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Abstract

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.
In: Acta Applicandae Mathematicae, Kluwer Academic Publishers 23: 65-93, (1991).
The Design of Linear Algebra and Geometry
David Hestenes
Abstract. Conventional formulations of linear algebra do not do justice to
the fundamental concepts of meet, join, and duality in projective geometry.
This defect is corrected by introducing Clifford algebra into the foundations
of linear algebra. There is a natural extension of linear transformations on
a vector space to the associated Clifford algebra with a simple projective
interpretation. This opens up new possibilities for coordinate-free compu-
tations in linear algebra. For example, the Jordan form for a linear trans-
formation is shown to be equivalent to a canonical factorization of the unit
pseudoscalar. This approach also reveals deep relations between the struc-
ture of the linear geometries, from projective to metrical, and the structure
of Clifford algebras. This is apparent in a new relation between additive
and multiplicative forms for intervals in the cross-ratio. Also, various fac-
torizations of Clifford algebras into Clifford algebras of lower dimension are
shown to have projective interpretations.
As an important application with many uses in physics as well as in
mathematics, the various representations of the conformal group in Clifford
algebra are worked out in great detail. A new primitive generator of the
conformal group is identified.
[Note: Some corrections have been added in brackets.]
1. Introduction
It is sometimes said that projective geometry is just linear algebra. Nevertheless, the two
subjects persist today as separate branches of mathematics. Although they grew up together
in the nineteenth century, projective geometry was left behind as linear algebra flourished
through diverse applications in the twentieth century. This is the second of two articles
aimed at bringing them back together in a mutually beneficial reconciliation. These articles
propose specific designs for a mathematical system to achieve that end.
Conventional linear algebra is based on the concepts of vector space and linear transfor-
mation. Projective geometry can be formulated within this framework, but only by intro-
ducing coordinate systems without intrinsic geometric significance. To achieve a coordinate-
free formulation of projective geometry, a richer algebraic system is needed. In the first
article [1] geometric algebra was proposed as the ideal system for that purpose. It was
shown to have exactly the right dimension and algebraic structure to describe all the meet
and join operations essential to projective geometry. This article extends the mathematical
designs of [1] along two lines: (1) The standard concept of a linear transformation is ex-
tended to provide a direct, coordinate-free formulation of projective transformations, and
this extension is shown to be valuable even in standard linear algebra. As an example,
the Jordan canonical form is shown to be equivalent to a particular factoring of the unit
pseudoscalar. (2) The concepts of ‘projective split’ and ‘conformal split’ are introduced
1
and shown to reveal a deep connection between projective structures and the structure of
Clifford algebras. They are used to simplify and coordinate relations between projective,
conformal, affine, and metric geometries. A new relation between additive and multiplica-
tive forms for intervals in the cross-ratio is derived. The conformal groups for metric spaces
of arbitrary dimension and signature are analyzed in great detail, establishing connections
between several alternative approaches in the literature.
Reference [1] is taken as prerequisite to this article, so the notations and nomenclature
introduced there can be taken for granted. An overview of that article and the present one
has been published in [2].
2. Outermorphism
This section reviews the concept of outermorphism developed in [3], with emphasis on its
significance for projective geometry. The treatment in [3] is more general and extensive
than the present one. Unfortunately, that may have obscured the essential simplicity and
utility of the outermorphism idea, so the aim here is to clarify the concept. To that end,
some of the results in [3] will be introduced without proof, while others are omitted.
The outermorphism is actually inherent in linear algebra. By making it explicit and
giving it a notation we can exploit it more easily. The conventional abstract formulation of
linear algebra follows the format of matrix algebra. This does not do justice to Grassmann’s
outer product or to duality, two fundamental concepts of projective geometry. The main
point of this section is that the outermorphism concept corrects this defect perfectly. Since
the outermorphism is inherent in linear algebra, it has been repeatedly rediscovered in
various guises. For example, as explained in [3], the so-called ’push-forward’ and ’pull-
back’ mappings of differential forms are outermorphisms. While preparing this article, it
was discovered that Whitehead [4] had independently defined the outermorphism of a blade
and was probably the first to do so. He evidently sensed the importance of the idea, but
he did not develop it far, and no one seems to have followed his lead.
As in [1], Vnis an n-dimensional vector space which generates a geometric algebra
Gn=G(Vn). Every linear transformation fon Vnhas a unique extension to a linear
transformation fon Gndefined by the following four properties:
f(AB)=(fA)(fB),(2.1)
fa=fa, (2.2)
fα=α, (2.3)
f(αA +βB)=αfA+βfB, (2.4)
where A, B are blades in Gn,ais a vector and α, β are scalars. Thus, fis linear on Gnand
distributive with respect to the outer product. It is called the outermorphism of fbecause,
according to (2.1), it preserves the outer product.
It follows from (2.1) and (2.2) that fis step-preserving, that is,
f(hMik)= hfMi
k(2.5)
2
for any multivector Min Gn, Moreover, for any factoring A=a1a2∧···a
rof an r-blade
Ainto vectors
fA=(fa1)(fa2)···(far).(2.6)
That is to say, fAdoes not depend on how Amight be factored into vectors. It will be
seen that this property is a great help in finding and describing the invariants of a linear
transformation.
For the sake of simplicity, we limit our attention here to transformations from Vnto Vn,
so fmaps Gninto Gn. In this case, (2.5) implies that fcan alter the pseudoscalar Ionly
by a scalar multiple. Indeed,
fI= (det f)I(2.7)
defines the determinant of f, which we write as det for det f. Note that the outermorphism
concept makes it possible to define the determinant of a linear transformation without
introducing a basis or matrices. As shown in [3], the matrix form of the determinant
is easily introduced when needed, but the determinant can often be evaluated without
introducing matrices.
Let h=gf be the product of linear transformations fand gon Vn. From (2.6) it is
easily proved that
h=gf . (2.8)
In other words, the outermorphism of a product is equal to the product of outermorphisms.
Some important facts about linear transformations follow immediately from (2.8). For
example,
gfI = (det f)gI = (det f)(det g)I.
Therefore,
det (fg) = (det f)(det g).(2.9)
It follows by iteration that, for positive integer k, det (fk) = (det f)k, and if f1exists,
then (det f1)(det f)=1.
Every linear transformation fhas an adjoint transformation f, and the adjoint can be
extended to an outermorphism which we denote by falso. The adjoint outermorphism f
can be defined directly by
hMfNi=h(fN)Mi=hNfMi(2.10)
assumed to hold for all M,N in Gn. The multivector derivative defined in [3] can be used
to get
fN=MhNfMi.(2.11)
Differentiation is helpful and much used in [3], especially to generate invariant relations,
but it is not essential and will be avoided in the following. It follows from (2.7) and (2.10)
that
det f=I1fI =hI1fI i=I1f I =detf . (2.12)
Unlike the outer product, the inner product is not generally preserved by an outermor-
phism. However, it is proved in [3] that the inner product obeys the fundamental transfor-
mation law
A·(fB)=f[(f A)·B]or(fB)·A=f[B·(fA)] ,(2.13)
3
for (step A)(step B). For clarity, the square brackets are used as parentheses here. Note
that both fand fare needed to describe how the inner product transforms. Also note
that, by symmetry, fand fcan be interchanged in (2.13).
The law (2.13) admits the important special case
A(fI)=f[(fA)I]or(fI)A=f[I(fA)] .(2.14)
For de t f6= 0, this gives us immediately an explicit expression for the inverse outermor-
phism:
f1A=f(AI)I1
det f=I1f(IA)
det f.(2.15)
Applied to ain Vn, of course, this gives us the inverse transformation
f1(a)=I
1
f(Ia)
det f.(2.16)
This reveals that the adjoint and double duality are essential relations underlying the
general matrix formula for f1, which can, in fact, be obtained easily from (2.16).
