18^{th}International Conference on the Application of Computer
Science and Mathematics in Architecture and Civil Engineering
K. G¨urlebeck and C. K¨onke (eds.)
Weimar, Germany, 07–09 July 2009

### INTRODUCTION TO CLIFFORD ANALYSIS OVER PSEUDO-EUCLIDEAN SPACE

G. Franssens^{∗}

*∗**Belgian Institute for Space Aeronomy*
*Ringlaan 3, B-1180 Brussels, Belgium*
E-mail: ghislain.franssens@aeronomy.be

Keywords: Clifford Analysis, pseudo-Euclidean space

Abstract. *An introduction is given to Clifford Analysis (CA) over pseudo-Euclidean spaceE*^{n}*of arbitrary signature. CA overE*^{n}*is regarded as a function theory of Clifford-valued functions*
*F* *satisfying an equation of the form∂F* =*−S, for a given functionS* *and with∂* *a first order*
*vector-valued differential (Dirac) operator. The formulation of CA overE*^{n}*presented here pays*
*special attention to its geometrical setting. This permits to identify tensors which qualify as*
*geometrical Dirac operators and to take a position on the naturalness of contravariant and*
*covariant versions of such a theory. In addition, a formal method is described to construct the*
*general solution to the aforementioned equation in the context of covariant CA overE*^{n}*.*

1 INTRODUCTION

The aim of this paper is to give an introduction to Clifford Analysis (CA) over pseudo-
Euclidean space *E** ^{n}*. CA over

*E*

*, called Ultrahyperbolic Clifford Analysis (UCA) for short, is a non-trivial extension of the more familiar Euclidean CA overR*

^{n}*, [1], [2], [3]. UCA is the proper mathematical setting for studying physics in Minkowski spaces with an arbitrary number of time dimensions*

^{n}*p*and space dimensions

*q. The particular case of Hyperbolic Clifford Analy-*sis (HCA), corresponding to

*p*= 1and

*q >*1, has direct relevance to physics, with in particular the case

*p*= 1,

*q*= 3providing a tailor-made function theory, applicable to electromagnetism and quantum physics, [4], [5].

Let Ω be an open region in *E** ^{n}*, V

*a real sesquilinear product space of signature (p, q),*

^{p,q}*m*,

*p*+

*q, andCl*(V

*)the universal Clifford algebra generated byV*

^{p,q}*. UCA is the study of (e.g., smooth) functions from Ω*

^{p,q}*→*

*Cl*(V

*), acted upon by a first order*

^{p,q}*Cl*(V

*)-valued vector differential operator*

^{p,q}*∂, called a Dirac operator. In particular, we want to derive integral*representations for functions

*F*satisfying first order equations such as

*∂F*=

*−S, with*

*S*smooth and of compact support. By choosing

*S*= 0, a special subset of functions is singled out, called (left) ultrahyperbolic Clifford holomorphic functions, and which can be thought of as a generalization of the familiar complex holomorphic functions in Complex Analysis.

We here give a formulation of UCA which pays attention to its geometrical setting relative
to pseudo-Euclidean space *E** ^{n}* of signature (r, s),

*n*,

*r*+

*s. More precisely, we focus on*how a given abstract spaceV

*is related – or if possible can be unrelated – to the (common) tangent space E*

^{p,q}*of*

^{r,s}*E*

*. To this end, we discuss the concept of soldering, relating a given linear space to the tangent space of a given manifold. We start off with independent signatures forV*

^{n}*andE*

^{p,q}*and then investigate the feasibility of this choice along the way. Further, we also show how to give an invariant meaning to the vector differential operator*

^{r,s}*∂*, a matter which is usually not addressed in CA. It is found that a Dirac operator in UCA acquires a natural geometrical meaning if it is defined as an appropriate tensor. As a consequence of this, any such Dirac operator itself defines a soldering, which implies that in a geometrical meaningful UCA, the spaces V

*andE*

^{p,q}*can not be unrelated. In addition, we show that there exists a natural asymmetry between covariant and contravariant Clifford Analysis over*

^{r,s}*E*

*. On the one hand, it is found that no geometrical invariant contravariant Clifford Analysis with a*

^{n}*first order*contravariant Dirac operator, taking values in

*Cl*(E

*), exists. On the other hand, a geometrical invariant covariant Clifford Analysis with a first order covariant Dirac operator, taking values in*

^{r,s}*Cl*(E

*), arises naturally.*

^{∗r,s}Based on the aforementioned geometrical setting, we then give a formal method to construct
an integral representation for covariant Clifford-valued functions *F** ^{∗}* satisfying

*∂*

^{∗}*F*

*=*

^{∗}*−S*

*, with*

^{∗}*∂*

*an anisotropic covariant Dirac operator, taking values in*

^{∗}*Cl*(V

*). It is found that, in order to completely characterize this representation, it is necessary to calculate the restriction*

^{∗p,q}*C*

_{x}

^{∗}_{0}

*|*

*of an ultrahyperbolic covariant Cauchy kernel*

_{δc}*C*

_{x}

^{∗}_{0}to the boundary

*δc*of a chosen region

*c. Any Cauchy kernelC*

_{x}

^{∗}_{0}in turn is obtained by operating with

*∂*

*on a fundamental solution of the ultrahyperbolic wave equation. The details of this calculation however lead far into distribution theory and are outside the scope of this introduction.*

^{∗}2 CLIFFORD ALGEBRAS OVER PSEUDO-EUCLIDEAN SPACE

We only consider real Clifford Algebras of finite dimension, which we wish to situate in a pseudo-Euclidean setting.

