table is obtained
with the help of [26].
Spinor Groups
Let (V, g) be a quadratic space over K.Ifu 2 V is
not null, then it is invertible as an element of
C‘(V, g) and the map v 7!uvu
1
is a reflection in
the hyperplane orthogonal to u. The orthogonal
group O(V, g) = O(V, g) = fR 2 GL(V) j R
g
R = gg is generated by the set of all such reflections.
A spinor group G is a subset of C‘(V, g) that is a
group with respect to multiplication induced by the
product in the algebra, with a homomorphism
: G ! GL(V) whose image contains the connected
component SO
0
(V, g) of the group of rotations of
(V, g). In the case of real quadratic spaces, one
considers also spinor groups that are subsets of C
C‘(V, g) with similar properties. By restriction, every
528 Clifford Algebras and Their Representations
representation of C‘(V, g)orC C‘(V, g)gives
spinor representations of the spinor groups it
contains.
Pin Groups
It is convenient to define a unit vector v 2 V
C‘(V, g)tobesuchthatv
2
= 1forV complex and
v
2
= 1or1forV real. The group Pin(V, g)is
defined as the subgroup of Cpin(V, g) consisting of
products of all finite sequences of unit vectors.
Defining now the twisted adjoint representation
f
Ad
by
f
Ad(a)v = (a)va
1
, one ontains the exact sequence
1 ! Z
2
! PinðV; gÞ!
e
Ad
OðV; gÞ!1 ½27
If dimV is even, then the adjoint representation
Ad(a)v = ava
1
also yields an exact sequence like
[27]; if it is odd, then the image of Ad is SO(V, g) and
the kernel is the four-element group f1, 1, , g.
Given an orthonormal frame (e
)in(V, g)and
a 2 Pin(V, g), one defines the orthogonal matrix
R(a) = (R
v
(a)) by
f
AdðaÞe
= e
v
R
v
ðaÞ½28
If (V, g) is complex, then the algebras C‘(V, g) and
C‘(V, g) are isomorphic; this induces an iso-
morphism of the groups Pin(V, g) and Pin(V, g).
If V = C
m
, then this group is denoted by Pin
m
(C). If
V = R
kþl
and g of signature (k, l), then one writes
Pin(V, g) = Pin
k, l
. A similar notation is used for the
groups spin, see below.
Spin Groups
The spin group Spin(V, g) = Pin(V, g) \ C‘
0
(V, g)is
generated by products of all sequences of an even
number of unit vectors. Since the algebras C‘
0
(V, g)
and C‘
0
(V, g) are isomorphic, so are the groups
Spin(V, g) and Spin(V, g). Since (a) = a for a 2
Spin(V, g), the twisted adjoint representation
reduces to the adjoint representation and yields the
exact sequence
1 ! Z
2
! SpinðV; gÞ!
Ad
SOðV; gÞ!1 ½29
For V = C
m
, the spin group is denoted by Spin
m
(C).
Since Spin
m
(C) G
1
(), the bilinear form B is
invariant with respect to the action of this group.
Spin
0
Groups
The connected component Spin
0
(V, g) of the group
Spin(V, g) coincides with Spin(V, g) if either the
quadratic space (V, g) is complex or real and kl = 0.
In signature (k, l), the connect group Spin
0
k, l
is
generated in C‘
0
k, l
by all products of the form
u
1
...u
2
p
v
1
...v
2
q
such that u
2
i
= 1andv
2
j
= 1.
The connected groups Spin
m:0
and Spin
0, m
are
isomorphic and denoted by Spin
m
. Since Spin
0
k, l
G
1
(), the Hermitian form A and the bilinear form
B are invariant with respect to the action of this
group. Moreover, for k þ l even, from [24] and
[28] there follows the transformation law of
multivectors formed from pairs of spinors,
A
1
p
ððaÞs;ðaÞtÞ
= A
v
1
...v
p
ðs; tÞR
v
1
1
ða
1
Þ...R
v
p
p
ða
1
Þ
Consider Spin
0
(V, g) and assume that either V is
complex of dimension 52 or real with k or l 5 2.