The inverse of the outermorphism fgiven by (2.15) is equal to the outermorphism of the
inverse f1. To prove that, we need only establish the distributive property
f1(AB)=(f
1
A)(f
1
B).(2.17)
For the proof it will be convenient to use the notation A=e
AI1and put (2.14) in the form
f[(fe
A)I]=Adet f. (2.18)
The proof uses the duality of inner and outer productsalong with (2.13) as follows. Assuming
(step A) + (step B)n,
(f1A)(f1B)=[(fe
A)I][(fe
B)I]
(det f)2=[(fe
A)I]·(fe
B)
(det f)2I
=f[(f[(fe
A)I]) ·e
B]
(det f)2I=f[A·e
B]I
det f=f[(AB)I1]
det f.
(2.19)
Thus (2.17) is established. Now let us turn to the matter of geometrical interpretation.
Outermorphisms can be supplied with a geometrical interpretation by adopting the
projective interpretations for blades introduced in [1]. Recall that vectors representing rays
in Vncan be identified with points in the projective space Pn1. Projective transformations
(or projectivities) are of two types, collineations and correlations. The collineations of Pn1
are simply linear transformations of Vnwith nonvanishing determinants. A correlation is
simply the composite of a collineation with a duality transformation (i.e., multiplication
by a pseudoscalar). Having discussed duality in [1], we can limit our attention here to
collineations without loss of generality.
The most basic fact about collineations is that they map points into points, lines into
lines, planes into planes, etc. This follows from the distributive property of outermorphisms,
4
since points, lines, and planes can be represented by blades of step 1, 2, 3, respectively.
Indeed, a collineation finduces an outermorphism of the equation xA= 0 into
x0A0=(fx)(fA)=0.
This is equivalent to the original equation for point, line, or plane provided fA6= 0 when
A6= 0, and that is assured by the fact that every nonzero blade is a factor of the pseudoscalar
I. For if blade A6= 0, there exists another (not unique) blade Bsuch that AB=I. Since
(fA)(fB) = (det f)I6=0,
the factor fAcannot vanish.
Thus, we see that the outermorphism fof a collineation fdirectly describes the induced
transformations of lines and planes. We can interpret fAas the transformation of a specific
line or plane according as blade Ahas step 2 or 3.
To establish the invariance of projective relations under arbitrary projectivities, we need
to prove that the fundamental meet and join relations are preserved under collineations.
Since we can identify the join with the outer product, the invariance of the join relation
is assured by the distributive property (2.1). To find the transformation law for the meet,
recall that for (step A) + (step B)n, the meet can be expressed in terms of the inner
product by Equation (3.6) of [1]:
AB=e
A·B. (2.20)
Proceeding as in the proof of (2.17), we have
(fA)(fB)=[(fA)I
1
]·(fB)= f[(f[( fA)I1]) ·B
=f[e
A·B](det f).
Thus, the meet obeys the transformation law
(fA)(fB) = (det f)f(AB).(2.21)
The factor (det f) here is not significant in nonmetrical applications of projective geometry,
since the definition of the meet is arbitrary within a scale factor. However, it can be
removed by a natural change in the definition of duality, defining duality on the left with
respect to the transformed pseudoscalar I0=f I = (det f)I. Then (2.15) implies that
the transformation A0=fA entails the induced transformation e
A0=fe
Aon the dual
e
A0=A0(I0)1, and (2.21) can be put in the alternative form
f(e
A·B)= e
A
0·B
0.(2.22)
This transformation law is mathematically equivalent to (2.13), but its geometric meaning
is much more transparent. It tells us that the ‘incidence properties’ in projective geometry
are invariant under collineations or, equivalently, that the ‘subspace intersection property’
is preserved by nonsingular linear transformations. For this reason, it should be counted
as one of the fundamental results of linear algebra, though it does not appear in standard
textbooks on the subject.
5
3. Invariant Blades
The outermorphism is a natural tool for characterizing the invariant subspaces of an arbi-
trary linear transformation. As an illustration, this section shows how to use that tool to
achieve a characterization equivalent to the Jordan canonical form.
The outermorphism gives us a natural generalization of the eigenvector concept. A blade
Aissaidtobeaneigenblade of fwith (scalar) eigenvalue λif
fA=λA . (3.1)
‘Projectively speaking’, this says that Ais a fixed point, line, or plane of a collineation f
according as the step of Ais 1, 2, or 3. Let us call Aa symmetric eigenblade if
fA=f A =λA . (3.2)
This generalizes to arbitrary linear transformations an important feature of the eigenvectors
of a symmetric transformation. Of course, the pseudoscalar Iis a symmetric eigenblade of
every linear transformation, with
fI=f I =µI , (3.3)
where µ=detf. Henceforth, it will be convenient to assume µ6=0.
If Ais an eigenblade of f, then its dual e
A=AI1is an eigenblade of f. This follows
trivially from (2.14); specifically, with (3.1) and (3.3), we have
fe
A=µ
λe
A. (3.4)
Since I=A1e
A=A1e
A,
fI=µI =(fA
1
)(fe
A)= µ
λ(fA
1
)e
A.
We cannot in general remove the wedge from the right side of this expression, but we can
conclude that there exists a blade Bwith the same step as Asuch that
fA=λA +B. (3.5a)
where Be
A=(B·A)I
1= 0. Dually, we have
fe
A=µ
λe
A+e
C. (3.5b)
where Ae
C=(A·C)I
1= 0. Clearly, Ais a symmetric eigenblade of fif and only if e
A
is also symmetric; moreover
fA=fe
A=µ
λe
A. (3.6)
A symmetric eigenblade will be called a proper blade of fif it has no factors which
are symmetric eigenblades of lower step. By iterating the argument yielding (3.6), we can
decompose the pseudoscalar Iinto a geometric product of proper blades Ik;thus,
I=I
1
I
2···I
m,(3.7a)
6
where, for k=1,2,...,m n,
fI
k=fI
k=µ
kI
k,(3.7b)
and
det f=µ=µ1µ2···µ
m.(3.7c)
The Ikare unique (within a scale factor) if the µkare distinct. Their supports are precisely
the invariant subspaces of the linear transformation f. If all the Ikhave step 1, then
m=nand (3.7b) is a complete spectral decomposition of f. However, for any Ikwith
(step Ik)>1, we must take the decomposition further to characterize fcompletely. That
we do next.
[Warning: The following analysis does not suffice to establish the Jordan form for an
arbitrary linear transformations, though the method of analysis may be of general interest
in linear algbera.]
Suppose that fhas a proper blade Amof step m>1, and let complex scalars be allowed
just to avoid discussing points of peripheral interest. Then the following result is equivalent
to the Jordan decomposition of f: there exists a unique scalar λand a nested sequence of
blades Aksuch that
fAk=λkAk(3.8)
for k=1,2,...,m. The term ‘nested’ means that Ak1is a factor of Ak. Let e
Ak=AkA1
m
be the dual with respect to Am; then (3.4) implies that
fe
Ak=λmke
Ak.(3.9)
The Akare unique and they completely characterize fon the support of Amk. In fact, the
Akcan be generated from the unique eigenvector a1=A1,by
A
k=a
1(fA
k1
),(3.10)
which iterates to
Ak=a1(f a1)(f2a1)∧···(fk1a
1).(3.11)
We can prove all this by showing that it is equivalent to the Jordan canonical form.