2.1 Preliminaries

Let R* ^{m}* denote

*m-dimensional coordinate space over*R. Let

*p, q*

*∈*N with

*m*=

*p*+

*q*and

*h*a real sesquilinear product structure, of signature(p, q), and given by a nondegenerate symmetric bilinear function

*h*: R

^{m}*×*R

^{m}*→*Rsuch that(v,w)

*7→*

*h*(v,w). Introduce the sesquilinear product spaceV

*,(R*

^{p,q}

^{m}*, h)*and its dualV

*,(R*

^{∗p,q}

^{m}*, h*

*).*

^{−1}Define(V* ^{p,q}*)

*,R,(V*

^{∧0}*)*

^{p,q}*,V*

^{∧1}*and denote by(V*

^{p,q}*)*

^{p,q}*,*

^{∧k}*∀k∈*Z

_{[2,m]}, the linear space of antisymmetric contravariant tensors of grade (i.e., order)

*k*over R. Elements of (V

*)*

^{p,q}*are called*

^{∧k}*k-vectors (0-vectors and*1-vectors are also called scalars and vectors, respectively) and elements of the graded linear spaceM,

*⊕*

^{m}*(V*

_{k=0}*)*

^{p,q}*overRare called multivectors. A*

^{∧k}*k-vector*v

*k*

*∈*(V

*)*

^{p,q}*has strict components(v*

^{∧k}

^{i}^{1}

^{...i}

^{k}*,*1

*≤i*1

*< . . . < i*

*k*

*≤m)*with respect to a basis

*B*

*,*

_{k}*{²*

_{i}_{1}

_{...i}

_{k}*,*1

*≤i*

_{1}

*< . . . < i*

_{k}*≤m}*for(V

*)*

^{p,q}*(B*

^{∧k}_{0},

*{1}) and can be represented*as (using the summation convention over unordered indices)

v* _{k}*= 1

*k!v*^{i}^{1}^{...i}^{k}*²*_{i}_{1}_{...i}* _{k}*. (1)
Define(V

*)*

^{∗p,q}*, R, (V*

^{∧0}*)*

^{∗p,q}*, V*

^{∧1}*and denote by(V*

^{∗p,q}*)*

^{∗p,q}*,*

^{∧k}*∀k*

*∈*Z

_{[2,m]}, the linear space of antisymmetric covariant tensors of grade

*k*overR. Elements of(V

*)*

^{∗p,q}*are called*

^{∧k}*k-covectors (0-vectors and*1-covectors are also called scalars and covectors, respectively) and elements of the graded linear space

*M*

*,*

^{∗}*⊕*

^{m}*(V*

_{k=0}*)*

^{∗p,q}*overRare called multicovectors. A*

^{∧k}*k-covector*v

^{∗}

_{k}*∈*(V

*)*

^{∗p,q}*has strict components(v*

^{∧k}*j*1

*...j*

*k*

*,*1

*≤j*1

*< . . . < j*

*k*

*≤m)*with respect to a cobasis

*B*

_{k}*, ©*

^{∗}*θ*^{j}^{1}^{...j}^{k}*,*1*≤j*_{1} *< . . . < j*_{k}*≤m*ª

for (V* ^{∗p,q}*)

*(B*

^{∧k}_{0}

*,*

^{∗}*{1}) and can be*represented as

v^{∗}* _{k}* = 1

*k!v*_{j}_{1}_{...j}_{k}*θ*^{j}^{1}^{...j}^{k}*.* (2)
Endow(V* ^{p,q}*)

*with a real sesquilinear product*

^{∧k}*·*: (V

*)*

^{p,q}

^{∧k}*×*(V

*)*

^{p,q}

^{∧k}*→*Rsuch that, with respect to a basis

*B*

*k*for(V

*)*

^{p,q}*,*

^{∧k}*∀k*

*∈*Z[1,m],

(v_{k}*,*w* _{k}*)

*7→*v

_{k}*·*w

*, 1*

_{k}*k!*

1

*k!v*^{i}^{1}^{...i}^{k}*w*^{j}^{1}^{...j}* ^{k}*det [h(²

_{i}

_{a}*,²*

_{j}*)]*

_{b}*,*

= 1

*k!*

1

*k!v*^{i}^{1}^{...i}^{k}*w*^{j}^{1}^{...j}^{k}*δ*^{l}_{i}^{1}_{1}^{...l}_{...i}^{k}_{k}*h*(²_{l}_{1}*,²*_{j}_{1})*. . . h*(²_{l}_{k}*,²*_{j}* _{k}*)

*,*

= 1

*k!v*^{l}^{1}^{...l}^{k}*w*^{j}^{1}^{...j}^{k}*h*(²_{l}_{1}*,²*_{j}_{1})*. . . h*(²_{l}_{k}*,²*_{j}* _{k}*)

*.*(3) Herein is

*δ*the

*k-covariant,k-contravariant generalized Kronecker tensor, [6, p. 142].*

Similarly, we endow (V* ^{∗p,q}*)

*with the induced real sesquilinear product*

^{∧k}*·*: (V

*)*

^{∗p,q}

^{∧k}*×*(V

*)*

^{∗p,q}

^{∧k}*→*Rsuch that, with respect to a basis

*B*

_{k}*for(V*

^{∗}*)*

^{∗p,q}*,*

^{∧k}*∀k*

*∈*Z

_{[1,m]},

(v^{∗}_{k}*,*w^{∗}* _{k}*)

*7→*v

^{∗}

_{k}*·*w

^{∗}*, 1*

_{k}*k!*

1

*k!v*_{i}_{1}_{...i}_{k}*w*_{j}_{1}_{...j}* _{k}*det£

*h*

*¡*

^{−1}*θ*^{i}^{a}*,θ*^{j}* ^{b}*¢¤

*,*

= 1

*k!*

1

*k!v*_{i}_{1}_{...i}_{k}*w*_{j}_{1}_{...j}_{k}*δ*^{i}_{l}^{1}_{1}_{...l}^{...i}_{k}^{k}*h** ^{−1}*¡