Then there are two unit orthogonal vectors
e
1
, e
2
2 V such that (e
1
, e
2
)
2
= 1. The vector
u(t) = e
1
cos t þ e
2
sin t is obtained from e
1
by rotation
in the plane span fe
1
, e
2
g by the angle t 2 R. The
curve t 7!e
1
u(t), 0 t , connects the elements
1and1ofSpin
0
(V, g). Its image in SO
0
(V, g), that
is, the curve t 7!Ad(e
1
u(t)), 0 t ,isclosed:
Ad(1) = Ad(1). This fact is often expressed by
saying that ‘‘a spinor undergoing a rotation by 2
changes sign.’’ There is no homomorphism – not
even a continuous map – f :SO
0
(V, g) ! Spin
0
(V, g)
such that Ad f = id.
Spin
c
Groups
For the purposes of physics, to describe charged
fermions, and in the theory of the Seiberg–Witten
invariants, one needs the Spin
c
groups that are spinorial
extensions of the real orthogonal groups by the group U
1
of ‘‘phase factors.’’ Assume V to be real and g of
signature (k, l) so that the sequence [29] can be
written as
1 ! Z
2
! Spin
k;l
! SO
k;l
! 1
Define the action of Z
2
= f1, 1g in Spin
k, l
U
1
so
that (1)(a, z) = ( a, z). The quotient (Spin
k, l
U
1
)=Z
2
= Spin
c
k, l
yields the extensions
1 ! U
1
! Spin
c
k;l
! SO
k;l
! 1
and
1 ! Spin
k;l
! Spin
c
k;l
! U
1
! 1
For example, Spin
3
= SU
2
and Spin
c
3
= U
2
.
Spin Groups in Dimensions <6
The connected components of spin groups asso-
ciated with orthogonal spaces of dimension 46 are
isomorphic to classical groups. They can be expli-
citly described starting from the following
observations.
Clifford Algebras and Their Representations 529
Consider the four-dimensional vector space
(of twistors) T over K, with a volume element
vol 2^
4
T. The six-dimensional vector space
V = ^
2
T has a scalar product g defined by
g(u, v)vol = 2u ^ v for u, v 2 V. The quadratic form
g(u, u) is the Pfaffian, Pf(u). If u 2 V is represented
by the corresponding isomorphism T
! T and a 2
End T,thenPf(aua
) = det aPf(u). The last for-
mula shows Spin
0
(V, g) = SL(T), so that Spin
6
(C) =
SL
4
(C). For K = R, the Pfaffian is of signature (3, 3), so
that Spin
0
3, 3
= SL
4
(R). A non-null vector v 2 V defines
a symplectic form on T
. The five-dimensional vector
space v
?
V is invariant with respect to the symplec-
tic group Sp(T
, u) = Spin
0
(v
?
,Pfjv
?
). This shows that
Spin
5
(C) = Sp
4
(C) and Spin
0
2, 3
= Sp
4
(R). Spin groups
for other signatures in real dimensions 6 and 5 are
obtained by considering appropriate real subspaces of
C
6
and C
5
, respectively. For example, [6] is used to
show that Spin
0
1, 5
= SL
2
(H).
Spin groups in dimensions 4 and lower are
similarly obtained from the observation that det is
a quadratic form on the four-dimensional space K(2)
and C‘
0
(K(2), det) = K(2) K(2).
Several spin groups are listed below.
The complex spin groups
Spin
2
ðCÞ = C
; Spin
3
ðCÞ= SL
2
ðCÞ
Spin
4
ðCÞ = SL
2
ðCÞSL
2
ðCÞ
Spin
5
ðCÞ = Sp
4
ðCÞ
Spin
6
ðCÞ = SL
4
ðCÞ
The real, compact spin groups
Spin
2
= U
1
; Spin
3
= SU
2
Spin
4
= SU
2
SU
2
; Spin
5
= Sp
2
ðHÞ
Spin
6
= SU
4
The groups Spin
0
k, l
for 1 4 k 4 l and k þ l 6
Spin
0
1;1
= R
; Spin
0
1;2
= SL
2
ðRÞ
Spin
0
1;3
= SL
2
ðCÞ
Spin
0
2;2
= SL
2
ðRÞSL
2
ðRÞ
Spin
0
1;4
= Sp
1;1
ðHÞ
Spin
0
2;3
= Sp
4
ðRÞ; Spin
0
1;5
= SL
2
ðHÞ
Spin
0
2;4
= SU
2;2
Spin
0
3;3
= SL
4
ðRÞ
See also: Dirac Operator and Dirac Field; Index
Theorems; Relativistic Wave Equations Including Higher
Spin Fields; Spinors and Spin Coefficients; Twistors.
