General Spinor Representation and Clifford Algebras – Framework

Abstract

The Clifford Algebra language for describing the immersion of a (pseudo-)Riemannian manifold of dimension p in R^n, n>p is developed and shown to be the adequate one. Any signature of these spaces is possible by choosing the corresponding signature of the Clifford algebra, thus the spinor representation is generalised to any signature and extended to any dimension. Then is investigated whether the Dirac equation can be introduced too. Some results are given, especially for a hypersurface the equivalence with the Dirac equation is proven, and for the other cases a weaker solution is found. The extended Gauß map and its stereographic projection is also worked out in the Clifford language, and it is shown how it could be lifted. Finally all the immersion formulæ known so far are derived. The application of this framework to the multidimensional inverse scattering transform is discussed in the conclusion.

Full text

GENERAL SPINOR REPRESENTATION AND
CLIFFORD ALGEBRAS - FRAMEWORK
Claude Pierre MASSÉ
Laboratoire de Physique Fondamentale, FRANCE
© Claude Pierre MASSÉ
The Cliord Algebra language for describing the immersion of a
(pseudo-)Riemannian manifold of dimension in is developed and shown to
be the adequate one. Any signature of these spaces is possible by choosing the
corresponding signature of the Cliord algebra, thus the spinor representation is
generalised to any signature and extended to any dimension. Then is investigated
whether the Dirac equation can be introduced too. Some results are given, especially for
a hypersurface the equivalence with the Dirac equation is proven, and for the other cases
a weaker solution is found. The extended Gauß map and its stereographic projection is
also worked out in the Cliord language, and it is shown how it could be lifted. Finally
all the immersion formulæ known so far are derived. The application of this framework
to the multidimensional inverse scattering transform is discussed in the conclusion.
Latest version: https://phy.clmasse.com/spin-clifford-framework.html
Introduction
The embedding or immersion of a surface in a at Euclidean space of at least
dimensions, both locally (in the small) and globally (in the large,) is a well understood topic of
geometry. It has been already studied in the classical period where the most important concepts
and nonlinear equations have been introduced. Recently the old Weierstraß-Enneper induction
formula has been refurbished and generalised by using the more modern Dirac equation [K96].
This being, not much progress have been done in extending the immersion to manifolds of
higher dimensions, a prerequisite to address realistic problems in physics. As will rapidly be
evident, this is due to formidable algebraic barriers, that need to be tackled with more powerful
calculational tools. Cliord algebras provide such tools, as shown here.
Konopelchenko’s formulation is basically analytic, and uses two complex variables. The
idea is that instead of an element of the minimal left ideal, the “ spinor ” satisfying the Dirac
equation should be taken in the sense of Hestenes’ operator spinor as an element of the special
Cliord group associated to the ambiant space [BLR’17, V’19]. For the simplest case of a surface
p
R
n
,
n
>
p
3

in the special Cliord group is isomorphic to or to the set of non-singular
matrices of the form
Taking as the matrix representation of the Cliord generators
it is rewriten as
It is but a compacted version, that we expand by using the isomorphism
generated by
where is the even subalgebra of Then really is an element of representing a mere
rotation.
So similarly as a surface is immersed in an Euclidean space of greater dimension, a Cliord
algebra is immersed in another algebra of higher order. It is a local homomorphism of Cliord
algebras realising a Cliord bundle. As all the developments will be local, issues like the
existence of a spin structure do not arise, and the related, somewhat opaque formalism is
bypassed.
Contrary to the traditional presentation order, it will be more direct to begin from the
end. We shall also be very general, the known special cases are given in the last section only as a
result. We begin by a compendium about the Cliord algebras in order to make available all
that is necessary to use them here §1. Then Cartan’s method of the moving frame is spelled out
and adapted to the Cliord algebras §2. The most important part is the thorough investigation
of the use of the Dirac equation for the representation of a submanifold by an immersion in an
Euclidean space §3. A essential concept in these problems is the Gauß map, that is worked out
and extended in §4. The last section is devoted to recover the concrete formula for particular
cases that are found in the literature §5. Finally we conclude by an outlook of the possible
application and open problems, especially for nonlinear partial dierential equations.
§1. CLIFFORD algebras
The strenuous eort of Hamilton to “ multiply vectors ” eventually led to one of the most
imitated mathematical structure : from Cliord original proposition, passing by the multenions
R
3, Γ0
3,0
R
∗⊗
SU
(2),
(
φ
−¯
χ
χ
¯
φ
)
,
φ
,
χ
∈
C
.
E
1=
i
(
0 1
1 0
)
,
E
2=
i
(
0 −
i
i
0
)
,
E
=
(
1 0
0 1
)
,
Ψ =
(
φ
′+
iφ
′′ −
χ
′+
iχ
′′
χ
′+
iχ
′′
φ
′−
iφ
′′
)
=
φ
′
E
+
χ
′′
E
1−
χ
′
E
2−
φ
′′
E
12 =
ψIEI
∈
C
ℓ2,0,
ψI
∈
R
.
C
ℓ
n
−1,0 ↪
C
ℓ0
n
,0,
Ei
↦
Eni
,
C
ℓ0
C
ℓ. Ψ Γ0
3,0