*θ*^{l}^{1}*,θ*^{j}^{1}¢

*. . . h** ^{−1}*¡

*θ*^{l}^{k}*,θ*^{j}* ^{k}*¢

*,*

= 1

*k!v*_{l}_{1}_{...l}_{k}*w*_{j}_{1}_{...j}_{k}*h** ^{−1}*¡

*θ*^{l}^{1}*,θ*^{j}^{1}¢

*. . . h** ^{−1}*¡

*θ*^{l}^{k}*,θ*^{j}* ^{k}*¢

*.* (4)

For a fixed contravariant vectoru*∈*V* ^{p,q}*, the map fromV

^{p,q}*→*Rsuch thatv

*7→h*(u,v) = u

*·*v, defines a canonical isomorphism between V

*and its dual V*

^{p,q}*. This canonical iso- morphism is the map*

^{∗p,q}*[*: V

^{p,q}*→*V

*such thatu*

^{∗p,q}*7→*u

*,*

^{∗}*h*(u, .). Its inverse is the map

*]*:V

^{∗p,q}*→*V

*such thatu*

^{p,q}

^{∗}*7→*u ,

*h*

*(u*

^{−1}

^{∗}*, .). The sesquilinear products (3) and (4) allow*to naturally extend the domain of the canonical isomorphism to higher order tensors. The maps

*[*and

*]*allow to “raise or lower the indices” of the components of tensors.

The map *[* or the map *]* generates in a canonical way the bilinear binary function *h., .i* :
(V* ^{∗p,q}*)

^{∧k}*×*(V

*)*

^{p,q}

^{∧k}*→*Rsuch that

(u^{∗}_{k}*,*v* _{k}*)

*7→ hu*

_{k}

^{∗}*,*v

_{k}*i*,u

^{∗}*(v*

_{k}*) =*

_{k}*]u*

^{∗}

_{k}*·*v

*=u*

_{k}

^{∗}

_{k}*·[v*

_{k}*.*(5) The function

*h., .i*may serve to uniquely determine the canonically associated cobasis

*B*

^{∗}*of a given basis*

_{k}*B*

*, using the relations*

_{k}*θ*^{j}^{1}^{...j}^{k}*,²*_{i}_{1}_{...i}* _{k}*®

=*δ*^{j}_{i}_{1}^{1}_{...i}^{...j}_{k}* ^{k}*,

*∀k∈*Z

_{[1,m]}. 2.2 Clifford Algebra overV

^{p,q}Define the bilinear associative exterior product *∧* : (V* ^{p,q}*)

^{∧k}*×*(V

*)*

^{p,q}

^{∧l}*→*(V

*)*

^{p,q}*,*

^{∧(k+l)}*∀k, l∈*Z_{[0,m]}, such that: (i) for*k*+*l* *≤m,*
(v_{k}*,*w* _{l}*)

*7→*v

_{k}*∧*w

*, 1*

_{l}*k!*

1

*l!v*^{i}^{1}^{...i}^{k}*w*^{i}^{k+1}^{...i}^{k+l}*δ*_{i}^{j}_{1}^{1}_{...i}^{...j}_{k+l}^{k+l}*²*_{j}_{1}_{...j}_{k+l}*,* (6)
and (ii) for *m < k*+*l,* v*k**∧*w*l* , 0. The exterior product extends to *∧* : M*×*M *→* Mby
distributivity over the direct sum. The exterior product is independent of any choice of bases.

Define a (left) contraction product y : V^{p,q}*×* (V* ^{p,q}*)

^{∧k}*→*(V

*)*

^{p,q}*such that (i) for*

^{∧k−1}*∀k* *∈*Z_{[1,m]},

(v,w* _{k}*)

*7→*vyw

*, 1*

_{k}*k!*

X*k*

*l=1*

(−1)^{l−1}*v*^{i}*w*^{j}^{1}^{...j}^{k}*h*(²_{i}*,²*_{j}* _{l}*)

*²*

_{j}_{1}

_{...}*j*b

*l*

*...j*

*k*

*,*(7) wherein

*j*b

*l*denotes the absence of

*j*

*l*, and (ii) for

*k*= 0,vyw

*k*, 0. The contraction product extends toy:V

^{p,q}*×*M

*→*Mby distributivity over the direct sum and reduces for

*k*= 1to the sesquilinear product ofV

*. The contraction product is independent of any choice of bases.*

^{p,q}Define a Clifford product, denoted by juxtaposition, first between a vector vand a*k-vector*
w* _{k}*as

vw* _{k}* , +vyw

*+v*

_{k}*∧*w

_{k}*,*(8) (−1)

*w*

^{k}*v ,*

_{k}*−v*yw

*+v*

_{k}*∧*w

_{k}*,*(9) and then extend it, by distributivity over the direct sum and associativity, toM

*×*M

*→*M. The linear space

*M*together with the Clifford product defined on

*M*is the contravariant universal real Clifford Algebra

*Cl*(V

*)generated byV*

^{p,q}*.*

^{p,q}Practical calculations are considerably simplified by choosing orthonormal bases *B** _{k}* ,

*{e*

_{i}_{1}

_{...i}

_{k}*,∀i*

_{1}

*< . . . < i*

_{k}*}, with respect to*

*h, for each*(V

*)*

^{p,q}*, as then e*

^{∧k}

_{i}_{1}

_{...i}*= e*

_{k}

_{i}_{1}

*. . .*e

_{i}*= e*

_{k}*i*1

*∧. . .∧*e

*i*

*k*ande

*i*e

*i*=

*±1.*

2.3 Clifford Algebra overV^{∗p,q}

Define the bilinear associative exterior product *∧* : (V* ^{∗p,q}*)

^{∧k}*×*(V

*)*

^{∗p,q}

^{∧l}*→*(V

*)*

^{∗p,q}*,*

^{∧(k+l)}*∀k, l∈*Z[0,m], such that: (i) for*k*+*l* *≤m,*
(v^{∗}_{k}*,*w^{∗}* _{l}*)

