INTRODUCTION TO CLIFFORD ANALYSIS OVER PSEUDO-EUCLIDEAN SPACE

18^thInternational 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 spaceEⁿ of arbitrary signature. CA overEⁿis 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 overEⁿpresented 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 overEⁿ.

1 INTRODUCTION

The aim of this paper is to give an introduction to Clifford Analysis (CA) over pseudo- Euclidean space Eⁿ. CA over Eⁿ, called Ultrahyperbolic Clifford Analysis (UCA) for short, is a non-trivial extension of the more familiar Euclidean CA overRⁿ, [1], [2], [3]. UCA is the proper mathematical setting for studying physics in Minkowski spaces with an arbitrary number of time dimensionspand space dimensionsq. The particular case of Hyperbolic Clifford Analy- sis (HCA), corresponding top= 1andq >1, has direct relevance to physics, with in particular the casep= 1, q = 3providing a tailor-made function theory, applicable to electromagnetism and quantum physics, [4], [5].

Let Ω be an open region in Eⁿ, V^p,q a real sesquilinear product space of signature (p, q), m , p+q, andCl(V^p,q)the universal Clifford algebra generated byV^p,q. UCA is the study of (e.g., smooth) functions from Ω → Cl(V^p,q), acted upon by a first order Cl(V^p,q)-valued vector differential operator∂, 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 choosingS = 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 Eⁿ of signature (r, s), n , r +s. More precisely, we focus on how a given abstract spaceV^p,q is related – or if possible can be unrelated – to the (common) tangent space E^r,s of Eⁿ. 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^p,q andE^r,s 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∂, 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^p,q andE^r,s can not be unrelated. In addition, we show that there exists a natural asymmetry between covariant and contravariant Clifford Analysis overEⁿ. On the one hand, it is found that no geometrical invariant contravariant Clifford Analysis with afirst order contravariant Dirac operator, taking values inCl(E^r,s), exists. On the other hand, a geometrical invariant covariant Clifford Analysis with a first order covariant Dirac operator, taking values inCl(E^∗r,s), arises naturally.

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 inCl(V^∗p,q). It is found that, in order to completely characterize this representation, it is necessary to calculate the restriction C_x^∗₀|_δcof an ultrahyperbolic covariant Cauchy kernelC_x^∗₀ to the boundaryδcof 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 functionh : R^m ×R^m → Rsuch that(v,w) 7→ h(v,w). Introduce the sesquilinear product spaceV^p,q ,(R^m, h)and its dualV^∗p,q ,(R^m, h⁻¹).

Define(V^p,q)^∧0 ,R,(V^p,q)^∧1 ,V^p,qand denote by(V^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)^∧k are called k-vectors (0-vectors and1-vectors are also called scalars and vectors, respectively) and elements of the graded linear spaceM,⊕^m_k=0(V^p,q)^∧koverRare called multivectors. A k-vectorvk ∈ (V^p,q)^∧k has strict components(v^i1...ik,1≤i1 < . . . < ik ≤m)with respect to a basisB_k ,{²_i1...ik,1≤i₁ < . . . < i_k ≤m}for(V^p,q)^∧k (B₀ ,{1}) and can be represented as (using the summation convention over unordered indices)

v_k= 1

k!v^i1...ik²_i1...ik. (1) Define(V^∗p,q)^∧0 , R, (V^∗p,q)^∧1 , V^∗p,q and denote by(V^∗p,q)^∧k, ∀k ∈ Z_[2,m], the linear space of antisymmetric covariant tensors of grade k overR. Elements of(V^∗p,q)^∧k are called k-covectors (0-vectors and1-covectors are also called scalars and covectors, respectively) and elements of the graded linear spaceM^∗ ,⊕^m_k=0(V^∗p,q)^∧k overRare called multicovectors. A k-covectorv^∗_k ∈(V^∗p,q)^∧khas strict components(vj1...jk,1≤j1 < . . . < jk≤m)with respect to a cobasis B_k^∗ , ©

θ^j1...jk,1≤j₁ < . . . < j_k ≤mª

for (V^∗p,q)^∧k (B₀^∗ , {1}) and can be represented as

v^∗_k = 1

k!v_j1...jkθ^j1...jk. (2) Endow(V^p,q)^∧kwith a real sesquilinear product·: (V^p,q)^∧k×(V^p,q)^∧k→Rsuch that, with respect to a basisBkfor(V^p,q)^∧k,∀k ∈Z[1,m],

(v_k,w_k) 7→ v_k·w_k , 1 k!