and multivectors, to Dirac matrices and Eddington’s -numbers, until fully exploited by
Hestenes. True to the dream of Hamilton, the Cliord algebras provides all the tools to handle
geometrical problems. Vectors represent the additive property of geometry, they see it as a
juxtaposition of parallelipipeds (or parallelitopes,) always in the same dimension. Cliord
algebras in contrast address the multiplicative properties. Their basic object is the sphere, and by
combining, or more precisely multiplying them, get spheres of higher dimensions. That is why
they are particularly well adapted to representations of rotation groups, be they real, complex, or
symplectic, and then to geometry in the philosophy of Klein’s Erlangen program.
We won’t need the full abstract theory of Cliord algebras here. It is presented in many
texts, the author recommends [L’01] for a pedagogical account and more details. Only what is
necessary in this memoir is presented. We consider real Cliord algebras associated to a bilinear
form in As the form can be made diagonal by a linear substitution of we restrict
ourselves to diagonal bilinear forms with or as elements. For plus signs and minus
signs, the Cliord algebra is denoted by and is called the signature. For the
sake of notation simplicity, in such context we write for a dimension with an unspecied
signature. We adopt the common convention in dierential geometry that the basis elements
of that correspond to a plus sign satisfy Lowercase indices denote indices
in while uppercase ones denote indices in or equivalently, compound indices that
contain several, one, or no single indices in
A Cliord algebra like the ones we use is generated multiplicatively by a set of
generating elements like etc. The Cliord product in noted simply by
juxtaposition, like matrices product. The algebra is closed by two types of relations :
The sign depends on the signature. A basis element of the full Cliord algebra can therefore
have a maximum of indices. A generic element of a Cliord algebra, or Cliord number, can
thus be written
An involutive anti-automorphism called reverse is dened in the following way :
That simply means the reverse of is Consequently we have
where is the number of distinct single indices in called the grade. For the practical
calculations it is useful to know that the sign change by reversion depends only on the grade
modulo and
In order to be able to make calculations that are valid for any signature, we dene
contravariant and covariant vector as usual:
E
R
n
.
R
n
,
+1 −1
n
+
n
−
C
ℓ
n
+,
n
−, (
n
+,
n
−)
n
Ei
C
ℓ
n
+,
n
−
EiEi
= −
E
.
R
n
,
C
ℓ
n
,
R
n
.
n
E
1, ⋯ ,
En
,
E
1
E
2=
E
12
EiEj
= −
EjEi
,
EiEi
= ±
E
. (1.1)
n
C
=
cE
+
ciEi
+
cijEij
+ ⋯ (1.2)
(
cKEK
)~
=
cK
~
EK
,~
~
EI
=
EI
,~
Ei
=
Ei
, (
EIEJ
)~
=~
EJ
~
EI
(1.3)
EiEj
⋯
EmEm
⋯
EjEi
.
~
EI
= (−)|
I
|(|
I
|−1)/2
EI
= (−)[|
I
|/2]
EI
(1.4)
|
I
|
I
,
4,
~
E
=
E
,~
Ei
=
Ei
,~
Eij
= −
Eij
,~
Eijk
= −
Eijk
. (1.5)

where the indicator is the -th diagonal element of the bilinear form, so that
Similarly, compound indices look like
so that in the same way
Beware the conventions for are dierent from usual for the Minkowski metric, in particular
there is never summation in its index.
We use some short hand notations :
are the component and the grade of a Cliord number.
are the unit pseudo scalars of and while
For numerals we use the special greek letter e.g.
The grade
has the properties of the trace. It is linear, and
while for all
The most important structure of the Cliord algebras for us, and that generalises the
spinor is the Cliord, or Lipschitz group It is the subset of the elements that transforms a
vector into another vector as
ηi
= −
EiEi
,
Ei
=
ηiEi
, (1.6)
ηii
EiEi
=
EiEi
= −
E
. (1.7)
EI
=
E
⋅
jk
i
⋅⋅ℓ,
EI
=
E
ℓ⋅⋅
i
⋅
kj
= (−)|
I
|(|
I
|−1)/2
ηiηjηkη
ℓ
EI
=:
ηIEI
,(1.8)
EIEI
=
EIEI
= (−)|
I
|
E
. (1.9)
η
(
cKEK
)
I
=
cI
,
⟨
C
⟩
r
=
∑
|
I
|=
r
(
C
)
IEI
(1.10)
I r
EP
=
E
1⋯
p
,
EN
=
E
1⋯
n
(1.11)
C
ℓ
pC
ℓ
n
,
EQ
=
Ep
+1⋯
n
. (1.12)
ι
,
Eι
4=
E
1234 (1.13)
0
⟨
C
⟩
:=
⟨
C
⟩
0(1.14)
⟨
AB
⟩
=
⟨
(
aIEI
)(
bJEJ
)
⟩
=
aIbJ
⟨
EIEJ
⟩
=
bJaI
⟨
EJEI
⟩
=
⟨
BA
⟩
, (1.15)
⟨
ABC
⟩
=
⟨
CAB
⟩
=
⟨
BCA
⟩
, (1.16)
r
,
⟨
~
C
⟩
r
= (−)
r
(
r
−1)/2
⟨
C
⟩
r
. (1.17)
Φ, Γ
n
.
V
=
viEi
V
′=
v
′
iEi

The inverse comes rst because we reserve the symbol for the wave function. But that is only
a matter of convention, the mathematics are the same by swapping since it must be
inversible from the denition. Though, at variance with the traditional twisted adjoint
representation, we use a slightly dierent one, that has the advantage of being more faithful :
Since this introduces an additional scaling. Finally the subgroup dened by
is the double cover of Actually, only the special Cliord group is
used, that is, the elements that belong to the even sub algebra of the Cliord algebra. For this
group, twisted and untwisted representations are identical. The reason is that the rotations have
to be arbitrary close to the identity, which is necessary to describe a continuous surface.
Similarly, a spinor formalism is used instead of a simpler vector one in order to represent non
orientable surfaces. A local change of orientation can be performed in the ambient space by a
proper rotation, while an improper one is required in the surface.
§2. CARTAN moving frame
In all the following, greek indices are used for the index range Latin ones
for while middle Latin ones are used for the full range
Further or overriding conventions are specied where they are used.
2A. Classical formulation
The coordinates of a simply connected domain of a manifold of dimension are noted
by is immersed in by the function
To each point of the immersed space, that is for each value of is attached an
orthogonal frame of In order to have a one-to-one correspondance with the
Cliord group, the basis vectors are not normalised to unity, but they all have the same length,
that is
The operation is the associated bilinear form of and is a real function over The rst
vectors are tangent to the immersed manifold, and the remaining ones are therefore
normal to it. We formally write
The frame eld is dierentiable, but instead of expressing the dierentials in a xed basis,
they are expressed in the moving frame itself [C’01]. Dierential -forms over are dened
V
′= Ψ−1
V
Ψ. (1.18)
Ψ
Ψ−1 ↔ Ψ
Ψ(
V
) = ~
Ψ
V
Ψ. (1.19)
~
ΨΨ =:
ρE
,
~
ΨΨ =
E Spin
(
n
),
SO
(
n
). Γ0
n
1, 2, ⋯ ,
p
,
a
,
b
, ⋯ ,
r
,
s
, ⋯
p
+ 1, ⋯ ,
n
,
i
,
j
,
k
, ℓ
1, 2, ⋯ ,
n
.
D
p
ξμ
, 
μ
= 1, ⋯ ,
p
.
D
R
n
x
:
D
→
R
n
,
ξ
= (
ξ
1, ⋯ ,
ξp
) ↦
x
(
ξ
) = (
x
1, ⋯ ,
xn
). (2A.1)
(
ξ
1, ⋯ ,
ξp
),
(
e
1, ⋯ ,
e
n
)
t
R
n
.
e
i
⋅
e
j
=
ρ
2
δj
i
. (2A.2)
⋅
R
nρ
D
.
p
q
=
n
−
p
n
=
p
⊕
q
.
1
ω
D