Further Reading
Adams JF (1981) Spin (8), triality, F
4
and all that. In: Hawking
SW and Roc
ˇ
ek M (eds.) Superspace and Supergravity.
Cambridge: Cambridge University Press.
Atiyah MF, Bott R, and Shapiro A (1964) Clifford modules.
Topology 3(suppl. 1): 3–38.
Baez JC (2002) The octonions. Bulletin of the American
Mathematical Society 39: 145–205.
Brauer R and Weyl H (1935) Spinors in n dimensions. American
Journal of Mathematics 57: 425–449.
Budinich P and Trautman A (1988) The Spinorial Chessboard,-
Trieste Notes in Physics. Berlin: Springer.
Cartan E
´
(1938) The´orie des spineurs. Actualite´s Scientifiques et
Industrielles, No. 643 et 701. Paris: Hermann (English
transl.:The Theory of Spinors. Paris: Hermann, 1966).
Chevalley C (1954) The Algebraic Theory of Spinors. New York:
Columbia University Press.
Clifford WK (1878) Applications of Grassmann’s extensive
algebra. American Journal of Mathematics 1: 350–358.
Clifford WK (1882) On the classification of geometric algebras.
In: Tucker R (ed.) Mathematical Papers by William Kingdon
Clifford, pp. 397–401. London: Macmillan.
Dirac PAM (1928) The quantum theory of the electron.
Proceedings of the Royal Society of London A 117: 610–624.
Eckmann B (1942) Gruppentheoretische Beweis des Satzes von
Hurwitz–Radon u¨ ber die Komposition quadratischer Formen.
Commentarii Mathematici Helvetici 15: 358–366.
Karoubi M (1968) Alge` bres de Clifford et K-the´orie. Annales
Scientifiques de l’E
´
cole Normale Superieure 4e`me se´r 1: 161–270.
Lipschitz RO (1886) Untersuchungen u¨ber die Summen von
Quadraten. Berlin: Max Cohen und Sohn.
Lounesto P (2001) Clifford Algebras and Spinors, 2nd edn.
London Math. Soc. Lecture Note Series, vol. 286. Cambridge:
Cambridge University Press.
Pauli W (1927) Zur Quantenmechanik des magnetischen
Elektrons. Z. Physik 43: 601–623.
Penrose R and MacCallum MAH (1973) Twistor theory: an
approach to the quantisation of fields and space-time. Physics
Report 6C(4): 241–316.
Porteous IR (1995) Clifford Algebras and the Classical Groups,
Cambridge Studies in Advanced Mathematics, vol. 50. Cam-
bridge: Cambridge University Press.
Postnikov MM (1986) Lie groups and Lie algebras. Mir: Moscow.
Sudbery A (1987) Division algebras (pseudo)orthogonal groups
and spinors. Journal of Physics A17: 939–955.
Trautman A (1997) Clifford and the ‘‘square root’’ ideas.
Contemporary Mathematics
203: 3–24.
Trautman A and Trautman K (1994) Generalized pure spinors.
Journal of Geometry and Physics 15: 1–22.
Wall CTC (1963) Graded Brauer groups. Journal fu¨r die Reine
und Angewandte Mathematik 213: 187–199.
530 Clifford Algebras and Their Representations
Cluster Expansion
R Kotecky´ , Charles University, Prague,
Czech Republic, and the University of Warwick, UK
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The method of cluster expansions in statistical
physics provides a systematic way of computing
power series for thermodynami c potentials (loga-
rithms of partition funtions) as well as correlat ions.
It originated from the works of Mayer and others
devoted to expansions for dilute gas.
Mayer Expansion
Consider a system of interacting particles with
Hamiltonian
H
N
ðp
1
; ...; p
N
; r
1
; ...; r
N
Þ
¼
X
N
i¼1
p
2
i
2m
þ
X
N
i; j¼1
ðr
i
r
j
Þ½1
where is a stable and regular pair potential.