*7→*v

_{k}

^{∗}*∧*w

^{∗}*, 1*

_{l}*k!*

1

*l!v*_{i}_{1}_{...i}_{k}*w*_{i}_{k+1}_{...i}_{k+l}*δ*_{j}^{i}_{1}^{1}^{...i}_{...j}^{k+l}_{k+l}*θ*^{j}^{1}^{...j}^{k+l}*,* (10)

and (ii) for*m < k* +*l,*v^{∗}_{k}*∧*w^{∗}* _{l}* ,0. The exterior product extends to

*∧*:M

^{∗}*×*M

^{∗}*→*M

*by distributivity over the direct sum. The exterior product is independent of any choice of cobases.*

^{∗}Define a (left) contraction product y : V^{∗p,q}*×*(V* ^{∗p,q}*)

^{∧k}*→*(V

*)*

^{∗p,q}*such that (i) for*

^{∧k−1}*∀k* *∈*Z_{[1,m]},

(v^{∗}*,*w^{∗}* _{k}*)

*7→*v

*yw*

^{∗}

^{∗}*, 1*

_{k}*k!*

X*k*

*l=1*

(−1)^{l−1}*v**i**w**j*1*...j**k**h** ^{−1}*¡

*θ*^{i}*,θ*^{j}* ^{l}*¢

*θ*^{j}^{1}^{...}^{j}^{b}^{l}^{...j}* ^{k}* (11)
and (ii) for

*k*= 0, v

*yw*

^{∗}

^{∗}*, 0. The contraction product extends to y : V*

_{k}

^{∗p,q}*×*M

^{∗}*→*M

*by distributivity over the direct sum and reduces for*

^{∗}*k*= 1to the sesquilinear product ofV

*. The contraction product is independent of any choice of cobases.*

^{∗p,q}Define a Clifford product, denoted by juxtaposition, first between a covector v* ^{∗}* and a

*k-*covectorw

_{k}*as*

^{∗}v* ^{∗}*w

^{∗}*, +v*

_{k}*yw*

^{∗}

^{∗}*+v*

_{k}

^{∗}*∧*w

^{∗}

_{k}*,*(12) (−1)

*w*

^{k}

^{∗}*v*

_{k}*,*

^{∗}*−v*

*yw*

^{∗}

^{∗}*+v*

_{k}

^{∗}*∧*w

^{∗}

_{k}*,*(13) and then extend it, by distributivity over the direct sum and associativity, toM

^{∗}*×*M

^{∗}*→*M

*. The linear space*

^{∗}*M*

*together with the Clifford product defined on*

^{∗}*M*

*is the covariant universal real Clifford Algebra*

^{∗}*Cl*(V

*)generated byV*

^{∗p,q}*.*

^{∗p,q}Practical calculations are considerably simplified by choosing orthonormal cobases *B*_{k}* ^{∗}* ,

*{t*

^{j}^{1}

^{...j}

^{k}*,∀j*

_{1}

*< . . . < j*

_{k}*}, with respect toh*

*, for each(V*

^{−1}*)*

^{∗p,q}*, as thent*

^{∧k}

^{j}^{1}

^{...j}*=t*

^{k}

^{j}^{1}

*. . .*t

^{j}*= t*

^{k}

^{j}^{1}

*∧. . .∧*t

^{j}*andt*

^{k}*t*

^{j}*=*

^{j}*±1.*

2.4 Real pseudo-Euclidean space*E*^{n}

Recall that the differential manifold of real*n-dimensional pseudo-Euclidean spaceE** ^{n}*is an
affine space over R, i.e., a principal homogeneous space with automorphism group the affine
group, given by the semi-direct product

*T*

*(R)o*

_{n}*GL*

*(R). At any point*

_{n}*x*

*∈*

*E*

*, the tangent space*

^{n}*V*

*,*

_{x}*T*

_{x}*E*and the dual cotangent space

*V*

_{x}*,*

^{∗}*T*

_{x}

^{∗}*E*are isomorphic (as linear spaces) toR

*. Further,*

^{n}*E*

*comes with a real sesquilinear product structure, given by a nondegenerate symmetric bilinear function*

^{n}*g*:

*V*

_{x}*×V*

_{x}*→*R,

*∀x*

*∈*

*E*

*, such that (v,w)*

^{n}*7→*

*g*(v,w), with

*g*of signature(r, s)and independent of

*x. The spaceE*

*is in addition tacitly endowed with a trivial flat connection, i.e., the standard rule for parallel transport. The latter property and the independence of*

^{n}*g*of

*x*make that all

*V*

*,*

_{x}*∀x*

*∈*

*E*

*, can be identified with a common single sesquilinear product spaceE*

^{n}*, (R*

^{r,s}

^{n}*, g). Similarly, all*

*V*

_{x}*,*

^{∗}*∀x*

*∈*

*E*

*, can be identified with E*

^{n}*,(R*

^{∗r,s}

^{n}*, g*

*).*

^{−1}Although it is possible to also impose an independent Clifford structure on E* ^{∗r,s}*, we will
refrain from doing so at this point. We will see further howE

*may inherit a Clifford struc- ture from V*

^{∗r,s}*in a natural way (then requiring (r, s) = (p, q)). One could also impose an independent Clifford structure onE*

^{∗p,q}*, but such structure appears to be uninteresting. It will be shown in the next section that the algebra*

^{r,s}*Cl*(E

*)does not generate a natural*

^{r,s}*Cl*(E

*)-valued Clifford Analysis.*

^{r,s}2.5 Rings

We will make use of the following rings.

(i) *F** ^{∞}* , (C

*(Ω,R)*

^{∞}*,*+,): the unital ring of smooth real-valued functions from Ω

*→*R, together with pointwise addition+and pointwise multiplication (denoted by juxtaposition).