through the structure equations of the space ( )
which, multiplying by on the right, yields
Here we have
Dierentiating the orthonormality conditions (2A.2) we get
Then assuming the equations (2A.3ab) are integrable, that is, is closed and the frame
eld denes a -dimensional immersed manifold, by dierentiating them and substituting the
expression of wherever possible, we get compatibility equations, called the structure equations
of the group of motions by Cartan :
Dierentiating further gives nothing new, unless the intrinsic and extrinsic part are kept
separate. Since there is no torsion. The structure equations (2A.7ab) are written as
in which is the curvature -form. Through dierentiation of these equations,
substituting the dierentials by their expression in the same equations, we get
These are the Bianchi identities. Similar identities are obtained in the normal space by dening a
second -form :
that is
d = d
ξμ
∂
μ
{
d
x
=
ωi
e
i
,
d
e
i
=
ωk
i
e
k
,(2A.3ab)
e
j
{
ωj
=
ρ
−2d
x
⋅
e
j
,
ωj
i
=
ρ
−2d
e
i
⋅
e
j
.(2A.4ab)
ωa
= 0. (2A.5)
ωj
i
+
ηijωi
j
= 2
ρ
−1d
ρ

δj
i
(no sum.) (2A.6)
d
x
p
d
e
i
{
d
ωi
=
ωμ
∧
ωi
μ
d
ωi
j
=
ωk
j
∧
ωi
k
.(2A.7ab)
ωa
= 0,
{
d
ωμ
=
ωλ
∧
ωμ
λ
d
ων
μ
=
ωλ
μ
∧
ων
λ
+
θν
μ
,(2A.8ab)
θν
μ
=
ωa
μ
∧
ων
a
2
d
ω
{
ωλ
∧
θν
λ
= 0
d
θν
μ
−
ωλ
μ
∧
θν
λ
+
θλ
μ
∧
ων
λ
= 0 =
d
θν
μ
.(2A.9ab)
2
τ
d
ωb
a
=
ωc
a
∧
ωb
c
+
τb
a
, (2A.10)
τb
a
=
ωμ
a
∧
ωb
μ
, (2A.11)
d
τb
a
−
ωc
a
∧
τb
c
+
τc
a
∧
ωb
c
= 0 =
d
τb
a
. (2A.12)

The -form actually corresponds to the Ricci tensor, and is called the Riemannian or Gaußian
torsion. (Here they are the same because the ambient space is at.) The symmetries of these forms
are
The fundamental forms of the submanifold are given as usual by the dierential forms
2B. CLIFFORD formulation
Now we go over to the Cliord algebra. The vectors are replaced by the corresponding
vectors of the algebra, that is :
Moreover, the frame is expressed in the standard basis through a function
with values in the special Cliord group :
The system becomes
In the last equation (2B.3b), there is still that pertains to the vector representation. In
fact it is the right invariant Maurer-Cartan form of the group and thus is an
element of the Lie algebra Its corresponding spinor representation is the right
invariant Maurer-Cartan form of the special Cliord group, or belonging to the
Lie algebra that is
To work out this expression, let us write
2
θ τ
θν
μ
+
ημνθμ
ν
, (2A.13)
τb
a
+
ηabτa
b
, (2A.14)
I = d
x
⋅ d
x
=
ρ
2
∑
μημωμωμ
, (2A.I)
II
a
= d(
ρ
−1
e
a
) ⋅ d
x
=
ρ
∑
μημωμ
aωμ
, (2A.II)
III
ab
= d(
ρ
−1
e
a
) ⋅ d(
ρ
−1
e
b
) =
∑
μωμ
aωμ
b
. (2A.III)
{
x
→
xiEi
=
X
e
i
→
ej
iEj
=
E
i
(2B.1ab)
(
E
1, ⋯ ,
En
) Ψ(
ξ
)
Γ0
n
E
i
=~
Ψ
Ei
Ψ. (2B.2)
{
d
X
=
ωμ
E
μ
d
E
i
=
ωk
i
E
k
(2B.3ab)
ω
R
∗⊗
SO
(
n
),
r
⊕
so
(
n
).
R
∗⊗
Spin
(
n
)
r
⊕
spin
(
n
),
Z
=
ρ
−1dΨ ~
Ψ = dΨΨ−1. (2B.4)
Ψ =
ρ
1/2 ^
Ψ, (2B.5)

where Inserting it in the expression we get
Now is a bivector. We denote it by Finally
Now the second structure equation of the group of motions (2B.3b) reads
Multiplying on the left by and on the right by it becomes
As is a bivector this is equivalent to
The commutator does not cancel only for the components of in or which give terms
in so that in the whole, equating the coecients of each we have
The second structure equation of the group of motions (2B.3b) is therefore
From here on, we no longer explicitely write conditions like they are implied and is
not dened. In pure spinor notation, this equation is equivalent to the obvious relation
also known as the Gauß-Weingarten equation. From the compatibility condition of the system
as usual by cross dierentiating, the scalar terms cancel and we get the so-called zero-curvature
conditions
^
Ψ ∈
Spin
(
n
).
Z
,
ρ
−1dΨ ~
Ψ =
ρ
−1d
ρ
+ d^
Ψ^
Ψ−1. (2B.6)
1
2
d^
Ψ^
Ψ−1 ∈
spin
(
n
) = span(
E
⋅
j
i
/2) ^
Z
.
Z
=
ρ
−1dΨ ~
Ψ = d
u
+^
Z
=:
ρ
−1d
ρ
+
∑
i
≠
j
zj
iE
⋅
i
j
. (2B.7)
1
2
1
2
1
2
d(~
Ψ
Ei
Ψ) = d~
Ψ
Ei
Ψ + ~
Ψ
Ei
dΨ =
ωk
i
~
Ψ
Ek
Ψ. (2B.8)
ρ
−1Ψ
ρ
−1 ~
Ψ
(
ρ
−1dΨ ~
Ψ)~