The blades Akdetermine mvectors akdefined by
ak=A1
k1Ak=A1
k1·Ak(3.12)
with the convention A0= 1. Conversely, the Akare determined by the akaccording to
Ak=Ak1ak=a1a2···a
k.(3.13)
Note that this implies the orthogonality relation
ai·aj=0 for i6=j. (3.14a)
To ascertain the action of fon the ak, we apply (3.8) to (3.13) to obtain
λAk=Ak1(fak).
7
This can be solved for fak=fakas follows:
λA1
k1Ak=λak=A1
k1·(Ak1(fak))
=(A
1
k1·A
k1
)fa
k+(1)k1(A1
k1·(fak))·A
k1
=fa
ka
k1.
The last step follows from the ‘Laplace expansion’
(a1
k∧···a
1
2a
1
1)·(fa
k)=(l)
k2
a
1
k2···a
1
2a
1
1(a
1
k1·fa
k)
=(l)
k2
a
1
k2.
Only one term in the expansion survives because
a1
j·(fak)=a
k
(fa
1
j)=0 for j<k1,
as follows from the vanishing of terms in the formula
Aj+1 ·a1
k=[A
j(fa
j
)] ·a1
k=Aj(a1
k·f aj)(Aj·a1
k)A(f aj).
For j=k1, this formula implies the relation
a1
k·(f ak1)=a
k1·(fa
1
k)=1
which was also used above.
Thus we have proved that the nested eigenblade formula (3.8) implies that
fak=λak+ak1(3.14b)
for 1 <km. Of course,
fa1=λa1(3.14c)
since a1=A1. Note that no assumptions about signature were used in the proof, although
it was necessary that A2
m6= 0. Conversely, (3.8) can be derived from (3.14b,c) simply by
inserting them into
fAk=(fa
1
)(fa
2
)∧···(fa
k).
Equations (3.14a,b,c) will be recognized as the Jordan canonical form for a linear transfor-
mation on a cyclic subspace. We have proved that on the support of Amthe Jordan form
is equivalent to the factoring of Aminto a nested sequence of eigenblades.
If only real scalars are allowed, the above decomposition of a cyclic subspace applies if the
characteristic polynomial has a root λof multiplicity m, in which case there is exactly one
eigenvector. However, when the characteristic polynomial of the subspace does not have
a real root, it has an irreducible factor of some multiplicity m. In this case, the subspace
necessarily has even dimension 2m. There are no eigenvectors in the subspace, but there
is, nevertheless, a scalar γand a nested sequence of blades A2kwith even step 2ksuch that
fA2k=γkA2k(3.15)
for k=1,2,... ,m. This assertion can be proved along lines similar to the previous case,
but we will not go into that here. It should be noted, though, that the determination
of the eigenblades is not sufficient, in this case, to characterize the linear transformation
completely.
8
4. Projective Splits
We have seen how the entire geometric algebra Gn+1 is needed to describe the join, meet,
and duality operations on the projective n-space Pnas well as their projective outermor-
phisms. This section shows that the multiplicative structure of geometric algebra reflects
deep properties of projective geometry and facilitates connections with affine and metric
geometry.
The reduction of projective geometry to affine geometry can be expressed as a relation
between a vector space Vn+1 and a vector space Vnof one less dimension. Ordinarily Vn
is taken to be a subspace of Vn+1, but geometric algebra admits a more profound way of
relating Vn+1 to Vnwhich will be investigated here. Let Gr
n+1 denote the subspace of all
multivectors of step rin Gn+1. The subspace
G+
n+1 =X
kG2k
n+1 (4.1)
of all multivectors with even step is a subalgebra called the even subalgebra of Gn+1.It
can be identified with the geometric algebra Gnof an n-dimensional vector space Vn,as
expressed by
Gn=G+
n+1 .(4.2)
This requires the specification of a unique relation between the vector spaces Vn+1 =G1
n+1
and Vn=G1
n. A geometrically significant way to do this is as follows. Let e0and xbe
vectors in Vn+1, then, for fixed e0with e2
06=0,
xe0=x·e0+xe0(4.3)
is a linear mapping of Vn+1 into G0
n+1 +G2
n+1, which defines a linear correspondence between
Vn+1 and G0
n+G1
nby the identification G0
n+1 =G0
nand
Vn=G1
n={xe0}.(4.4)
This gives Vna projective interpretation as the pencil of all lines through the point e0.
Moreover, it determines a split of G2
n+1 into a direct sum,
G2
n+1 =G1
n+G2
n.(4.5)
where, in accordance with (4.4) G1
nconsists of all bivectors in G2
n+1 which anticommute
with e0while G2
nconsists of the bivectors which commute with e0. The split of any bivector
Fin G2
n+1 is thus given by
F=Fe
1
0e
0=(F·e
1
0)e
0+(Fe
1
0)e
0,(4.6)
where
(F·e1
0)e0=(F·e
1
0)e
0
is in G1
n=Vnand
(Fe1
0)e0=(Fe
1
0)·e
0
9
is in G2
n.
The linear mapping (4.3) Vn+1 into Gnalong with the identification of G+
n+1 with Gn
according to (4.4) and (4.5) will be called a projective split of Vn+1, or if you will, of Gn+1.
In view of its geometrical significance, the projective split can be regarded as a canonical
relation between geometric algebras of different dimension. Indeed, the underlying geomet-
rical idea played a crucial role in Clifford’s original construction of geometric algebras [5].
However, Clifford’s motivating ideas have been largely ignored in subsequent mathematical
applications of Clifford algebras. The projective split idea was first explicitly formulated
and applied to physics in [6]. Accordingly, flat spacetime is represented by a vector space
V4with Minkowski metric, and its associated geometric algebra G4is called the spacetime
algebra. The ‘splitting vector’ e0is taken to be the timelike vector of some inertial frame,
and it determines a projective split of V4and G4into space and time components. This is
called a spacetime split. Each inertial system determines a unique spacetime split. This
relates the invariant properties of objects in spacetime to their ‘observable’ representations
in inertial systems. For example, an electromagnetic field is represented by an invariant
bivector-valued function F=F(x)onV
4
. In that case, the split (4.6) describes the frame-
dependent splitting of Finto electric and magnetic components. The spacetime split is thus
a fundamental relation in physics, though its projective character usually goes unrecognized.
The projective split (4.3) can be put in the form
xe0=x0(1 + x),(4.7)
where x0=x·e0and boldface is used to denote a vector
x=xe0
x·e0
(4.8)
in Vn. Clearly xe0and x·e0or just xrelative to e0amounts to a representation of the
‘point’ xby ‘homogeneous coordinates’.
For ‘points’ a, b in Vn+1, the projective split (4.7) relates products in Gn+1 to products in
Gn;thus,
ab =(ae0)(e0b)
e2
0
=a0b0
e2
0
(1 + a)(1 + b)
=a0b0
e2
0
(1 a·b+ab+ba).
Separating inner and outer products in Gn+1 we obtain
a·b=a0b0
e2
0
(1 a·b)(4.9)
and
ab=a0b0
e2
0
(ab+ab)
=a0b0
e2
0
(ab+1
2(a+b)(ab)).(4.10)
10
It is important to distinguish inner and outer products in Gn+1 from inner and outer products
in Gn. The distinction is made here by using boldface for vectors in Gn.