Namely, we assume that there exists B 0 such that
X
N
i;j¼1
ðr
i
r
j
ÞBN ½2
for all N = 2, 3, ... and all (r
1
, ..., r
N
) 2 R
3N
, and
that
CðÞ¼
Z
e
ðrÞ
1
d
3
r < 1½3
for some >0 (and hence all >0). Basic
thermodynamic quantities are given in terms of the
grand-canonical partition function
Zð;; VÞ¼
X
1
N¼0
z
N
N!
Z
R
3N
V
N
e
H
N
Q
d
3
p
i
Q
d
3
r
i
h
3N
¼
X
1
N¼0
N
N!
Z
V
N
e
P
i;j
ðr
i
r
j
Þ
Y
d
3
r
i
½4
In the second expression we absorbed the factor
resulting from the integration over impulses into
(configurational) activity = (2m=h
2
)
3=2
z. In par-
ticular, the pressure p and the density are defined
by the thermodyn amic limits (with V !1 in the
sense of Van Hove)
pð; Þ¼
1
lim
V!1
1
jVj
log Zð; ;VÞ½5
and
ð; Þ¼ lim
V!1
1
jVj
@
@
log Z ð; ;VÞ½6
Mayer seri es are the expansions of p and in powers
of :
pð; Þ¼
X
1
n¼1
b
n
n
½7
and
ð; Þ¼
X
1
n¼1
nb
n
n
½8
Mayer’s idea for a systematic computation of
coefficients b
n
was based on a reformulation of
partition function Z(, , V) in terms of cluster
integrals. Introducing the function
f ðrÞ¼e
ðrÞ
1 ½9
and using G[N] to denote the set of all graphs on N
vertices {1, ..., N}, we get
Zð; ;VÞ¼
X
1
N¼0
N
N!
Z
V
N
Y
N
i;j¼1
1 þ f ðr
i
r
j
Þ
Y
d
3
r
i
¼
X
1
N¼0
N
N
X
g2G½N
wðgÞ½10
where
wðgÞ¼
Z
V
N
Y
fi;jg2g
f ðr
i
r
j
Þ
Y
d
3
r
i
½11
Observing that the weight w is multiplicative in
connected components (clusters) g
1
, ..., g
k
of the
graph g,
wðgÞ¼
Y
k
‘¼1
wðg
‘
Þ½12
we can rewrite
Zð; ; VÞ¼
X
1
N¼0
N
N!
X
fg
l
g
Y
g2G
wðgÞ½13
with the sum running over all disjoint collections fg
l
g
of connected graphs with vertices in {1, ..., N}. A
straightforward exponential expansion can be used to
show that, at least in the sense of formal power series,
log Zð; ; VÞ¼
X
1
n¼1
n
n!
X
g2C½n
wðgÞ½14
Cluster Expansion 531
where C[ n] is the set of all connected graphs on n
vertices. Using b
(V)
n
to denote the coefficients
b
ðVÞ
n
¼
1
jVj
1
n!
X
g2C½n
wðgÞ½15
and observing that the limits lim
V !1
(1=jVj)w(g)of
cluster integrals exist, we get b
n
= lim
V !1
b
(V)
n
.The
convergence of Mayer series can be controlled directly
by combinatorial estimates on the coefficients b
(V)
n
.Asa
result, the diameter of convergence of the series [7] and
[8] can be proved to be at least (C()e
2Bþ1
)
1
.Aless
direct proof is based on an employment of linear
integral Kirkwood–Salsburg equations in a suitable
Banach space of correlation functions.
Similar combinatorial methods are availa ble also
for evaluation of coefficients of the virial expansion
of pressure in powers of gas density,
pð; Þ¼
X
1
n¼1
n
n
½16
obtained by inverting [8] (notice that b
1
= 1) and
inserting it into [7]. One is getting
n
= lim
V !1
(V)
n
with
ðVÞ
n
¼
1
jVj
1
n!
X
g2B½n
wðgÞ½17
where B[n] C[n] is the set of all 2-connected
graphs on {1, ..., n}; namely, those graphs that
cannot be split into disjoint subgraphs by erasing
one vertex (and all adjacent edges). The diameter of
convergence of the virial expansion turns out to be
no less than (C()e(e
2B
þ 1))
1
.