(ii)*D* ,(C_{c}* ^{∞}*(E

^{n}*,*R)

*,*+,): the ring of smooth real-valued functions with compact support from

*E*

^{n}*→*R. Denote by

*D*

*the linear space of distributions as the continuous dual of*

^{0}*D.*

We will also use the multiplication between a smooth function*ψ* *∈ F** ^{∞}* and a distribution

*f*

*∈ D*

*, which is defined as the distribution*

^{0}*ψf*=

*f ψ*given by,

*∀ϕ∈ D,*

*hψf, ϕi*,*hf, ψϕi,* (14)
which is legitimate since*ψϕ∈ D.*

3 CLIFFORD ANALYSIS OVER*E** ^{N}*
3.1 Definition

Contravariant Clifford Analysis (CA) over pseudo-Euclidean space*E** ^{n}*is the study of func-
tions from Ω

*→*

*Cl*(V

*), together with a first order vector differential operator*

^{p,q}*∂. Covari-*ant Clifford Analysis (CA

^{*}) over pseudo-Euclidean space

*E*

*is the study of functions from Ω*

^{n}*→Cl*(V

*), together with a first order covector differential operator*

^{∗p,q}*∂*

*.*

^{∗}Within CA over *E** ^{n}*, we want in particular derive an integral representation for functions

*F*

*∈C*

*(Ω, Cl(V*

^{∞}*))satisfying*

^{p,q}*∂F* =*−S,* (15)

with *S* *∈* *C*_{c}* ^{∞}*(E

^{n}*, Cl*(V

*)). Similarly, within CA*

^{p,q}^{*}over

*E*

*, we want to derive an integral representation for functions*

^{n}*F*

^{∗}*∈C*

*(Ω, Cl(V*

^{∞}*))satisfying*

^{∗p,q}*∂*^{∗}*F** ^{∗}* =

*−S*

^{∗}*,*(16)

with*S*^{∗}*∈C*_{c}* ^{∞}*(E

^{n}*, Cl*(V

*)).*

^{∗p,q}From a mathematical point of view, it is not necessary to relate the space V* ^{p,q}* to E

*(or V*

^{r,s}*toE*

^{∗p,q}*) any further. Their only connection so far is through the functions that we study, which: (i) have as domain a subset of the affine space*

^{∗r,s}*E*

*underlyingE*

^{n}*andE*

^{r,s}*and (ii) have as codomain the Clifford algebra generated by eitherV*

^{∗r,s}*orV*

^{p,q}*.*

^{∗p,q}For further convenience, we denote by*{²*_{i}*}*a general basis forV* ^{p,q}* and by©

*θ*

*ª*

^{j}the canon-
ically associated cobasis forV* ^{∗p,q}* with respect to

*h. Similarly, let{ε*

_{µ}*}*be a general basis for E

*and*

^{r,s}*{ϑ*

^{ν}*}*the canonically associated cobasis forE

*with respect to*

^{∗r,s}*g.*

3.2 Solderings

Sometimes a physics application might require that an additional relation between the spaces
V* ^{p,q}* and E

*is to be imposed and that*

^{r,s}*m*=

*n. Such a relation is commonly defined in the*literature by a smooth bijective map E

^{r,s}*→*V

*such that*

^{p,q}*u*

*7→*

*v*,

*hχ, ui*with either (a)

*χ*

*∈*V

^{p,q}*⊗*E

*or (b)*

^{∗r,s}*χ∈*E

^{∗r,s}*⊗*V

*. With respect to the above bases,*

^{p,q}*hχ, ui*is defined in terms of the bilinear binary function (5), relative to

*g, as for (a) and (b) respectively,*

*hχ, ui* =

*χ*^{i}* _{ν}*(²

_{i}*⊗ϑ*

*)*

^{ν}*, u*

^{µ}*ε*

*®*

_{µ},*χ*^{i}_{ν}*u*^{µ}*²*_{i}*⊗ hϑ*^{ν}*,ε*_{µ}*i*=*χ*^{i}_{κ}*u*^{κ}*²*_{i}*.* (17)
*hχ, ui* =

*χ*_{ν}* ^{j}*(ϑ

^{ν}*⊗²*

*)*

_{j}*, u*

^{µ}*ε*

*®*

_{µ},*u*^{µ}*χ*_{ν}^{j}*hϑ*^{ν}*,ε*_{µ}*i ⊗²** _{j}* =

*u*

^{κ}*χ*

_{κ}

^{j}*²*

_{j}*.*(18) In the physics literature, this map is usually called a

*soldering*as it “solders” V

*to*

^{p,q}*E*

*by establishing a bijective relation between the linear space V*

^{n}*and the common tangent space E*

^{p,q}*. We assume here that*

^{r,s}*χ*is independent of

*x∈E*

*.*

^{n}soldering type (a) type (b)
1 E^{p,q}*→*V^{p,q}*χ∈*V^{p,q}*⊗*E^{∗p,q}*χ∈*E^{∗p,q}*⊗*V* ^{p,q}*
2 E

^{p,q}*→*V

^{∗p,q}*χ∈*V

^{∗p,q}*⊗*E

^{∗p,q}*χ∈*E

^{∗p,q}*⊗*V

*3 E*

^{∗p,q}

^{∗p,q}*→*V

^{p,q}*χ∈*V

^{p,q}*⊗*E

^{p,q}*χ∈*E

^{p,q}*⊗*V

*4 E*

^{p,q}

^{∗p,q}*→*V

^{∗p,q}*χ∈*V

^{∗p,q}*⊗*E

^{p,q}*χ∈*E

^{p,q}*⊗*V

^{∗p,q}Table 1: All possible solderings relatingV* ^{p,q}*to

*E*

*.*

^{n}The inverse soldering is the mapV^{p,q}*→* E* ^{r,s}* such that

*v*

*7→*

*u*,

*hχ*

^{−1}*, vi*with (a)