Equation (4.10) relates two different forms for the ‘Plucker coordinates’ of a line passing
through points aand b. It tells us that abrepresents a line in Vnwith direction ab
passing through the point 1
2(a+b). We can put (4.10) in the form
ab=a0b0
e2
0
(1 + d)(ab),(4.11)
where
d=(ba)·(ab)
1(4.12)
can be regarded as the ‘directed distance’ from the origin to the line. For three distinct
points a, b, c on the same line, we have abc= 0, and application of (4.11) yields the
invariant ratio ac
bc=a0
b0µac
bc.(4.13)
This is a ‘projective invariant’ in two senses: It is independent of the chosen projective
split, and it is invariant under collineations. Thus, for a collineation fthe linearity of the
outermorphism immediately implies
f(ac)
f(bc)=ac
bc.(4.14)
The interval ratio (ac)/(bc) is not a projective invariant, but it is an affine invariant
because a0/b0is (see below). The classical cross-ratio for four distinct points a, b, c, d on a
line is given by µac
bc¶µbd
ad=µac
bc¶µbd
ad.(4.15)
This is a projective invariant in both senses mentioned above. Using (4.15), all the well-
known implications of the cross-ratio are easily derived. The considerable advantage of
using geometric algebra here should be obvious. The division in (4.15) is well-defined
without abuse of notation. This is an important example of a projective invariant which is
not so simply expressed in terms of meet and join products alone.
The essential relation between affine and projective geometry is a projective split with
respect to a preferred vector e0. The equation
x·e0=λ(4.16)
determines a 1-parameter family of hyperplanes in Vn+1 with normal e0. The affine group
of Vnis the group of collineations on Vn+1 which leaves the projective split by e0and, hence,
the hyperplane equation (4.16), invariant. Thus, every affine transformation fsatisfies the
equation
e0·(fx)=x·(fe
0
)=x·e
0.(4.17)
This is equivalent to the condition that e0is a fixed point of the adjoint transformation f:
fe0=e0.(4.18)
11
(Henceforth, the underbar and overbar notations will be used to denote linear operators as
well as their outermorphisms.)
Of special interest in affine geometry is the translation operator Tadefined for a·e0=0
by
Tax=x+ax ·e0=x+e0·(xa).(4.19)
Its adjoint transformation is
Tax=x+e0(a·x)=x+a·(xe
0
),(4.20)
which obviously satisfies the e0invariance condition (4.18). The translation operator in-
duces an outermorphism of lines in Pn:
Ta(xy)=xy+a(yx ·e0xy ·e0)
=xy+(xya)·e
0.
For kpoints, this generalizes to
Ta(x1∧···∧x
k)=x
1∧···x
k+(1)k(x1∧···x
ka)·e
0.(4.21)
This applies to all vectors in Vn+1 whether or not they ‘lie’ on a single invariant hyperplane.
The linear transformation (4.19) is actually a shear on Vn+1; however, it projects to a
translation on VnTo show that explicitly, write x0=Taxand note that x0·e0=x·e0,so
the projective split (4.8) gives the translation
x0=x+a.(4.22)
By applying the outermorphism (4.20) to xe0, we ascertain the important fact that the
projective split is not preserved by the outermorphism.
The advantage of the projective split for affine geometry is precisely the advantage of
homogeneous coordinates, namely, it reduces translations to linear transformations. In the
next section we show how geometric algebra helps with a further simplification, reducing
translations to rotations.
The projective split has important implications for metric geometry as well as affine
geometry. For metrical applications we restrict the scalars to real numbers and we must
take signature into account. We write
Vn=V(p, q)(4.23)
when we wish to indicate that the vector space Vnhas the signature (p, q) imposed by its
associated geometric algebra Gn=G(p, q).
When the signature is taken into account in a split by e0, the split relation (4.2) takes
the more specific form
G(p, q)=G
+
(q+1,p)whene
2
0
>0,(4.24)
or
G(p, q)=G
+
(p, q +1) when e
2
0<0.(4.25)
12
To characterize orthogonal transformations, it is convenient to follow [3] and introduce the
concept of ‘versor.’ A versor Rin G(p, q) is any element which can be factored into a
geometric product
R=v1v2... v
k(4.26)
of unit vectors viin V(p, q). The factorization (4.26) is by no means unique, but there is
always one with kn. The multivector Ris even or odd according to whether kis even
or odd. This determines its parity under the main involution:
R=(1)kR. (4.27)
An even versor R=Ris called a rotor.
The versors in G(p, q) form a multiplicative group Pin(p, q) commonly called a ‘pin group’
[7]. The rotors comprise a subgroupSpin(p, q) called a ‘spin group’.
Now we can simply state what may be regarded as the fundamental theorem of metric
geometry: Every orthogonal transformation Ron V(p, q) can be written in the canonical
form
Rx=RxR1=Rx(R)1,(4.28)
where the underbar distinguishes the linear operator Rfrom its corresponding versor R.
This is called a rotation if R=Ris a rotor.
A remark on terminology is in order. The word ‘versor’ comes from Clifford [5] who got it
from Hamilton. Clifford explains that it is derived from the word ‘reverse’ in the expression
‘reverse direction’. The term is employed here with the same motivation, but it is defined
differently. According to (4.26), the most elementary versor is a vector R=v, in which
case (4.28) takes the form
Vx=vxv1.(4.29)
This represents a reflection Vin a hyperplane with normal v; its net effect is simply to
reverse the direction of vectors collinear with v. Thus, each unit vector vcan be interpreted
as an operator which ‘reverses’ direction; whence ‘versor’. From this it should be clear
that the factorization (4.26) corresponds to the Cartan–Dieudonn´e theorem that every
orthogonal transformation of Vncan be expressed as a product of at most nsymmetries.
The main fact of interest here is that by a projective split with respect to e0Spin(p, q)
is related to Spin(q+1,p)ife
2
0= 1 or to Spin(p, q +1) if e
2
0=1. Specifically, every rotor
in Gn+1 can be factored into a product
S=S0R, (4.30a)
where Ris a rotor in Gnand there exists a vector uin Vn+1 such that
S0=ue0.(4.30b)
The vector uis determined by noting that Sdetermines a rotation of e0into a vector
v=Se0S1(4.31)
and requiring that S0also rotates e0into v.Thenwehave
v=S
0
e
0
S
1
0=S
2
0
e
1
0,
13
where the last step follows from assuming that S0has the form (4.30b). Solving for S0we
get
S0=(ve0)1/2=(v+e0)e0
|v+e0|=v(v+e0)
|v+e0|.(4.32)
Thus, u=(v+e
0
)|v+e
0|
1is a unit vector ‘halfway between’ vand e0.NowRis given
by
R=S1
0S, (4.33)
and it follows that
Re0R1=e0.(4.34)
This implies that Ris in Spin(p, q)asclaimed.
Since (4.30b) has the form of the projective split (4.3), we can regard (4.30a) as a
projective split of the spin group. For the spacetime algebra G(1,3), Ref. [5] derives (4.30a)
as a spacetime split of a Lorentz transformation into a boost (or pure Lorentz transforma-
tion) and a spatial rotation. This has many important physical applications. Mathemati-
cians have also used the split (4.30a) to study the transitivity of spin groups on spheres
(e.g. Chap. 21 of [6]) without, however, recognizing its general connection with projective
geometry.