1

k!v^i1...ikw^j1...jkdet [h(²_ia,²_jb)],

= 1

k!

1

k!v^i1...ikw^j1...jkδ^l_i¹₁^...l_...i^k_kh(²_l1,²_j1). . . h(²_lk,²_jk),

= 1

k!v^l1...lkw^j1...jkh(²_l1,²_j1). . . h(²_lk,²_jk). (3) Herein isδthek-covariant,k-contravariant generalized Kronecker tensor, [6, p. 142].

Similarly, we endow (V^∗p,q)^∧k with the induced real sesquilinear product · : (V^∗p,q)^∧k× (V^∗p,q)^∧k→Rsuch that, with respect to a basisB_k^∗ for(V^∗p,q)^∧k,∀k ∈Z_[1,m],

(v^∗_k,w^∗_k) 7→ v^∗_k·w^∗_k , 1 k!

1

k!v_i1...ikw_j1...jkdet£ h⁻¹¡

θ^ia,θ^jb¢¤

,

= 1

k!

1

k!v_i1...ikw_j1...jkδⁱ_l¹_1...l^...i_k^kh⁻¹¡

θ^l1,θ^j1¢

. . . h⁻¹¡

θ^lk,θ^jk¢ ,

= 1

k!v_l1...lkw_j1...jkh⁻¹¡

θ^l1,θ^j1¢

. . . h⁻¹¡

θ^lk,θ^jk¢

. (4)

For a fixed contravariant vectoru∈V^p,q, the map fromV^p,q →Rsuch thatv7→h(u,v) = u·v, defines a canonical isomorphism between V^p,q and its dual V^∗p,q. This canonical iso- morphism is the map [ : V^p,q → V^∗p,q such thatu 7→ u^∗ , h(u, .). Its inverse is the map ] :V^∗p,q →V^p,qsuch thatu^∗ 7→ u ,h⁻¹(u^∗, .). 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_ki,u^∗_k(v_k) =]u^∗_k·v_k=u^∗_k·[v_k. (5) The functionh., .imay serve to uniquely determine the canonically associated cobasisB^∗_k of a given basisB_k, using the relations­

θ^j1...jk,²_i1...ik®

=δ^j_i1¹_...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) fork+l ≤m, (v_k,w_l)7→v_k∧w_l , 1 k!

1

l!v^i1...ikw^ik+1...ik+lδ_i^j₁¹_...i^...j_k+l^k+l²_j1...jk+l, (6) and (ii) for m < k+l, vk∧wl , 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)^∧k−1 such that (i) for

∀k ∈Z_[1,m],

(v,w_k)7→vyw_k , 1 k!

Xk

l=1

(−1)^l−1vⁱw^j1...jkh(²_i,²_jl) ²_j1...jbl...jk, (7) whereinjbl denotes the absence ofjl, and (ii) fork = 0,vywk , 0. The contraction product extends toy:V^p,q×M→Mby distributivity over the direct sum and reduces fork = 1to the sesquilinear product ofV^p,q. The contraction product is independent of any choice of bases.

Define a Clifford product, denoted by juxtaposition, first between a vector vand ak-vector w_kas

vw_k , +vyw_k+v∧w_k, (8) (−1)^kw_kv , −vyw_k+v∧w_k, (9) and then extend it, by distributivity over the direct sum and associativity, toM×M→M. The linear spaceM together with the Clifford product defined on M is the contravariant universal real Clifford AlgebraCl(V^p,q)generated byV^p,q.

Practical calculations are considerably simplified by choosing orthonormal bases B_k , {e_i1...ik,∀i₁ < . . . < i_k}, with respect to h, for each (V^p,q)^∧k, as then e_i1...ik = e_i1. . .e_ik = ei1∧. . .∧eik andeiei =±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) fork+l ≤m, (v^∗_k,w^∗_l)7→v_k^∗∧w^∗_l , 1 k!