Ei
+
Ei
(
ρ
−1dΨ ~
Ψ) = ~
ZEi
+
EiZ
=
ωk
iEk
. (2B.9)
^
Z
ρ
−1d
ρEi
− [ ^
Z
,
Ei
] =
ωk
iEk
. (2B.10)
^
Z E
⋅
j
iE
⋅
i
j
,
Ej
,
Ei
2
zj
≠
i
i
=
ωj
i
2
Z
=
ρ
−1d
ρ
+
∑
i
≠
j
ωj
iE
⋅
i
j
.(2B.11ab)
1
2
d
E
i
=
ρ
−1d
ρ
E
i
+ 2
zk
≠
i
i
E
k
.(2B.12)
k
≠
i
,
zi
i
dΨ =
Z
Ψ, (2B.GW)
{
∂
α
Ψ =
Zα
Ψ
∂
β
Ψ =
Zβ
Ψ(2B.13ab)
∂
α
^
Zβ
− ∂
β
^
Zα
= [ ^
Zα
,^
Zβ
], (2B.14)

or in components,
There are sets of them. In fact, expanding the second structure equation of the group of
motion (2A.7b), and substituting for we recover this same equation, it is the spinor
expression of the integrability. This derivation is parallel to the one of the second structure
equation, the cross dierentiation corresponding to the exterior dierentiation.
The same symmetry relations as for can be derived directly for As is a bivector we
have
then
so that
For separating the extrinsic and intrinsic parts, we use the involution represented by
and we dene
satisfying
and from the Jacobi identities
Using these commutation relations, the zero curvature condition (2B.14) splits into
The rst of these equations (2B.GR) contains the Gauß and the Ricci equations, while the
second one (2B.CM) contains the Codazzi-Mainardi equations. They are automatically satised
whenever is derived from a function so that in this case the integrability condition is
only concentrated in the rst structure equation of the group of motions (2A.7a), which is part
of the Gauß equations. This is the strength of the moving frame method.
∂
αzj
i
/
β
− ∂
βzj
i
/
α
= 2
zk
i
/
αzj
k
/
β
− 2
zk
i
/
βzj
k
/
α
. (2B.15)
( )
p
22
z ω
,
ω Z
.^
Z
Z
+~
Z
=
ρ
−1d
ρ
, (2B.16)
zj
i
+
ηijzi
j
= 0,
i
≠
j
, (2B.17)
zj
iE
⋅
i
j
=
zi
jE
⋅
j
i
, (2B.18)
EP
,
^
Z
+= { ^
Z
,
EP
}
E
−1
P
,
Z
+=^
Z
++ d
u
,
Z
−=^
Z
−= [
Z
,
EP
]
E
−1
P
(2B.19)
1
21
21
2
Z
=
Z
++
Z
−, (2B.20)
[^
Z
+,
EP
] = [
Z
+,
EP
] = { ^
Z
−,
EP
} = 0, (2B.21)
[[ ^
Z
+,^
Z
+],
EP
] = [[ ^
Z
−,^
Z
−],
EP
] = {[ ^
Z
+,^
Z
−],
EP
} = 0. (2B.22)
∂
α
^
Z
+
β
− ∂
β
^
Z
+
α
= [ ^
Z
+
α
,^
Z
+
β
] + [ ^
Z
−
α
,^
Z
−
β
], (2B.GR)
∂
α
^
Z
−
β
− [ ^
Z
+
α
,^
Z
−
β
] = ∂
β
^
Z
−
α
− [ ^
Z
+
β
,^
Z
−
α
]. (2B.CM)
Z
Ψ(
ξ
),

The compound curvature and torsion tensor is dened by
where made of all the terms in corresponds to and made of all the terms in
corresponds to We have the usual expression
so that the Jacobi identity
directly gives the second Bianchi identity
which is in two independent parts
if is made of the terms in and of the terms in
Finally, the fundamental forms become
§3. DIRAC equation
Now we investigate the relation between an immersed submanifold into an Euclidean
space and a Dirac-like equation, using the moving frame in the language of Cliord algebras.
3A. Conformal submanifold
The purpose is to extend the known results for a surface [I] to any immersed -manifold
with The former case is much simpler because every regular surface admits a conformal
Rαβ
+
Fαβ
:= −∂
α
^
Z
+
β
+ ∂
β
^
Z
+
α
+ [ ^
Z
+
α
,^
Z
+
β
] = −[ ^
Z
−
α
,^
Z
−
β
], (2B.23)
Rαβ E
⋅
ν
μθαβ
,
Fαβ E
⋅
b
a
ταβ
.
Rαβ
+
Fαβ
= [∂
α
−^
Z
+
α
, ∂
β
−^
Z
+
β
], (2B.24)
[∂
γ
−^
Z
+
γ
,
Rαβ
+
Fαβ
] + [∂
α
^
Z
+
α
,
Rβγ
+
Fβγ
] + [∂
β
−^
Z
+
β
,
Rγα
+
Fγα
] =
0
(2B.25)
∂
γ
(
Rαβ
+
Fαβ
) + ∂
α
(
Rβγ
+
Fβγ
) + ∂
β
(
Rγα
+
Fγα
)
= [ ^
Z
+
γ
,
Rαβ
+
Fαβ
] + [ ^
Z
+
α
,
Rβγ
+
Fβγ
] + [ ^
Z
+
β
,
Rγα
+
Fγα
],
(2B.B2)
∂
γRαβ
− [ ^
Z
∥
γ
,
Rαβ
] + ∂
αRβγ
− [ ^
Z
∥
α
,
Rβγ
] + ∂
βRγα
− [ ^
Z
∥
β
,
Rγα
] =
0
, (2B.B2r)
∂
γFαβ
− [ ^
Z
⊥
γ
,
Fαβ
] + ∂
αFβγ
− [ ^
Z
⊥
α
,
Fβγ
] + ∂
βFγα
− [ ^
Z
⊥
β
,
Fγα
] =
0
,(2B.B2f)
^
Z
∥
E
⋅
ν
μ
^
Z
⊥
E
⋅
b
a
.
I = −
⟨
d
X
d
X
⟩
=
ρ
2
∑
μημωμωμ
, (2B.I)
II
a
= −
⟨
d(
ρ
−1
E
a
)d
X
⟩
= 2
ρ
∑
μημzμ
a
/
α
d
ξαωμ
, (2B.II)
III
ab
= −
⟨
d(
ρ
−1
E
a
)d(
ρ
−1
E
b
)
⟩
= 4
∑
μzμ
a
/
αzμ
b
/
β
d
ξα
d
ξβ
. (2B.III)
p
p
> 2.