Abstract Polymer Models
An application of the ideas of Mayer expansions to
lattice models is based on a reformulation of the
partition function in terms of a polymer model,a
formulation akin to [13] above. Namely, the partition
function is rewritten as a sum over collections of
pairwise compatible geometric objects – polymers.
Most often, the compatibility means simply their
disjointness.
While the reformulation of ‘‘physical partition
function’’ in terms of a polymer model (including the
definition of compatibility) depends on particularities
of a given lattice model and on the considered region of
parameters – high-temperature, low-temperature, large
external fields, etc. – the essence and results of cluster
expansion may be conveniently formulated in terms of
an abstract polymer model.
Let G = (V, E) be any (possibly infinite) countable
graph and suppose that a map w : V !C is given.
Vertices v 2 V are called abstract polymers, with
two abstract polymers connected by an edge in the
graph G called incompatible. We shall refer to w(v)
as to the weight of the abstract polymer v. For any
finite W V, we consider the induced subgraph
G[W]ofG spanned by W and define
Z
W
ðwÞ¼
X
IW
Y
v2I
wðvÞ½18
Here the sum runs over all collections I of
compatible abstract polymers – or, in other words,
the sum is over all independent sets I of vertices in
W (no two vertices in I are connected by an edge).
The partition function Z
W
(w) is an entire function
in w = {w(v)}
v2W
2 C
jWj
and Z
W
(0) = 1. Hence, it is
nonvanishing in some neighborhood of the origin
w = 0 and its logarithm is, on this neighbourhood, an
analytic function yielding a convergent Taylor series
log Z
W
ðwÞ¼
X
X2XðWÞ
a
W
ðXÞw
X
½19
Here, X(W) is the set of all multi-indices X : W !
{0 1, ...}andw
X
=
Q
v
w(v)
X(v)
. Inspecting the formula
for a
W
(X) in terms of corresponding derivatives of
log Z
W
(w), it is easy to show that the Taylor coefficients
a
W
(X) actually do not depend on W : a
W
(X) = a
supp
X(X), where supp X = {v 2 V: X (v) 6¼ 0}. As a result,
one is getting the existence of coefficients a(X) such that
log Z
W
ðwÞ¼
X
X2XðWÞ
aðXÞw
X
½20
for every finite W V.
The coefficients a(X ) can be obtained explicitly.
One can pass from [18] to [20] in a similar way as
passing from [10] to [13]. The starting point is to
replace the restriction to compatible collect ions of
abstract polymers in the sum [18] by the factor
Q
v; v
0
2W
(1 þ F(v; v
0
)) with
Fðv; v
0
Þ¼
0ifv and v
0
are compatible
1 otherwise ðv and v
0
connected by an edge from GÞ
8
>
<
>
:
½21
and to expand the product afterwards. The resulting
formula is
aðXÞ¼ðX!Þ
1
X
HGðXÞ
ð1Þ
jEðHÞj
½22
Here, G(X) is the graph with jXj=
P
jX(v)j vertices
induced from G[supp X] by replacing each of its
vertices v by the complete graph on jX(v)j vertices
and X! is the multifactorial X!=
Q
v2supp X
X(v)!. The
sum is over all connected subgraphs H G(X)
spanned by the set of vertices of G(X) and jE(H)j
is the number of edges of the graph H.
532 Cluster Expansion
A useful property of the coefficients a(X) is their
alternating sign,
ð1Þ
jXjþ1
aðXÞ0 ½23
More important than an explicit form of the
coefficients a(X) are the convergence criteria for the
series [20]. One way to proceed is to find direct
combinatorial bounds on the coefficients as expressed
by [22]. While doing so, one has to take into account the
cancelations arising in view of the presence of terms of
opposite signs in [22]. Indeed, disregarding them would
lead to a failure since, as it is easy to verify, the number
of connected graphs on jXj vertices is bounded from
below by 2
(jXj1)(jXj2)=2
. An alternative approach is to
prove the convergence of [20] on polydisks D
W, R
=
{w : jw(v)jR (v)forv 2 W} by induction in jWj,
once a proper condition on the set of radii R = {R(v);
v 2 V} is formulated. The most natural for the inductive
proof (leading in the same time to the strongest claim)
turns out to be the Dobrushin condition:
There exists a function r : V ![0 ; 1) such that, for
each v 2 V
RðvÞrðvÞ
Y
v
0
2NðvÞ
1 rðv
0
ÞðÞ½24
Here N( v ) is the set of vertices v
0
2 V adjacent in
graph G to the vertex v.