5. Conformal and Metric Geometry
We have seen that the projective split by a vector determines a geometrically significant
relation between the geometric algebras Gnand Gn+1. This section shows that a split by a
bivector determines an equally significant relation between Gnand Gn+2. Let e0be a fixed
unit 2-blade in G2
n+2 and let xbe a generic vector in Vn+1 =G1
n+2. A linear ‘split’ of Vn+2
into vector spaces V2and Vnis determined by the equation
xe0=x·e0+xe0=x0+ρx,(5.1)
where
V2={x0=x·e0=e0·x}(5.2)
and
Vn={ρx=xe0=e0x}.(5.3)
The significance of the scale factor ρwill be discussed later. The vector space V2generates
a geometric algebra G2with pseudoscalar e0,andV
ngenerates a geometric algebra Gn.We
have the commutative relations
x0x=xx0,(5.4a)
e0x=xe0.(5.4b)
Hence, all elements of G2commute with the elements of Gn, and we can express Gn+2 as
the Kronecker product
Gn+2 =Gn⊗G
2
.(5.5)
14
This is the fundamental multiplicative decomposition theorem for geometric algebras. Equa-
tion (5.5) has been treated previously only as a formal algebraic relation. The present ap-
proach gives it a geometric meaning by developing it as split by a bivector. Note that the
vector space Vndefined by (5.3) has a projective interpretation: The 2-blade e0represents
a fixed line and x=xe0represents a plane containing that line; therefore Vnrepresents
a pencil of planes intersecting in a common line. We will not pursue the implications of
this fact for projective geometry. We will concentrate on metrical implications.
The decomposition theorem (5.5) was proved in Clifford’s original article [5] for anti-
Euclidean signature. The signature does not play a crucial role in the proof, as the above
approach shows. However, when the theorem is used to classify geometric algebras by
multiplicative structure, the signature is important, as has been emphasized recently by
several authors ([8], [9], [10]). From the above, we find that (5.5) separates into three
different cases depending on the signature of G2:
G(p+1,q+1)=G(p, q)⊗G(1,1),(5.6)
G(q+2,p)=G(p, q)⊗G(2,0),(5.7)
G(q, p +2)=G(p, q)⊗G(0,2) .(5.8)
Note that e2
0= 1 in (5.6) and e2
0=1 in (5.7) and (5.8); this determines the relative
places of pand qon the two sides of the equations. The algebras G(1,1) and G(2,0) are
isomorphic, though not geometrically equivalent, and both are not isomorphic to G(0,2).
There follows a host of theorems about the algebraic equivalence of algebras with different
signature [8–10], but that will not concern us here. We will be concerned only with the
geometric significance of (5.6), which we abbreviate by
Gn+2 =Gn⊗G(1,1) (5.9)
when the particular signature (p, q) is not a matter of concern. Anticipating the results to
follow, let us refer to this as a conformal split of Gn+2 by the unit 2-blade e0.
From our previous considerations, we know that every orthogonal transformation of
Vn+2 =V(p+1,q+ 1) has the canonical form
x0=Gx(G)1,(5.10)
where Gis an element of the versor group Pin(p+1,q+ 1). The theorem of central interest
in the remainder of this section is that the orthogonal group O(p+1,q + 1) is isomorphic
to the complete conformal group C(p, q)onV
n=V(p, q); therefore Pin(p+1,q+1)isthe
spin representation of C(p, q). We will use the conformal split to prove this theorem and
analyze the structure of the conformal group and its spin representation in detail.
Representations of conformal groups in Clifford algebras have been discussed by many
authors, notably Angles [11], Lounesto and Latvamaa [12], Ahlfors [13], Maks [14], and
Crumeyrolle [15]. The present study aims to show that the conformal split idea simplifies
algebraic manipulations, clarifies geometric meanings, and reveals connections among alter-
native approaches. A complete, efficient and systematic treatment of the conformal group
should be helpful in applications to physics. The most thorough discussion of the physical
significance of the conformal group for spacetime has been given by Kastrup [16].
Our plan of study for the remainder of this Section is as follows. First, we analyze the
algebra G(1,1) and Pin(1,1) thoroughly. Though the results are well-known, the analysis is
15
necessary to establish notation for subsequent application. Next, we establish homogeneous
coordinates for Vnin Gn+2 and the general relation of the Pin group to the conformal group.
Finally, we determine a canonical factorization of the groups into elementary generators.
5.1. STRUCTURE OF G(1,1) AND Pin(1,1)
An orthonormal basis {e1,e
2}for the vector space V2=V(1,1) is defined by the properties
e2
1=1,e
2
1
=1,e
1
·e
2
=0,(5.11)
A null basis {e+,e
}for V(1,1) is defined by
e±=1
2(e1±e2).(5.12)
It has the properties
e2
+=e2
=0 (5.13)
and
e+e=1
2(1 + e0).(5.14)
This latter relation separates into
2e+·e=1=2e
+
·e
1=2e
·e
2(5.15)
and
2e+e=e0=e2e1.(5.16)
The bivector e0is the unit pseudoscalar for V2. It has the important properties
e2
0=1,(5.17)
e0e±=±e±=e±e0.(5.18)
The null basis has the advantage of being unique (up to a scale factor), since it characterizes
intrinsic structure of V(1,1), namely its ‘isotropic cone.’ For this reason, we use the null
basis whenever feasible. However, null vectors are not elements of the versor group, so we
use e1and e2to represent reflections.
For comparison with the literature, it should be understood that G(1,1) is isomorphic to
the algebra L2(R)ofreal2×2 matrices, that is,
G(1,1) 'L2(R)(5.19)
To make the isomorphism explicit, we write a generic multivector Min G(1,1) in the
expanded form
M=1
2[A(1 + e0)+B(e
1+e
2
)+C(e
1e
2
)+D(1 e0)] ,(5.20)
where A,B,C,Dare scalars. The basis elements have the matrix representation
[e1]=·01
10
¸
,[e
2
]=·01
10
¸
,[e
0
]=·10
01
¸
.(5.21)
16
Consequently (5.20) has the matrix representation
[M]=·AB
CD
¸
.(5.22)
The matrix representation preserves the geometric product, that is
[MN]=[M][N].(5.22)
The conjugate M
ecan be defined by
M
e=(M
)
,(5.24)
where the dagger denotes reversion in G(1,1). The involute is given by
M=e0Me0.(5.25)
It is readily verified that
det [M]=M
eM. (5.26)
Therefore, we have the isomorphisms
{M|MM
e6=0}'GL2(R),(5.27)
{M|MM
e=1}'SL2(R),(5.28)
to the general linear and special linear groups.
The group (5.28) is a 3-parameter group. Its structure is revealed by the following
canonical decomposition: If MM
e= 1, then there are values of scalar parameters λ, α, β in
the interval [−∞,] such that (modulo ±1)
M=KαTβDλ,(5.29)
where
Kα=eαe=1+αe,(5.30)
Tβ=eβe+=1+βe+,(5.31)
Dλ=1
2(1 + e0)λ+1
2(1 e0)λ1=eφe0(5.32)
for λ=eφ. An alternative to (5.29) is the so-called Iwasawa decomposition
M=VαTβDλ,(5.33)
where, for α= cosh θ,
Vα=α+e1(1 α2)1/2=eθe1.(5.34)
The Iwasawa decomposition has been much used in the theory of SL2(R) group represen-
tations [17]. However, it will be seen that (5.29) is more significant in the present context,
because e+,eand e0are geometrically unique generators, whereas e1is not.
17
For the matrix representation of (5.29), we get
[M]=[K
α
][Tβ][Dλ]
=·10
α1
¸·1β
01
¸·λ0
0λ
1¸
=·λβλ
1
αλ (1 + αβ)λ1¸.(5.35)
For Vλgiven by (5.34) with λ=αthis gives the matrix representation
[Vλ]=[K
α
T
β
D
λ
]=·λβλ
1
αλ λ ¸,(5.36)
where λ2=α/β =(1+αβ). This shows the equivalence of (5.29) and (5.33).