*χ*

^{−1}*∈*V

^{∗p,q}*⊗*E

*or (b)*

^{r,s}*χ*

^{−1}*∈*E

^{r,s}*⊗*V

*. With respect to the above bases,*

^{∗p,q}*hχ*

^{−1}*, vi*is defined in terms of the bilinear binary function (5) now relative to

*h, as for (a) and (b) respectively,*

*χ*^{−1}*, v*®

= D¡

*χ** ^{−1}*¢

_{ν}*j*

¡*θ*^{j}*⊗ε**ν*

¢*, v*^{i}*²**i*

E
,*v** ^{i}*¡

*χ** ^{−1}*¢

_{ν}*j*

*θ*^{j}*,²**i*

®*⊗ε**ν* =*v** ^{k}*¡

*χ*

*¢*

^{−1}

_{ν}*k* *ε**ν*(19)*,*

*χ*^{−1}*, v*®

= D¡

*χ** ^{−1}*¢

_{µ}*j*

¡*ε*_{µ}*⊗θ** ^{j}*¢

*, v*

^{i}*²*

_{i}E ,¡

*χ** ^{−1}*¢

_{µ}*j**v*^{i}*ε*_{µ}*⊗*
*θ*^{j}*,²** _{i}*®

=¡
*χ** ^{−1}*¢

_{µ}*k**v*^{k}*ε*_{µ}*,(20)*
wherein

*χ*^{i}* _{κ}*¡

*χ** ^{−1}*¢

_{κ}*j* = *δ*^{i}* _{j}* and

*χ*

^{k}*¡*

_{µ}*χ*

*¢*

^{−1}

_{ν}*k* =*δ*^{ν}_{µ}*,* (21)

*χ*_{κ}* ^{i}*¡

*χ*

*¢*

^{−1}

_{κ}*j* = *δ*^{i}* _{j}* and

*χ*

_{µ}*¡*

^{k}*χ*

*¢*

^{−1}

_{ν}*k* =*δ*^{ν}_{µ}*.* (22)

A soldering*χ*also relates higher tensor spaces in a natural way. In particular, the sesquilinear
product structures*g* and*h*become related by

*g** _{µν}* =

*h*

_{ij}*χ*

^{i}

_{µ}*χ*

^{j}

_{ν}*.*(23)

Hence, if a soldering*χ*is present, then only two members of the triple(g, χ, h)can be chosen
freely. The earlier requirement that *m* = *n* avoids that either *g* or *h* becomes degenerated.

Further, by Sylvester’s law of inertia, both sesquilinear product structures must have the same signature. Hence, once a soldering is present, we will write(r, s) = (p, q).

More generally, a soldering can be specified by any of the eight smooth bijective maps given in Table 1. We will use this observation in the next subsection.

3.3 Dirac operators

We want here introduce a Dirac operator as a geometrical invariant object, rather than as an ad hoc operator.

I.a Let *d*_{V E}*∈* V^{p,q}*⊗*E* ^{p,q}* and

*d*

_{V}

^{∗}

_{E}*∈*V

^{∗p,q}*⊗*E

*be nondegenerate (i.e., maximal rank) tensors. The tensors*

^{p,q}*d*

*and*

_{V E}*d*

_{V}

^{∗}*have representatives,*

_{E}*d**V E* , *d** ^{iκ}*(²

*i*

*⊗ε*

*κ*)

*,*(24)

*d**V*^{∗}*E* , *d*_{j}* ^{κ}*¡

*θ*^{j}*⊗ε**κ*

¢*.* (25)

Since tangent vectors at a point of a manifold, such as *ε** _{κ}*, are defined as derivations on

*F*

*, they are partial differential operators, [6, p. 117]. Hence the tensors (24)–(25) are also first order partial derivative operators, acting on*

^{∞}*F*

*through the Pfaff derivatives*

^{∞}*ε*

*κ*, [6, p.138].

This becomes even more apparent if we choose a natural (i.e., a coordinate) basis*{∂/∂x*^{κ}*}*in
E* ^{p,q}*. Therefore, the tensors

*d*

*and*

_{V E}*d*

_{V}

^{∗}*can be identified as anisotropic Dirac operators, taking values inV*

_{E}*andV*

^{p,q}*, respectively.*

^{∗p,q}In the presence of a soldering*χ*of type 1.a (Table 1), the tensors*d**V E*and*d**V*^{∗}*E* are mapped
to the respective tensors*d*^{S}_{EE}*∈*E^{p,q}*⊗*E* ^{p,q}* and

*d*

^{S}

_{E}*∗*

*E*

*∈*E

^{∗p,q}*⊗*E

*, having representatives*

^{p,q}*d*^{S}* _{EE}* = ¡

*χ*

*¢*

^{−1}

_{λ}*i* *d** ^{iκ}*(ε

_{λ}*⊗ε*

*)*

_{κ}*,*(26)

*d*_{E}^{S}^{∗}* _{E}* =

*χ*

^{j}

_{ν}*d*

_{j}*(ϑ*

^{κ}

^{ν}*⊗ε*

*κ*)

*.*(27) Eq. (26) shows that

*d*

^{S}*is a second order differential operator acting on*

_{EE}*F*

*and hence does not qualify as a Dirac operator, taking values inE*

^{∞}*. Moreover, the tensor*

^{p,q}*d*

*itself defines a soldering of type 3.a, which corresponds to a type 1.a soldering*

_{V E}*χ*=

*[*

*g*

*d*

*V E*, having representa- tive

*χ*

^{i}*=*

_{ν}*d*

^{iκ}*g*

*. Under its own soldering,*

_{κν}*d*

*is mapped to*

_{V E}*d*

^{S}*= (g*

_{EE}*)*

^{−1}*(ε*

^{λκ}

_{λ}*⊗ε*

*), which is the Laplacian for scalar functions (since the connection coefficients are zero on*