parameterisation, so that the -forms are constant multiples of This is not true in higher
dimensions. We then begin with an immersed conformal manifold.
Given a -dimensional conformal manifold immersed in with a conformal
parameterisation of it by we have a moving frame dened up to a rotation in
the normal space. The function is then given. In this section, indices like
are all distinct.
The dierential forms are chosen as
where is rst taken arbitrary. The remaining forms are obtained from the rst structure
equations of the group of motions (2A.7a) that expands to
in addition to
from (2A.6). Comparing the coecients of the independent basis -forms
immediately gives
Then from (3A.4) we further deduce
by making use of the relation (2A.6) Indeed this is shown by taking the cyclic
permutation of the indices and subtracting two of the obtained equations from the third, or by
exchanging the indices alternatively in six steps.
We proceed by calculating
The term in vanishes by symmetry, and we insert the expressions (3A.5, 3A.7)
1
ωμ
d
ξμ
.
p
R
n
(
ξ
1, ⋯ ,
ξp
),
e
i
,
Ψ :
D
→ Γ0
nα
,
β
,
γ
, …
[
ωμ
=
ρ
′d
ξμ
,
ωa
= 0, (3A.1)
ρ
′
⎡
⎣
ρ
′−1∂
νρ
′d
ξν
∧ d
ξμ
=
ωμ
ν
/
α
d
ξν
∧ d
ξα
,
0 =
ωa
ν
/
α
d
ξν
∧ d
ξα
,(3A.2)
ωμ
μ
/
α
=
ρ
−1∂
αρ
. (3A.3)
2 d
ξα
∧ d
ξβ
ωγ
α
/
β
=
ωγ
β
/
α
, (3A.4)
ωβ
α
/
β
= (
ρρ
′)−1∂
α
(
ρρ
′), (3A.5)
ωa
α
/
β
=
ωa
β
/
α
. (3A.6)
ωγ
α
/
β
= 0 (3A.7)
ωβ
α
= −
ηαβωα
β
.
/∂Ψ :=
Eμ
∂
μ
Ψ =
Eμ
[
ρ
−1∂
μρ
+
zj
i
/
μE
⋅
i
j
]
Ψ
=
Eα
[
ρ
−1∂
αρ
+
zα
β
/
αE
⋅
β
α
+
zα
a
/
αE
⋅
a
α
+
zβ
γ
/
αE
⋅
γ
β
+
zβ
a
/
αE
⋅
a
β
+
zb
a
/
αE
⋅
a
b
]
Ψ.
(3A.8)
1
2
1
21
21
2
zβ
a
/
α

The contraction of the second term yields a term similar to the rst one, so that counting
carefully we get
Now if we set
we nd the Dirac-like equation
and the -forms
The submanifold so represented is then parameterised as
where stands for an arbitrary path form to
3B. Conserved currents
The standard procedure is used to get the conserved current. We multiply the Dirac
equation (3A.15) on the left by then by and in addition we take the reverse of the
equation thus obtained :
The tilde on is dropped because it is a mere change of sign. As commutes with both
and either adding or subtracting these equations according to the parity of gives
/∂Ψ =
{
Eα
[
ρ
−1∂
αρ
+ (
ρρ
′)−1∂
β
(
ρρ
′)
E
⋅
β
α
+
zb
a
/
αE
⋅
a
b
]
−
∑
αzα
a
/
αEa
}
Ψ. (3A.9)
1
21
21
2
/∂Ψ =
{
Eμ
[
(2 −
p
)
ρ
−1∂
μρ
+ (1 −
p
)
ρ
′−1∂
μρ
′+
zb
a
/
μE
⋅
a
b
]
−
∑
μzμ
a
/
μEa
}
Ψ. (3A.10)
1
21
21
2
ρ
′=
ρ
(
p
−2)/(1−
p
), (3A.11)
Aα
:=
zb
a
/
αE
⋅
a
b
, (3A.12)
1
2
M
:=
∑
μzμ
a
/
μEa
=:
maEa
, (3A.13)
{
Eμ
[
E
∂
μ
−
Aμ
] +
M
}
Ψ =
0
, (3A.14)
1
ωμ
=
ρ
(
p
−2)/(1−
p
)d
ξμ
. (3A.15)
R
(
ξ
) =
R
(
ξ
0) +
∫
Γ
ρ
(
p
−2)/(1−
p
)(~
Ψ
Eμ
Ψ)d
ξ
′
μ
, (3A.16)
Γ
ξ
0
ξ
.
EP
~
Ψ,
~
Ψ
EP
{
Eμ
[∂
μ
−
zb
a
/
μE
⋅
a
b
] +
maEa
}Ψ =
0
, (3B.1a)
1
2
~
Ψ{[←
∂
μ
+
zb
a
/
μE
⋅
a
b
]
Eμ
+
maEa
}
EP
Ψ =
0
. (3B.1b)
1
2
EPE
⋅
b
aEμ
EP
,
p
~
Ψ
EPEμ
(∂
μ
Ψ) + (∂
μ
~
Ψ)
EPEμ
Ψ =
0
, (3B.2)
∂
μ
(~
Ψ
EPEμ
Ψ) =
0
. (3B.2’)

Dening the tangent pseudo-vector currents
we have the continuity equations
They are the same as the Noether currents associated to the right multiplication of by an
element of the Cliord group, but we don’t bother to write a Lagrangian. From this
conservation law, as usual taking dierent types of terms apart, we draw the equations
that are equivalent to the continuity equations (3B.4). But unlike the case of a surface, if
these currents are not vectors and cannot be used to construct a manifold.
3C. Converse
We have shown that every conformal submanifold is associated to a Dirac equation of
witch one solution represents it. It was the easy part, now we have to prove the converse, that all
solutions of the Dirac equation with given functions and represents a
submanifold, which one may not be conformal. So we have
from which, in addition to the ones above (3B.5), we deduce the following equations
For a surface, integrability is linked to the integral along a closed loop. Then theorems
give the value of this integral as a function of an integral on the surface surrounded by the loop.
If it is zero, the integral along a path gives a well dened immersion function. Loosely speaking,
for higher dimensions this does not work because the surface integral depends on the partial
derivatives along directions normal to it. To have it work, these ones should be taken into
account, but the dimensions of the two integrals don’t match. For this reason, an expansion in
powers series of the coordinates have to be used so that there can be a term by term cancellation.
There are dierent ways to achieve this, but they are deducible from one another. Investigating
rst the simpler case of a hypersurface ( ) we proceed by looking for -forms with
To have integrability, it suces that they satisfy the structure equations of the space
J
μ
:= ~
Ψ
EPEμ
Ψ, (3B.3)
∂
μ
J
μ
=
0
. (3B.4)
Ψ
∑
αzα
β
/
α
=
ρ
−1∂
βρ
, (3B.5a)
1
2
za
α
/
β
=
za
β
/
α
, (3B.5b)
p
> 2
Ψ ∈ Γ0
M Aμ
{
Eμ
[
E
∂
μ
−
ab
a
/
μE
⋅
a
b
] +
maEa
}
Ψ =
0
, (3C.1)
1
2
ηαzα
β
/
γ
+
ηβzβ
γ
/
α
+
ηγzγ
α
/
β
= 0, (3C.2a)
zb
a
/
μ
=
ab
a
/
μ
, (3C.2b)
∑
μzμ
a
/
μ
=
ma
, (3C.2c)
q
= 1, 1
ωμ
ωn
= 0.