An important feature of (5.29) is that it separates distinct and unique 1-parameter sub-
groups Kα,T
β,D
λ, as well as such 2-parameter subgroups as TβDλ. Of these, it will be
seen below that only Dλbelongs to Pin(p+1,q + 1); however, Kαand Tβplay a role in
connecting Pin(1,1) to Pin(p, q).
The spin group of G(1,1) consists of
Spin(1,1) = {e0,D
λ},(5.37)
where Dλis given by (5.32). The rotor Dλrepresents the 1-parameter family of ‘Lorentz
transformations’ of V2. By (5.18),
Dλe±D1
λ=D2
λe±=λ±2e±.(5.38)
As noted in (5.25), the rotor e0represents the involution
e0e±e0=e±=e
±.(5.39)
The rest of Pin(1,1) is generated by e1, which represents the reflection
e1e±e1=e.(5.40)
The reflection represented by e2=e0e1is a composite of (5.39) and (5.40), and all other
reflections of V(1,1) cam be obtained by composites with rotations.
5.2. HOMOGENEOUS COORDINATES AND THE CONFORMAL SPLIT OF
Pin(p+1,q+1)
Like the projective split, the conformal split (5.1) can be regarded as assigning homogeneous
coordinates to the ‘points’ in Vn. There are two extra degrees of freedom in Vn+2, but one
of them can be eliminated by requiring x2= 0 for points xin Vn+2 representing points in
Vn. Then the conformal split (5.1) gives us
x2=(xe0)(e0x)=(xe
0+x·e
0
)(xe0x·e0)=0,
18
whence
(x·e0)2=(xe
0
)
2=ρ
2
x
2.(5.41)
Unique homogeneous coordinates for xwill be determined by a suitable choice of scale
factor ρ. From (5.16) we get
1
2x·e0=x·(e+e)=(x·e
+
)e
(x·e
)e
+.(5.42)
Squaring and using (5.15) and (5.41), we obtain
(x·e0)2=4x·e+x·e=ρ2x2.
Evidently, the simplest choice of scale factor is
ρ=2x·e
+,(5.43)
for then
x·e=x2x·e+.
Inserting (5.42) and (5.43) into (5.1), we get the conformal split in the form
xe0=2x·e
+
(e
+x
2
e
++2x)ρX , (5.44)
where
x=xe0
x·e+
(5.45)
expresses xin terms of ‘homogeneous coordinates’ x·e+and xe0. Itisofinterestto
note, using (5.20) and (5.21), that (5.44) has the 2 ×2 matrix representation
[X]=·xx
2
ax
¸
.(5.46)
This particular kind of split into homogeneous coordinates should be generally useful in
that part of projective geometry dealing with a null quadratic form (‘Cayley’s absolute’).
We show below that it is very useful in conformal and metric affine geometry.
To make the conformal split of the orthogonal transformation (5.10) explicit, we introduce
the abbreviation b
G=e0(G)1(5.47)
and write (5.10) in the form
GX b
G=σX0.(5.48)
For x2= 0, we can use (5.44) to put this in the explicit form
G(e+x2e++x)b
G=σ[e+[g(x)]2e++g(x)] (5.49)
where
x0=g(x)(5.50)
19
is a transformation on Vninvolving a scale change
σ=σg(x)=x
0·e
+
x
0·e
+
.(5.51)
Equation (5.49) is the main theorem relating the group Pin(p+1,q+1) = {G}to the con-
formal group C(p, q)={g}. It has been studied by Angles [11], but without the conformal
split to simplify the analysis.
To prove the main theorem, it is necessary to establish that every function gdetermined
by Gin (5.49) is conformal and that every conformal transformation can be obtained in this
way. Instead of approaching the proof directly, we achieve it as a byproduct of analyzing
the structural relationships between the two groups in detail. The analysis can be carried
out in two complementary ways.
One way is to note, by considering (5.20), that every multivector Gin Gn+2 can be
expressed in the form
G=Ae+e1+Be++Ce+Dee1,(5.52a)
where the ‘coefficients’A, B, C, D are in Gn. The operations of reversion and involution
yield
G=De+e1+Be++Ce+Aee1(5.52b)
and
G=Ae+e1Be+Ce+Aee1.(5.52c)
Equation (5.52a) differs from (5.20) only in the replacement of scalars by elements of Gn.
Since the elements of G(1,1) commute with those of Gn, this replacement applies as well to
the matrix representation (5.22). Thus, we have the isomorphism
Gn+2 =Gn⊗G(1,1) 'L2(Gn),(5.53)
where the right side denotes the algebra of 2×2 matrices with elements in Gn. Accordingly,
for the matrix representations of (5.52a,b,c) we have
[G]=·AB
CD
¸
,[G
]=·D
B
C
A
¸,[G
]=·A
B
C
D
¸.(5.54)
The matrix representation has the advantage of compactness and easy multiplication, but
the disadvantage of suppressing the relation to vectors in G(1,1) and so complicating geo-
metric interpretation. This kind of matrix representation for the conformal group has been
employed by Ahlfors [13] and Maks [14].
The requirement that Gis an element of Pin(p+1,q+ 1) puts restrictions on the coeffi-
cients in (5.52a). First, Gmust satisfy the versor condition
GG=|G|2,(5.55)
where |G|2is a nonvanishing scalar (possibly negative, unless the signature of Vnis Eu-
clidean). In the matrix representation (5.54), this condition is expressed by
[GG]=·AD+BCAB+BA
CD+DCCB+DA¸=|G|2·10
01
¸
.(5.56)
20
Second, Equation (5.10) or, equivalently, (5.49) must be satisfied. The matrix representa-
tion of (5.47) can be written
b
G=·b
Db
B
b
Cb
A¸.(5.57)
with the notation
b
A=(A
)
|G|
2.(5.58)
Then (5.49) gives us the condition
[GXG]=·(Ax+B)( b
D+xb
C)(Ax+B)( b
B+xb
A)
(Cx+D)( b
D+xb
C)(Cx+D)( b
B+xb
A)¸
=σ·g(x)[g(x)]2
1g(x)¸.(5.59)
where σis a scalar and g(x)∈V
n
. From (5.56) and (5.59) we can easlly read off a list of
necessary and sufficient conditions on A, B, C, D. In particular, from (5.59) we find that
σ=(Cx+D)( b
D+xb
C)(5.60)
and gcan be expressed in the homographic form
g(x)=(Ax+B)(Cx+D)1.(5.61)
Ahlfors [13] calls this a obius transformation. The big advantage of the versor represen-
tation in P(p+1,q + 1) over the elegant M¨obius form for a conformal transformation is
that composition of transformations is reduced to a versor product. This, in turn, simplifies
proofs and applications of a whole host of theorems, such as the canonical decomposition
theorem proved below.
To prove that (5.61) is indeed a conformal transformation, we compute its differential
g(a)= g(a,x) by taking the directional derivative. Using the fact a·∇x=a, we first
compute
a·∇(Cx+D)
1=a·∇"xb
C+b
D
σ#="ab
C(Cx+D)a·∇σ
σ#(Cx+D)
1,
whence
a·∇(Cx+D)
1=(Cx+D)
1
Ca(Cx+D)
1.(5.62)
Now,
a·∇g=[Aa(Ax+B)(Cx+D)1Ca](Cx+D)1
=(xC
+D
)
1
[(xC+D)A(xA+B)C]a(Cx+D)1,
where g=ghas been used. Applying the special conditions on the ‘coefficients’, we obtain
g(a)=a·∇g=(xb
C+ˆ
D)
1
a(Cx+D)
1.(5.63)
21
This immediately yields the defining condition for a conformal transformation:
[g(a)] ·[g(b)] = a·b
σ2.(5.64)
Thus, the proof that gis conformal is complete.