_{κ}*E*

*).*

^{n}Eq. (27) shows that *d*^{S}_{E}*∗**E* is a first order differential operator acting on *F** ^{∞}*. Hence, the
covector operator

*d*

^{S}

_{E}*∗*

*E*qualifies as a natural anisotropic Dirac operator, taking values inE

*. Also, any covector Dirac operator*

^{∗p,q}*d*

*V*

^{∗}*E*defines itself a soldering of type 4.a, which corresponds to a type 1.a inverse soldering

*χ*

*=*

^{−1}*d*

_{V}

^{∗}*, having representative(χ*

_{E}*)*

^{−1}

_{i}*=*

^{ν}*d*

_{i}*. Under its own soldering,*

^{ν}*d*

_{V}

^{∗}*is thus mapped to*

_{E}*d*

_{E}

^{S}*∗*

*E*=

*ϑ*

^{κ}*⊗ε*

*, which is the standard isotropic covector Dirac operator. In this special case,*

_{κ}*d*^{S}_{E}*∗**E* =*δ,* (28)

which shows that the isotropic covector Dirac operator, taking values in E* ^{∗p,q}*, is just the 1-
covariant,1-contravariant Kronecker tensor

*δ*

*∈*(E

^{p,q}*⊗*E

*)*

^{∗p,q}*∩*(E

^{∗p,q}*⊗*E

*).*

^{p,q}I.b Let *d*_{EV}*∈* E^{p,q}*⊗*V* ^{p,q}* and

*d*

_{EV}

^{∗}*∈*E

^{p,q}*⊗*V

*be nondegenerate tensors, with repre- sentatives*

^{∗p,q}*d** _{EV}* ,

*d*

*(ε*

^{κi}

_{κ}*⊗²*

*)*

_{i}*,*(29)

*d*_{EV}* ^{∗}* ,

*d*

^{κ}*¡*

_{j}*ε*_{κ}*⊗θ** ^{j}*¢

*.* (30)

Similarly, the tensors*d** _{EV}* and

*d*

_{EV}*can be identified as anisotropic Dirac type operators, taking values inV*

^{∗}*andV*

^{p,q}*, respectively.*

^{∗p,q}In the presence of a soldering *χ* of type 1.b, the tensors *d** _{EV}* and

*d*

_{EV}*are mapped to the respective tensors*

^{∗}*d*

^{S}

_{EE}*∈*E

^{p,q}*⊗*E

*and*

^{p,q}*d*

^{S}

_{EE}*∗*

*∈*E

^{p,q}*⊗*E

*, having representatives*

^{∗p,q}*d*^{S}* _{EE}* =

*d*

*¡*

^{κi}*χ*

*¢*

^{−1}

_{λ}*i*(ε_{κ}*⊗ε** _{λ}*)

*,*(31)

*d*_{EE}^{S}*∗* = *d*^{κ}_{j}*χ*_{ν}* ^{j}*(ε

_{κ}*⊗ϑ*

*)*

^{ν}*.*(32) Again, the vector operator

*d*

^{S}*is not a Dirac operator, while the covariant operator*

_{EE}*d*

^{S}

_{EE}*∗*

qualifies as an anisotropic Dirac operator. Under its own soldering, *d**EV** ^{∗}* is also mapped to

*ε*

_{κ}*⊗ϑ*

*=*

^{κ}*δ.*

II.a Let *d*_{V E}^{∗}*∈* V^{p,q}*⊗*E* ^{∗p,q}* and

*d*

_{V}

^{∗}

_{E}

^{∗}*∈*V

^{∗p,q}*⊗*E

*be nondegenerate tensors, with representatives*

^{∗p,q}*d*_{V E}* ^{∗}* ,

*d*

^{i}*(²*

_{κ}

_{i}*⊗ϑ*

*)*

^{κ}*,*(33)

*d*

_{V}

^{∗}

_{E}*,*

^{∗}*d*

*¡*

_{jκ}*θ*^{j}*⊗ϑ** ^{κ}*¢

*.* (34)

Since cotangent vectors at a point of a manifold, such as*ϑ** ^{κ}*, are not derivations on

*F*

*, they are not partial differential operators [6, p. 135]. Hence the tensors*

^{∞}*d*

*V E*

*and*

^{∗}*d*

*V*

^{∗}*E*

*can not be identified as Dirac operators.*

^{∗}In the presence of a soldering *χ*of type 1.a, the tensors*d**V E** ^{∗}* and

*d*

*V*

^{∗}*E*

*are mapped to the respective tensors*

^{∗}*d*

^{S}

_{EE}*∗*

*∈*E

^{p,q}*⊗*E

*and*

^{∗p,q}*d*

^{S}

_{E}*∗*

*E*

^{∗}*∈*E

^{∗p,q}*⊗*E

*, having representatives*

^{∗p,q}*d*^{S}_{EE}* ^{∗}* = ¡

*χ** ^{−1}*¢

_{λ}*i* *d*^{i}* _{κ}*(ε

_{λ}*⊗ϑ*

*)*

^{κ}*,*(35)

*d*^{S}_{E}*∗**E** ^{∗}* =

*χ*

^{j}

_{ν}*d*

*(ϑ*

_{jκ}

^{ν}*⊗ϑ*

*)*

^{κ}*.*(36) Eq. (35) shows that

*d*

^{S}

_{EE}*is a first order differential operator acting on*

^{∗}*F*

*. Hence, the covector operator*

^{∞}*d*

^{S}

_{EE}*∗*qualifies as an anisotropic Dirac operator, taking values inE

*. Under its own soldering,*

^{∗p,q}*d*

_{V E}*is also mapped to*

^{∗}*ε*

_{κ}*⊗ϑ*

*=*

^{κ}*δ.*

Eqs. (34) and (36) show that *d*^{S}_{E}^{∗}_{E}* ^{∗}* is not a differential operator. Any tensor