Using X to denote the set of all multi-indices
X : V !{0; 1, ...} with finite jXj=
P
jX(v)j and
saying that X 2X is a cluster if the graph G(supp
X) is connected, we can summarize the cluster
expansion claim for an abstract polymer model in
the following way:
Theorem (Cluster expansion). There exists a func-
tion a : X!R that is nonvanishing only on clusters,
so that for any sequence of diameters R satisfying
the condition [24] with a sequence {r(v)}, the
following holds true:
(i) For every finite W V, and any contour weight
w 2D
W, R,
one has Z
W
(w) 6¼ 0 and
log Z
W
ðwÞ¼
X
X2XðWÞ
aðXÞw
X
(ii)
P
X2X : suppX3v
ja(X)jjwj
X
log(1 r(v)).
Notice that, we have got not only an absolute
convergence of the Taylor series of log Z
W
in the closed
polydisk D
W, R
, but also the bound (ii) (uniform in W)
on the sum over all terms containing a fixed vertex v.
Such a bound turns out to be very useful in applications
of cluster expansions. It yields, eventually, bounds on
various error terms, avoiding a need of an explicit
evaluation of the number of clusters of ‘‘given size.’’
The restriction to compatible collections of polymers
can be actually relaxed. Namely, replacing [25] by
Z
W
ðwÞ¼
X
W
0
W
Y
v2W
0
wðvÞ
Y
v;v
0
2W
0
Uðv; v
0
Þ½25
with U (v, v
0
) 2 [0, 1] (soft repulsive interaction) , and
the condition [24] by
RðvÞrðvÞ
Y
v
0
6¼v
1 rðv
0
Þ
1 Uðv; v
0
Þrðv
0
Þ
½26
one can prove that the partition function Z
W
(w)
does not vanish on the polydisk D
W, R
implying thus
that the power series of log Z
W
(w) converges
absolutely on D
W, R
.
Polymers that arise in typical applications are
geometric objects endowed with a ‘‘support’’ in the
considered lattice, say Z
d
, d 1, and their weights
satisfy the condition of translation invariance. Cluster
expansions then yield an explicit power series for the
pressure (resp. free energy) in the thermodynamic
limit as well as its finite-volume approximation.
To formulate it for an abstract polymer model, we
assume that for each x 2 Z
d
, an isomorphism
x
: G !G is given and that with each abstract polymer
v 2 V a finite set (v) Z
d
is associated so that
(
x
(v)) =(v) þ x for every v 2 V and every x 2 Z
d
.
For any finite W V and any multi-index X,let
(W) = [
v2W
(v)and(X) =(supp(X)). On the
other hand, for any finite Z
d
,letW() = {v 2
V : (v) }. Assuming also that the weight w : V !C
is translation invariant – that is, w(v) = w(
x
(v)) for
every v 2 V and every x 2 Z
d
– we get an explicit
expression for the ‘‘pressure’’ of abstract polymer model
in the thermodynamic limit
p ¼ lim
!1
1
jj
log Z
WðÞ
ðwÞ¼
X
X:ðXÞ30
aðXÞw
X
jðXÞj
½27
In add ition, the finite-volume approximation can be
explicitly evaluated, yieldi ng
log Z
WðÞ
ðwÞ
¼ pj jþ
X
X:ðXÞ\
c
6¼;
aðXÞw
X
jðXÞ\j
jðXÞj
½28
Using the claim (ii), the second term can be bounded
by const. j@j.
Cluster Expansions for Lattice Models
There is a variety of applications of cluster expan-
sions to lattice models. As noticed above, the first
step is always to rewrite the model in terms of a
polymer representation.