Now we turn to an alternative way to analyze the group structure, exploiting the fact
(4.26) that every versor element Gcan be generated as a product of vectors in Vn+2. First,
we determine the explicit imbedding of Pin(p, q) in Pin(p+1,q+ 1) By the condition (5.2),
V2is a subspace of Vn+2 and vectors a, b in Vn+2 are in the subspace orthogonal to V2if
a·e0=b·e0= 0. According to (5.3), then,
a=ae0=e0a(5.65)
and
ab =ab ,(5.66)
where a=ae0=ae0,b=be0are vectors in Vn. This determines how Pin(p, q) fits
into Pin(p+1,q + 1) as a subgroup. Specifically, for every versor
R=v1v2...v
k(5.67a)
in Pin(p, q), there are kunit vectors vi=vie0in Vn+2 such that
v1v2...v
k=e
0Rfor kodd, (5.67b)
v1v2...v
k=Rfor keven. (5.67c)
From (5.43) we note that
ab=ab,(5.68)
which tells us that all bivectors in Gnare also bivectors in Gn+2, even though (5.3) says
that the vectors in Gnare 3-vectors in Gn+2. This implies that the Lie algebra of Spin(p, q)
is contained in the Lie algebra of Spin(p+1,q+ 1), but we will not be using that important
fact.
Now we are prepared to systematically classify the conformal transformations generated
by vectors in Pin(p+1,q + 1). First, we readily check that the imbedding of Pin(p, q)
defined by (5.67a) generates the orthogonal group O(p, q) when employed in the main
theorem (5.49). Indeed, we find b
R=(R
)
1and
g(x)=Rx(R
)
1(5.69)
as required.
5.3. INVERSIONS, INVOLUTIONS AND DILATATIONS
Next we classify the conformal transformations generated by Pin(1,1), which is also a
subgroup of Pin(p+1,q + 1). For a reflection of Vn+2 generated by e1, using (5.40) we
obtain
e1(e+x2e++x)e1=x2(e+x2e++x1),(5.70a)
22
whence
g(x)=x
1=x
x
2.(5.70b)
Therefore, e1is a versor representation of inversion in Vn. Of course (5.70b) is not defined
along null lines x2= 0. Similarly,
e0(e+x2e++x)e0=(e+x2e+x).(5.71a)
Here σ=1, and
g(x)=x.(5.71b)
Therefore, e0is a versor representation of involution in Vn. Since e2=e0e1, it generates
the composite of (5.70b) and (5.71b). Therefore, e2represents an involutory inversion.
For Dλdefined by (5.32), with the help of (5.38) we obtain
Dλ(e+x2e++x)D1
λ=λ2[e++(λ
2
x)
2
e
+λ
2
x],(5.72a)
whence
g(x)=λ
2
x.(5.72b)
Therefore, Dλis the spin representation of a dilatation by λ2. From (5.32) it is also readily
established that every unit vector in G(1,1) with positive signature can be expressed in the
form
e=Dλe1D1
λ=e2φe0e1(5.73)
or the form e=e0ee0. It follows that G(1,1) is a double representation of the group of
inversions, involutions and dilatations on V(p, q).
5.4. TRANSLATIONS, TRANSVECTIONS, AND DIVERSIONS
Our final task is to classify the conformal transformation generated by versors with non-
vanishing components in both V2and its orthogonal complement. According to (5.1), any
such vector vcan be written in form
v=e+a, (5.74)
where e=(v·e
0
)e
0and a=ae0is inVn. Also, the versor condition
v2=e2+a26=0 (5.75)
must be satisfied, but it can be scaled to any convenient value. We have two distinct cases
to consider. Case I:e2= 0 implies a26= 0, so we can choose e=e+or e. Case II:e26=0,
so we can choose e=e1or e2, since
e+a=Dλ(e1+a)D1
λ(5.76)
by (5.73) enables us to get the other possibilitiesby composition with dilatations.
For Case Ilet us write v=e+a1and consider the rotor
Ta=av =1+ae+.(5.77)
23
Anticipating its interpretation as the spin representation of a translation, let us refer to Ta
as a translator. By (5.18) and (5.65) we have ae+=ae0e+=ae+, so we can write
Ta=1+ae
+=1+e
+
a=1+ae+=Ta(5.78)
showing that T(a)is parameterized by vectors in Vn. Applied in (5.49), this yields
Ta(e+x2e++x)Ta=e+(x+a)
2
e
++(x+a),(5.79a)
thus determining the translation
g(x)=x+a,(5.79b)
as anticipated.
The translators form a group with the properties
TaTb=Ta+b(5.80)
T1
a=Ta=e0Tae0.(5.81)
Note that if a2=b2, then a+bis a null vector. Hence, (5.80) shows that the translator
(5.78) applies to null vectors, though in that case it cannot be factored into a productof
two vectors as in (5.77). However, (5.80) shows that it can be factored into a product of
four vectors.
Now define
Ka=e1Tae1=1ae=1+ae
=K
a,(5.82)
where (5.40) and (5.18) have been used. Thus, the interchange of e+with ein (5.78) is
the equivalent of an inversion followed by a translation and another inversion, so we can
use (5.70a) and (5.79a) to evaluate
Ka(e+x2e++x)Ka=x2e1[e+(x
1+a)
2
e
++(x
1+a)]e1
=x2(x1+a)2[e+(x
1+a)
2
e
++(x
1+a)
1
].(5.83a)
In this case,
σ(x)=x
2
(x
1+a)
2=1+2a·x+a
2
x
2,(5.83b)
and
g(x)=(x
1+a)
1=x(1 + ax)1=x+ax2
σ(x),(5.83c)
This kind of conformal translation is called a transversion. This completes the analysis of
Case I.
The analysis of Case Ican be summarized as follows. From (5.77) we conclude that the
vector v=a1Ta=e++a1represents a translation-reflection. Similarly, v=a1Ka=
e+a1represents a transvection-reflection. Turning to Case II, where v=e1+a,we
first ask whether vcan be generated from e1by a translation. Computing
Tae1T1
a=e1+aa2e+,(5.84)
24
we see that the answer is yes only if a2= 0. So we exclude a2= 0 from this case to assure
that it is completely distinct from the other possibilities.
Now for Case II let us compute the conformal transformation generated by the rotor
Va=ve1=(e
1+a)e
1=1+ae1=1+ae
2.(5.85)
The simplest way to do that is to write e2=e+eand, by comparison of (5.52a) with
(5.61), read off the result
g(x)=(x+a)(1 ax)1.(5.86)
This elementary type of conformal transformation and the fact that it is continuously
connected to the identity does not seem to have been noted previously in the literature,
probably because it cannot be generated by the more obvious classical conformal transfor-
mations. Let us call it a diversion.
[On the contrary, P. Lounesto has pointed out that the diversion (5.85) can be factored
into the product Va=λTaDλKawhere λ2=1+a
2
.]
The effect of interchanging e1and e2in (5.85) is merely to change the sign of the first a
in (5.86), but this is a new kind of diversion not reducible to (5.86). According to (5.76), we
can obtain every other diversion by composition with a dilatation and its inverse. Hence,
the most general form for a diversion is
Va=1+ae, (5.87)
and we may call the diversion positive if e2=1ornegative if e2=1.
An interesting result is obtained from the following product of two diversions:
(1 + ae2)(1 + be1)=(1+ab)e+e1+(a+b)e
++(ba)e
+(1ab)ee1.(5.88)
Note that all the coefficients on the right are null if a2=b2. This example was first ad-
duced by Maks [14] as a counterexample to the common belief that every special conformal
transformation can be generated by rotations, translations, inversions and dilatations [in
any order] . However, we have shown above that the diversion is a more fundamental coun-
terexample, and its existence does not require an indefinite metric. [Delete the preceding
sentence.] This concludes our analysis of Case II.