We introduce the components of the dierential forms :
with which the structure equations read
and we expand the searched forms in powers of the coordinates :
The prefactor is chosen to get rid of meaningless derivative terms. Mind it is not a Taylor
expansion, for the coecients are not constant. To the zeroth order we get
The second equation is satised if
owing to (3B.5b), and this is the only hypothesis we need. The rst equation then gives
A most general solution is
with symmetric in And for in all generality we can similarly take
We go on to rst order :
The second equation is a constraint for which until now was free. It is a linear system of
equations of unknowns. However the equations are not all
independent, as is easily seen by taking adding all the equations obtained by
circular permutation of then replacing the from (3C.10a, 3C.10b). The cancel out
[
d
ωμ
=
ων
∧
ωμ
ν
0 =
ων
∧
ωn
ν
(3C.3ab)
ωμ
=:
ωμ
/
α
d
ξα
,
ωi
j
=:
ωi
j
/
α
d
ξα
= 2
zi
j
/
α
d
ξα
+
δi
jρ
−1d
ρ
, (3C.4)
⎡
⎣
∂
αωμ
/
β
− ∂
βωμ
/
α
= 2
zμ
ν
/
βων
/
α
+
ρ
−1∂
βρ

ωμ
/
α
− 2
zμ
ν
/
αων
/
β
−
ρ
−1∂
αρ

ωμ
/
β
,
0 = 2
zn
ν
/
βων
/
α
− 2
zn
ν
/
αων
/
β
,(3C.5ab)
ωμ
/
α
=
ρ
−1(
ϖμ
/
α
+
ϖμ
σ
/
αξσ
+
ϖμ
{
στ
}/
αξσξτ
+
ϖμ
{
στρ
}/
αξσξτξρ
+ ⋯). (3C.6)
1
21
3!
[
∂
αϖμ
/
β
− ∂
βϖμ
/
α
+
ϖμ
α
/
β
−
ϖμ
β
/
α
= 2
zμ
ν
/
βϖν
/
α
− 2
zμ
ν
/
αϖν
/
β
,
0 = 2
zn
ν
/
βϖν
/
α
− 2
zn
ν
/
αϖν
/
β
.(3C.7ab)
ϖμ
/
α
=
δμ
α
(3C.8)
ϖμ
α
/
β
−
ϖμ
β
/
α
= 2
zμ
α
/
β
− 2
zμ
β
/
α
. (3C.9)
ϖμ
α
/
β
= 2
zμ
α
/
β
+ 2
sμ
{
αβ
}(3C.10a)
sμ
{
αβ
}
α
,
β
.
ϖμ
α
/
α
,
ϖμ
α
/
α
= 2
zμ
α
/
α
+ 2
sμ
{
αα
}. (3C.10b)
[
∂
αϖμ
σ
/
β
− ∂
βϖμ
σ
/
α
+
ϖμ
{
ασ
}/
β
−
ϖμ
{
βσ
}/
α
= 2
zμ
ν
/
βϖν
σ
/
α
− 2
zμ
ν
/
αϖν
σ
/
β
,
0 = 2
zn
ν
/
βϖν
σ
/
α
− 2
zn
ν
/
αϖν
σ
/
β
.(3C.11ab)
sμ
{
νλ
}
p
2(
p
− 1)/2
p
2(
p
+ 1)/2
σ
≠
α
,
σ
≠
β
,
α
,
β
,
σ
,
ϖ s

by their symmetry and the by the atness condition (2B.15). But they are not incompatible,
the number of equations is then reduced to The unknowns that can be
chosen as parameters are of the type and The general solution is
and
but from the symmetry of we have the conditions on
At the second order
with the general solutions of the rst equation (3C.14a)
and
It is obvious that each symmetry condition (3C.13) eliminates a dierent equation (3C.14a), so
that there is no net eect on the number of constraints. The coecient matrix of this system is
the same as above, and there are as many such systems, and as many series of unknowns as
values. All considered, there are then equations and unknowns.
We can continue to higher and higher orders, we always get the same types of terms. By
choosing appropriate functions for the parameters, solutions can be found that are dened
everywhere, even at the points where the matrix of coecients is singular, and for each order.
The known case of a surface is of this type, by taking and so on. The
problem is not the lack of solution, but the plethora of them, even for a surface. But already for
three dimensions the calculations become excruciating, and when the number of
equation is multiplied by so that it does not work. To get something simpler and universal,
we must relax the condition that the tangent vectors to the manifold should belong to the space
spanned by the or as we shall say, be parallel, since the constraint is a consequence of
and so this is no longer true.
For that we use another, equivalent method. If we nd independent vectors such that
z
p
(
p
+ 1)(
p
− 1)/3.
sμ
{
σμ
}
sγ
{
αβ
}+
sα
{
βγ
}+
sβ
{
γα
}.
ϖμ
{
ασ
}/
β
= ∂
βϖμ
σ
/
α
+ 2
zμ
ν
/
βϖν
σ
/
α
+ 2
sμ
{
αβ
}
σ
, (3C.12a)
ϖμ
{
ασ
}/
α
= ∂
αϖμ
σ
/
α
+ 2
zμ
ν
/
αϖν
σ
/
α
+ 2
sμ
{
αα
}
σ
, (3C.12b)
ϖμ
{
στ
}/
α
,
s
∂
αϖμ
τ
/
σ
+ 2
zμ
ν
/
αϖν
τ
/
σ
+ 2
sμ
{
σα
}
τ
= ∂
αϖμ
σ
/
τ
+ 2
zμ
ν
/
αϖν
σ
/
τ
+ 2
sμ
{
τα
}
σ
. (3C.13)
[
∂
αϖμ
{
στ
}/
β
− ∂
βϖμ
{
στ
}/
α
+
ϖμ
{
αστ
}/
β
−
ϖμ
{
βστ
}/
α
= 2
zμ
ν
/
βϖν
{
στ
}/
α
− 2
zμ
ν
/
αϖν
{
στ
}/
β
,
0 = 2
zn
ν
/
βϖν
{
στ
}/
α
− 2
zn
ν
/
αϖν
{
στ
}/
β
,(3C.14ab)
ϖμ
{
αστ
}/
β
= ∂
βϖμ
{
στ
}/
α
+ 2
zμ
ν
/
βϖν
{
στ
}/
α
+ 2
sμ
{
α
/
β
}
στ
, (3C.15a)
ϖμ
{
αστ
}/
α
= ∂
αϖμ
{
στ
}/
α
+ 2
zμ
ν
/
αϖν
{
στ
}/
α
+ 2
sμ
{
α
/
α
}
στ
. (3C.15b)
τ
p
2(
p
+ 1)(
p
− 1)/3
p
3(
p
+ 1)/2
s
1
{21} = 0,
s
2
{12} = 0,
q
> 1,
q
,
E
μ
,
ωa
= 0,
I
α
∂
β
I
α
− ∂
α
I
β
=
0
, (3C.16)