Cluster Expansion 533
High-Temperature Expansions
Let us illustrate this point in the simplest case of the Ising
model. Its partition function in volume Z
d
,with
free boundary conditions and vanishing external field, is
Z
ðÞ¼
X
exp
X
x;y2
jxyj¼1
x
y
8
>
<
>
:
9
>
=
>
;
½29
Using the identity
e
x
y
¼ cosh þ
x
y
sinh ½30
it can be rewritten in the form
Z
ðÞ¼2
jj
ðcosh Þ
jBðÞj
X
B
ðtanh Þ
jBj
½31
Here, the sum runs over all subsets B of the set B()of
all bonds in (pairs of nearest-neighbor sites from )
such that each site is contained in an even number of
bonds from B.Using( B) to denote the set of sites
contained in bonds from B, we say that B
1
, B
2
B()
are disjoint if (B
1
) \ (B
2
) = ;. Splitting now B into a
collection B= {B
1
, ..., B
k
} of its connected components
called (high-temperature) polymers and using B()to
denote the set of all polymers in ,wearegetting
Z
ðÞ¼2
jj
ðcosh Þ
jBðÞj
X
BBðÞ
Y
B2B
ðtanh Þ
jBj
½32
with the sum running over all collections B of mutually
disjoint polymers. This expression is exactly of the
form [18], once we define compatibility of polymers
by their disjointness. Introducing the weights
wðBÞ¼ tanh ðÞ
jBj
½33
and taking the set B() of all polymers in for W,
we get the polymer representation Z
() =
2
jj
(cosh )
jB()j
Z
B()
(w).
To apply the cluster expansion theorem, we have to
find a function r such that the right-hand side of [24] is
positive and yields thus the radius of a polydisk of
convergence. Taking r(B) =
jBj
with a suitable ,weget
Y
B
0
2NðBÞ
ð1 rðB
0
ÞÞ e
2jBj
½34
allowing to choose R(B) = r(B)e
2jBj
= (e
2
)
jBj
.
Indeed, to verify [34] we just notice that the number
of polymers of size n containing a fixe d site is
bounded by
n
with a suitable constant . Thus,
X
B
0
:ðB
0
Þ3x
jB
0
j
X
1
n¼1
n
n
1 ½35
once is sufficiently small, and thus
X
B
0
2Nð B Þ
jBj
jðBÞj jBj½36
yielding [34] (1 t > e
2t
for t < 1=2). To have w 2
D
W, R
(for any W) is, for R(B) = (e
2
)
jBj
, sufficient
to take
0
with tanh
0
= e
2
.
As a consequence, for
0
we can use the
cluster expansion theorem to obtain a convergent
power series in powers of tanh . In particular,
using (X) = [
B2suppX
(B), we get the pressure by
the explicit formula
pðÞ¼
log 2 þ d log ðcosh Þþ
X
X:ðXÞ3x
aðXÞ
jðXÞj
w
X
½37
for any fixed x 2 Z
d
(by translation invariance of
the contributing terms, the choice of x is irrelevant).
The function p() is analytic on the region
0
since it is obtained as a uniformly absolutely
convergent series of analytic terms ( tanh )
jXj
.
This type of high-temperature cluster expansion
can be extended to a large class of models with
Boltzmann factor in the form exp {
P
A
U
A
()},
where = (
x
; x 2 Z
d
) is the configuration with
a priori on-site probability distribution (d
x
) and
U
A
, for any finite A Z
d
, are the multi-site
interactions (depending only on (
x
; x 2 A)). Using
the Mayer trick we can rewrite
exp
X
A
U
A
ðÞ
()
¼
Y
A
1 þ f
A
ðÞðÞ½38
with f
A
() = exp {U
A
()} 1. Expanding the
product we will get a polymer representation with
polymers A consisting of connected collections
A= (A
1
, ..., A
k
) with weights
wðAÞ ¼
Z
Y
A2A
f
A
ðÞ
Y
x2[
A2A
A
ðd
x
Þ½39
under appropriate bounds on the interactions U
A
and for small enough, using (A) to denote the set
[
A2A
A, we get,
X
A:ðAÞ3x
jwðAÞj 1 ½40
This assumption allows, as before in the case of the
high-temperature Ising model, to apply the cluster
expansion theorem yielding an explicit series expan-
sion for the pressure.
Correlations
Cluster expansions can be ap plied for evaluation of
decay of correlations. Let us consider, for the class
of models discussed above, the expectation
hi
¼
1
Z
Z
ðÞe
H
ðÞ
Y
x2
ðd
x
Þ½41
534 Cluster Expansion