5.5. THE CONFORMAL DECOMPOSITION THEOREMS
The main theorem (5.49) establishes the double covering of the conformal group C(p, q)by
the versor group Pin(p+1,q + 1). Similarly, Spin(p+1,q+ 1) is a double covering of the
special conformal group SC(p, q), consisting of all conformal transformations continuously
connected to the identity. The fundamental decomposition theorem for SC(p, q) can be
expressed as follows: The rotor Grepresenting any combination of rotations, translations,
transversions, and dilatations can [usually] be put in the canonical form
G=±KaTbDλR, (5.89)
where the factors are defined by (5.82), (5.78), (5.32) and (5.67c). [The counterexample of
Maks shows that the order of factors may matter in special cases.]
25
This result is easily proved with the help of the following relations to reorder and combine
factors
DλTa=Ta0Dλwhere a0=λ2a, (5.90)
RTa=Ta0Rwhere a0=R(R)1.(5.91)
From our preceding analysis, it follows that any element of SC(p, q) can be represented as
a product of (5.89) with the positive and negative diversions represented by (5.87).
The most general decomposition theorem for C(p, q) asserts that any element can be
represented in Pin(p+1,q +1)byversorsoftheformVG,e
0
VG,e
1
VG or e0e1VG,
where Vis a diversion and Ghas the form (5.89), but with Rof the general form (5.67a)
representing an orthogonal transformation.
Subgroups of the conformal group can be represented in geometric algebras of lower
dimension. The group Spin(p+1,q+ 1) belongs to the even subalgebra G+
n+2 =Gn+1,so
its action on vectors can be represented in Gn+1 by a projective split. Under a projective
split by e2, Equation (5.44) gives
xe2=xe0e1=1
2(1 e0)+1
2(1 + e0)x2
1+x1,(5.92a)
where the scale has been fixed by setting x·e+= 1, and the projected point in Vnis now
represented by
x1=(xe
0
)·e
1=xe
1+(x·e
1
)e
0.(5.92b)
Defining
G=e1b
Ge1=e2(G)1e2,(5.93a)
we multiply (5.49) by e1and use (5.92a) to put it in the form
G[1
2(1 e0)+1
2(1 + e0)x2
1+x1]G
=σ£1
2(1 e0)+1
2(1 + e0)[g(x1)]2+g(x1)¤.(5.93b)
This is the representation in Gn+1. Here it is understood that Gis in Spin(p+1,q+1), so g
is in SC(p, q). Equation (5.93b) is the conformal representation obtained by Lounesto and
Latvamaa [12]. Evidently it has no advantage over (5.49). It is mentioned here to show
how alternative representations are related.
The metric affine group A(p, q) is a particularly important subgroup of C(p, q). It is the
group of orthogonal transformations and translations of Vn=V(p, q), so it is represented
by versors in the canonical form
S=TaR. (5.94)
It is represented in the orthogonal group of Vn+2 as the stability subgroup of the null vector
e+. That is to say that Sbelongs to the subgroup of versors satisfying
Se+ˆ
S=e+.(5.95)
Since e+commutes with S, we can simplify (5.49) for this subgroup by multiplying it by
e+and using (5.13) and (5.14) to get
S(1 + e0+xe+)ˆ
S=1+e
0+g(x)e
+.
26
Note that the scale factor σ= 1 is fixed in the subgroup. Since
Se0ˆ
S=e0,
we have the further reduction to the simple canonical form
S(1 + xe+)ˆ
S=1+g(x)e
+,(5.96a)
where ˆ
S=b
RTa=(R
)
1
T
a(5.96b)
is the spin representation of
g(x)=Rxb
R+a,(5.96c)
the general form for a metric affine transformation.
Note that the entire group representation (5.96a,b,c) can be obtained formally from Gn
by extending the algebra to include an element e+which commutes with all other elements
and satisfies e2
+= 0. Clifford himself extended the quaternions in exactly that way to
describe the Euclidean group of rigid rotations and translations in three dimensions [18].
His unfortunate early death prevented him from integrating that idea into his geometric
extension of Grassman algebra; there can be little doubt that he would have done so. Here
we see that the appropriate integration of Clifford’s great ideas entials the identification of
e+as a vector in a space of higher dimension. This is more appropriate than alternative
representations of the affine group in geometric algebra, because of its embedding in the
full conformal group.
The affine group representation (5.96a,b,c) has important applications to physics. When
the base algebra Gnis the spacetime algebra G(1,3), it is a representation of the Poincare
group. It is also of interest to note that the factorization
G(1,3) = G(2,0) ⊗G(1,1) (5.97)
implies that the Lorentz group for spacetime is isomorphic to the conformal group for
Euclidean 2-space.
References
1. Hestenes, D. and Ziegler, R.: Projective geometry with Clifford algebra, Acta Appl.
Math. 23 (1991), 25–63.
2. Hestenes, D.: Universal geometric algebra, Simon Stevin 62 (1988), 253–274.
3. Hestenes, D. and Sobczyk, G.: Clifford Algebra to Geometric Calculus, D. Reidel,
Dordrecht, 1985.
4. Whitehead, A. N.: A Treatise on Universal Algebra with Applications, Cambridge
University Press, Cambridge, 1898 (Reprint: Hafner, New York, 1960), p. 249.
5. Clifford, W. K.: Applications of Grassmann’s Extensive Algebra, Amer. J. Math.1
(1878), 350–358.
6. Hestenes, D.: Spacetime Algebra, Gordon and Breach, New York, 1966, 1987.
7. Porteous, 1.: Topological Geometry, 2nd edn., Cambridge University Press, Cam-
bridge, 1981.
27
8. Salingaros, N.: On the classification of Clifford algebras and their relations to spinors
in ndimensions, J. Math. Phys.23 (1982), 1.
9. Li, D., Poole, C. P., and Farach, H. A.: A general method of generating and classifying
Clifford algebras, J. Math. Phys.27 (1986), 1173.
10. Dimakis, A.: A new representation for spinors in real Clifford algebras, in J. S. R.
Chisholm and A. K. Common (eds.), Clifford Algebras and their Applications in Math-
ematical Physics, D. Reidel, Dordrecht, 1986.
11. Angles, P.: Construction de revetements du groupe conforme d’un espace vectoriel
muni d’une metrique de type (p, q), Ann. Inst. Henri Poincare 33 (1980), 33.
12. Lounesto, P. and Latvamaa, E.: Conformal transformations and Clifford algebras,
Proc. Amer. Math. Soc. 79 (1980), 533.
13. Ahlfors, L. V.: Clifford numbers and M¨obius transformations in Rn,inJ.S.R.
Chisholm and A. K. Common (eds.), Clifford Algebras and their Applications in Math-
ematical Physics, D. Reidel, Dordrecht, 1986.
14. Maks, J. G.: Modulo (1,1) periodicity of Clifford algebras and generalized (anti-)
obius transformations, Thesis, Delft University of Technology, The Netherlands.
15. Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras, Kluwer Academic
Pubs., Dordrecht, 1990, Chap. 12.
16. Kastrup, H. A.: Gauge properties of the Minkowski space, Phys. Rev.150 (1966),
183.
17. Lang, S.: SL2(R), Addison-Wesley, Reading, Mass., 1975.
18. Clifford, W. K.: Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. IV
(1973), 381–395.
28
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