then there exist an immersion function such that On the other hand we should
have
so that
Next we also expand this vectors in power series
with the correspondance
and the denition
So we have to look for the vectors. According to the foregoing, we make the rst guess
and evaluate (3C.16) order by order. At the zeroth one we get
and chose the solutions
Again at the rst order
and we can take the solutions
R I
α
= ∂
α
R
.
d
R
=
ωi
/
α
E
i
d
ξα
, (3C.17)
I
α
=
ωi
/
α
E
i
. (3C.18)
I
α
=
K
α
+
K
α
|
σξσ
+
K
α
|{
στ
}
ξσξτ
+
K
α
|{
στρ
}
ξσξτξρ
+ ⋯ , (3C.19)
1
21
3!
K
α
=
ϖi
/
α
^
E
i
,
K
α
|
σ
=
ϖi
σ
/
α
^
E
iξσ
,
K
α
|{
στ
}=
ϖi
{
στ
}/
α
^
E
iξσξτ
,
K
α
|{
στρ
}=
ϖi
{
στρ
}/
α
^
E
iξσξτξρ
, ⋯
(3C.20)
^
E
i
:=
ρ
−1
E
i
. (3C.21)
K
K
α
=^
E
α
(3C.22)
∂
β
K
α
+
K
α
|
β
= ∂
α
K
β
+
K
β
|
α
, (3C.23)
2
zμ
α
/
β
^
E
μ
+
K
α
|
β
= 2
zμ
β
/
α
^
E
μ
+
K
β
|
α
, (3C.23’)
K
α
|
β
= 2
zμ
β
/
α
^
E
μ
, (3C.24a)
K
α
|
α
= 2
zμ
α
/
α
^
E
μ
. (3C.24b)
∂
β
K
α
|
σ
+
K
α
|{
βσ
}= ∂
α
K
β
|
σ
+
K
β
|{
ασ
}, (3C.25)
∂
β
(2
zμ
σ
/
α
^
E
μ
) +
K
α
|{
βσ
}= ∂
α
(2
zμ
σ
/
β
^
E
μ
) +
K
β
|{
ασ
}, (3C.25')
∂
β
(∂
α
^
E
σ
− 2
za
σ
/
α
^
E
a
) +
K
α
|{
βσ
}= ∂
α
(∂
β
^
E
σ
− 2
za
σ
/
β
^
E
a
) +
K
β
|{
ασ
}, (3C.25'')
−∂
β
(2
za
σ
/
α
^
E
a
) +
K
α
|{
βσ
}= −∂
α
(2
za
σ
/
β
^
E
a
) +
K
β
|{
ασ
}, (3C.25''')

which have the correct symmetry. At the second order something happens :
as due to the cancellation we have
and similarly for the remaining orders. Finally we have found the nite and regular series
with which (3C.16) can be veried directly. We have proven the converse provided there is no
restriction on the value of the tangent vectors. This solution is not the same as the one given
above for a conformal manifold, even for the same solution of the Dirac equation.
To conclude this section, under the previous assumption that is everywhere an element
of the special Cliord group, a submanifold is represented by every solution of a Dirac equation,
whatever the dimensions and the signatures. In addition for a hypersurface, under the proviso
that the series converges and except for possible isolated singular points, this is also true with
parallel tangent vectors. As we used the only hypothesis (3B.5b), this can be viewed as the direct
consequence of the current conservation alone.
§4. Extended GAUSS map
4A. Definition
Suppose the local tangent space of the immersed submanifold is spanned by Then
its normal vectors are the In the Cliord algebra, the normal space can be represented by a
single object : It is equivalent and more convenient to use its dual, the
pseudo-scalar of the immersed manifold, since this avoids the reference to the
ambient space. Then in the general case, the pseudo-scalar is
Since we are only interested in the direction, the norm is immaterial, so we shall take
We call this the extended Gauß map. (We reserve the term generalised Gauß map to undened
quadratic forms in order to avoid confusions, since it is used indiscriminately in the literature.)
From this expression we see that multiplying on the left by an element of the group
K
α
|{
στ
}= −∂
α
(2
za
σ
/
τ
^
E
a
), (3C.26)
∂
β
K
α
|{
στ
}+
K
α
|{
βστ
}= ∂
α
K
β
|{
στ
}+
K
β
|{
αστ
}, (3C.27)
−∂
βα
(2
za
σ
/
τ
^
E
a
) +
K
α
|{
βστ
}= −∂
αβ
(2
za
σ
/
τ
^
E
a
) +
K
β
|{
αστ
}, (3C.27')
K
α
|{
στρ
}=
0
, (3C.28)
I
α
=^
E
α
+ 2
zμ
σ
/
α
^
E
μξσ
− ∂
α
(
za
σ
/
τ
^
E
a
)
ξσξτ
, (3C.29)
Ψ
{
E
μ
}.
E
a
.
E
p
+1 ⋯
E
n
=
E
p
+1⋯
n
.
E
1⋯
E
p
=
E
P
~
Ψ
E
1Ψ ⋯ ~
Ψ
Ep
Ψ =
ρp
−1 ~
Ψ
EP
Ψ. (4A.1)
^
E
P
:=
ρ
−1 ~
Ψ
EP
Ψ = Ψ−1
EP
Ψ. (4A.2)
Ψ

whose Lie algebra is or of the group
whose Lie algebra is the Gauß map is unaected, so that it is an element of the
homogeneous space of
It is the real oriented Grassmann manifold while is the isotropy group.
(Like and are understood as and ) Because of the invariance under
the Gauß map can be represented by
with the notations
Globally, the extended Gauß map has real components, but they satisfy
quadratic constraints. The real dimension of the Gauß map manifold,
i.e. the range of the Gauß map, is then
Locally, it is represented by which is the dierential of the coset space and
transforms under the adjoint representation of the isotropy group. We then have the linear space
decomposition
with
4B. Stereographic projection
It is always interesting to stereographically project the Gauß map onto The denition
of this extended stereographic projection requires a foray into the realm of orthogonal matrices. It is
essentially a function from an orthogonal group to its Lie algebra, and is actually a restriction of
the inverse Cayley transform.
Because a matrix is block-diagonalisable, it can be split into the product of
mutually commuting matrices. So let be the matrix of and let be the orthogonal
matrix that diagonalises it :
Lor
(
p
) =
Spin
(
p
) span{
Eμν
/2},
Y M
(
q
) =
Spin
(
q
)
span{
Eab
/2},
Spin
(
n
)
∼ , (4A.3)
Spin
(
n
)
Spin
(
p
) ×
Spin
(
q
)
SO
(
n
)
SO
(
p
) ×
SO
(
q
)
~
Gp
,
q
,
Lor
(
p
) ×
Y M
(
q
)
n
,
p q p
+,
p
−
q
+,
q
−.
Lor
(
p
) ×
Y M
(
q
),
^
E
P
= exp{−
U
/2}
EP
exp{
U
/2} =
R
−1
EP
=
EPR
(4A.4)
U
:=
uμaEμa
,
R
:= e
U
. (4A.5)
n
(
n
− 1)/2
[
p
(
p
− 1) +
q
(
q
− 1)]/2
[
n
(
n
− 1) −
p
(
p
− 1) −
q
(
q
− 1)] =
pq
. (4A.6)
1
2
W
= span{
Eμa
/2},
spin
(
n
) =
lor
(
p
) ⊕
W
⊕
ym
(
q
), (4A.7)
[
lor
,
lor
] ⊂
lor
, [
ym
,
ym
] ⊂
ym
, [
lor
,
ym
] = {
0
},
[
lor
,
W
] ⊂
W
, [
ym
,
W
] ⊂
W
, [
W
,
W
] ⊂
lor
⊕
ym
.
(4A.8)
W
.
SO
(
n
)
R
SO
(
n
),
M