*d*

_{V}

^{∗}

_{E}*defines a soldering of type 2.a, which corresponds to a type 1.a soldering*

^{∗}*χ*=

*]*

_{h}*d*

_{V}

^{∗}

_{E}*, having repre- sentative*

^{∗}*χ*

^{i}*=*

_{ν}*h*

^{ik}*d*

*. Under its own soldering,*

_{kν}*d*

_{V}

^{∗}

_{E}*is mapped to*

^{∗}*d*

^{S}

_{E}*∗*

*E*

*=*

^{∗}*g*

*¡*

_{λκ}*ϑ*^{λ}*⊗ϑ** ^{κ}*¢
,
which is the “dual Laplacian” (with respect to a natural basis,

*d*

^{S}

_{E}

^{∗}

_{E}*=*

^{∗}*ds*

^{2}, the square of the infinitesimal line element in

*E*

*).*

^{n}II.b Let *d*_{E}^{∗}_{V}*∈* E^{∗p,q}*⊗*V* ^{p,q}* and

*d*

_{E}

^{∗}

_{V}

^{∗}*∈*E

^{∗p,q}*⊗*V

*be nondegenerate tensors, with representatives*

^{∗p,q}*d*_{E}^{∗}* _{V}* ,

*d*

_{κ}*(ϑ*

^{i}

^{κ}*⊗²*

*)*

_{i}*,*(37)

*d*

_{E}

^{∗}

_{V}*,*

^{∗}*d*

*¡*

_{κj}*ϑ*^{κ}*⊗θ** ^{j}*¢

*.* (38)

In the presence of a soldering *χ*of type 1.b, the tensors*d**E*^{∗}*V* and*d**E*^{∗}*V** ^{∗}* are mapped to the
respective tensors

*d*

^{S}

_{E}*∗*

*E*

*∈*E

^{∗p,q}*⊗*E

*and*

^{p,q}*d*

^{S}

_{E}*∗*

*E*

^{∗}*∈*E

^{∗p,q}*⊗*E

*, having representatives*

^{∗p,q}*d*^{S}_{E}^{∗}* _{E}* =

*d*

_{κ}*¡*

^{i}*χ*

*¢*

^{−1}

_{µ}*i*(ϑ^{κ}*⊗ε**µ*)*,* (39)

*d*_{E}^{S}*∗**E** ^{∗}* =

*d*

_{κj}*χ*

_{ν}*(ϑ*

^{j}

^{κ}*⊗ϑ*

*)*

^{ν}*.*(40) Again, the covector operator

*d*

^{S}

_{E}

^{∗}*qualifies as an anisotropic Dirac operator, taking values inE*

_{E}*. The tensor*

^{∗p,q}*d*

^{S}

_{E}*∗*

*E*

*is not a Dirac operator.*

^{∗}In summary, we have the following geometrical Dirac operators:

(i) taking values inV* ^{∗p,q}*, and irrespective of the presence of a soldering

*χ,*

*∂*_{V}^{∗}* _{E}* ,

*d*

_{V}

^{∗}*=*

_{E}*d*

_{j}*¡*

^{κ}*θ*^{j}*⊗ε** _{κ}*¢

*,* (41)

*∂*_{EV}* ^{∗}* ,

*d*

_{EV}*=*

^{∗}*d*

^{κ}*¡*

_{j}*ε*_{κ}*⊗θ** ^{j}*¢

; (42)

(ii) taking values inE* ^{∗p,q}*, and in the presence of an inverse soldering

*χ*

*, (a) of type 1.a,*

^{−1}*∂*_{EE}^{(1a)}*∗* ,

*χ*^{−1}*, d*_{E}^{∗}* _{V}*®

=¡

*χ** ^{−1}*¢

_{ν}*i*

¡*θ*^{i}*⊗ε** _{ν}*¢

*, d*_{µ}* ^{j}*(ϑ

^{µ}*⊗²*

*)® , ¡*

_{j}*χ** ^{−1}*¢

_{ν}*i**d*_{µ}* ^{j}*

*θ*

^{i}*,²*

*j*

®(ε*ν* *⊗ϑ** ^{µ}*) = ¡

*χ*

*¢*

^{−1}

_{ν}*k**d*_{µ}* ^{k}*(ε

*ν*

*⊗ϑ*

*)*

^{µ}*,*(43)

*∂*_{EE}^{(2a)}*∗* ,

*χ*^{−1}*, d**V E*^{∗}

®=¡

*χ** ^{−1}*¢

_{ν}*i*

¡*θ*^{i}*⊗ε**ν*

¢*,*(²*j**⊗ϑ** ^{µ}*)

*d*

^{j}*® , ¡*

_{µ}*χ** ^{−1}*¢

_{ν}*i* *d*^{j}* _{µ}*

*θ*

^{i}*,²*

*®*

_{j}(ε_{ν}*⊗ϑ** ^{µ}*) = ¡

*χ*

*¢*

^{−1}

_{ν}*k* *d*^{k}* _{µ}*(ε

_{ν}*⊗ϑ*

*) ; (44) (b) of type 1.b,*

^{µ}*∂*_{EE}^{(1b)}*∗* ,

*χ*^{−1}*, d**E*^{∗}*V*

® =¡

*χ** ^{−1}*¢

_{ν}*i*

¡*ε**ν**⊗θ** ^{i}*¢

*, d*_{µ}* ^{j}*(ϑ

^{µ}*⊗²*

*j*)® , ¡

*χ** ^{−1}*¢

_{ν}*i**d*_{µ}* ^{j}*

*θ*

^{i}*,²*

*®*

_{j}(ε_{ν}*⊗ϑ** ^{µ}*) = ¡

*χ*

*¢*

^{−1}

_{ν}*k* *d*_{µ}* ^{k}*(ε

_{ν}*⊗ϑ*

*)*

^{µ}*,*(45)