Then
For non compact groups the blocks may look like
but mutatis mutandis, the reasoning and the results are the same.
The inverse Cayley transform is a projection of from an orthogonal group to
in its Lie algebra given by one of these equal expressions :
where is the unit matrix. The numerator and the denominator commute, which justify the
fraction notation. We take the last expression because cancellations will occur. Suppose rst that
is block-diagonal. Then we have
MRM
−1 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
cos
θ
1− sin
θ
10 0 ⋯
sin
θ
1cos
θ
10 0 ⋯
0 0 cos
θ
2− sin
θ
2…
0 0 sin
θ
2cos
θ
2…
⋮ ⋮ ⋮ ⋮ ⋱
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
cos
θ
1− sin
θ
10 0 ⋯
sin
θ
1cos
θ
10 0 ⋯
0 0 1 0 …
0 0 0 1 …
⋮ ⋮ ⋮ ⋮ ⋱
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 0 0 0 ⋯
0 1 0 0 ⋯
0 0 cos
θ
2− sin
θ
2…
0 0 sin
θ
2cos
θ
2…
⋮ ⋮ ⋮ ⋮ ⋱
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⋯ =:
R
′
1
R
′
2⋯
(4B.1)
R
=
M
−1
R
′
1
R
′
2⋯
M
= (
M
−1
R
′
1
M
)(
M
−1
R
′
2
M
) ⋯ =:
R
1
R
2⋯ (4B.2)
SO
(
n
+,
n
−),
(
cosh(
θ
) sinh(
θ
)
sinh(
θ
) cosh(
θ
)
)
, (4B.3)
R
SO
(
n
)
W
so
(
n
)
W
:= = = , (4B.4abc)
R
−
E
R
+
E
R
−
R
−1
2
E
+
R
+
R
−1
R
1/2 −
R
−1/2
R
1/2 +
R
−1/2
E
R

It is clear that is block-diagonal too, and is a sum
in which the contain only the -th block and everywhere else. Now we can write them as
the inverse Cayley transform of matrices containing the corresponding block, on
the remaining diagonal and everywhere else. It would be cumbersome to write it explicitly, so
it is left as an exercice. The key is that in the numerator the diagonal ’s cancel out. As seen
above, the product of all these matrices is just :
So we arrive at
If is not block-diagonal, it can be written as
where is so, and is an orthogonal matrix, then
W
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 − sin(
θ
1/2) 0 0 ⋯
sin(
θ
1/2) 0 0 0 ⋯
0 0 0 − sin(
θ
2/2) …
0 0 sin(
θ
2/2) 0 …
⋮ ⋮ ⋮ ⋮ ⋱
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
×
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
cos(
θ
1/2) 0 0 0 ⋯
0 cos(
θ
1/2) 0 0 ⋯
0 0 cos(
θ
2/2) 0 …
0 0 0 cos(
θ
2/2) …
⋮⋮⋮⋮⋱
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
−1
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 − tan(
θ
1/2) 0 0 ⋯
tan(
θ
1/2) 0 0 0 ⋯
0 0 0 −tan(
θ
2/2) …
0 0 tan(
θ
2/2) 0 …
⋮⋮⋮⋮⋱
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (4B.5)
W
W
=
∑
i
W
i
(4B.6)
W
ii
0
SO
(
n
)
R
i
1
0
1
R
R
=
∏
i
R
i
. (4B.7)
W
=
∑
i
= . (4B.8)
R
1/2
i
−
R
−1/2
i
R
1/2
i
+
R
−1/2
i
∑
i
(
R
1/2
i
−
R
−1/2
i
)
∏
j
≠
i
(
R
1/2
j
+
R
−1/2
j
)
∏
k
(
R
1/2
k
+
R
−1/2
k
)
R
R
=
M
−1
R
′
M
=
∏
i
M
−1
R
′
i
M
, (4B.9)
R
′
M
M
−1
W
′
M
=
M
−1
∑
i
M
, (4B.10)
R
′1/2
i
−
R
′−1/2
i
R
′1/2
i
+
R
′−1/2
i

which is the same formula as above (4B.8). Expanding the products, remarking with the
matrices that
the initial expression (4B.4c) is recovered. This additive property of the Cayley transform seems
to be unknown.
Back in the Cliord algebras, the bivector similarly decomposes into a sum of
orthogonal simple bivectors
each corresponding to the so that we get
The expansion of the numerator can be written as
for the coecients which arises from cancellation of identical terms because of
the minus sign. And the denominator is
with terms. Finally, by expanding the exponential functions all the products
cancel out, leaving only the bivector or scalar terms, so we see that this is equal to the compact
form
where the decomposition also does not appear explicitly.
The parameters of consist of polar angles, or moduli when they are
projected, and the others are azimutal angles contained in the diagonalizing orthogonal rotation.
The closed plane has then at most distinct innities corresponding to
The direct Cayley transform is given by
R
1/2
i
R
−1/2
j
+
R
−1/2
i
R
1/2
j
=
R
1/2
i
R
1/2
j
+
R
−1/2
i
R
−1/2
j
, (4B.11)
U
U
=
∑
i
Ui
,
UiUi
= ±(
θi
)2
E
, [
Ui
,
Uj
] =
0
, (4B.12)
Wi
,
W
=
∑
i
Wi
, [
Wi
,
Wj
] =
0
, (4B.13)
W
=
∑
i
= . (4B.14)
e
Ui
/2 − e−
Ui
/2
e
Ui
/2 + e−
Ui
/2
∑
i
[(e
Ui
/2 − e−
Ui
/2)
∏
j