1

l!v_i1...ikw_ik+1...ik+lδ_jⁱ₁^1...i_...j^k+l_k+lθ^j1...jk+l, (10)

and (ii) form < 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)^∧k−1 such that (i) for

∀k ∈Z_[1,m],

(v^∗,w^∗_k)7→v^∗yw^∗_k, 1 k!

Xk

l=1

(−1)^l−1viwj1...jkh⁻¹¡

θⁱ,θ^jl¢

θ^{j1...jbl...jk} (11) and (ii) fork = 0, v^∗yw^∗_k , 0. The contraction product extends to y : V^∗p,q×M^∗ → M^∗ by distributivity over the direct sum and reduces fork = 1to the sesquilinear product ofV^∗p,q. The contraction product is independent of any choice of cobases.

Define a Clifford product, denoted by juxtaposition, first between a covector v^∗ and a k- covectorw_k^∗as

v^∗w^∗_k , +v^∗yw^∗_k+v^∗∧w^∗_k, (12) (−1)^kw^∗_kv^∗ , −v^∗yw^∗_k+v^∗∧w^∗_k, (13) and then extend it, by distributivity over the direct sum and associativity, toM^∗ ×M^∗ → M^∗. The linear spaceM^∗together with the Clifford product defined onM^∗is the covariant universal real Clifford AlgebraCl(V^∗p,q)generated byV^∗p,q.

Practical calculations are considerably simplified by choosing orthonormal cobases B_k^∗ , {t^j1...jk,∀j₁ < . . . < j_k}, with respect toh⁻¹, for each(V^∗p,q)^∧k, as thent^j1...jk =t^j1. . .t^jk = t^j1∧. . .∧t^jk andt^jt^j =±1.

2.4 Real pseudo-Euclidean spaceEⁿ

Recall that the differential manifold of realn-dimensional pseudo-Euclidean spaceEⁿis an affine space over R, i.e., a principal homogeneous space with automorphism group the affine group, given by the semi-direct productT_n(R)oGL_n(R). At any pointx ∈ Eⁿ, the tangent space V_x , T_xE and the dual cotangent space V_x^∗ , T_x^∗E are isomorphic (as linear spaces) toRⁿ. Further,Eⁿcomes with a real sesquilinear product structure, given by a nondegenerate symmetric bilinear function g : V_x ×V_x → R, ∀x ∈ Eⁿ, such that (v,w) 7→ g(v,w), with g of signature(r, s)and independent ofx. The spaceEⁿ 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 g of x make that allV_x, ∀x ∈ Eⁿ, can be identified with a common single sesquilinear product spaceE^r,s , (Rⁿ, g). Similarly, all V_x^∗, ∀x ∈ Eⁿ, can be identified with E^∗r,s,(Rⁿ, 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^∗r,smay inherit a Clifford struc- ture from V^∗p,q in a natural way (then requiring (r, s) = (p, q)). One could also impose an independent Clifford structure onE^r,s, but such structure appears to be uninteresting. It will be shown in the next section that the algebraCl(E^r,s)does not generate a naturalCl(E^r,s)-valued Clifford Analysis.

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^∞(Eⁿ,R),+,): the ring of smooth real-valued functions with compact support fromEⁿ →R. Denote byD⁰the linear space of distributions as the continuous dual ofD.

We will also use the multiplication between a smooth functionψ ∈ F^∞ and a distribution f ∈ D⁰, which is defined as the distributionψf =f ψgiven by,∀ϕ∈ D,

hψf, ϕi,hf, ψϕi, (14) which is legitimate sinceψϕ∈ D.

3 CLIFFORD ANALYSIS OVERE^N 3.1 Definition

Contravariant Clifford Analysis (CA) over pseudo-Euclidean spaceEⁿis the study of func- tions from Ω → Cl(V^p,q), together with a first order vector differential operator ∂. Covari- ant Clifford Analysis (CA^*) over pseudo-Euclidean space Eⁿ is the study of functions from Ω→Cl(V^∗p,q), together with a first order covector differential operator∂^∗.

Within CA over Eⁿ, we want in particular derive an integral representation for functions F ∈C^∞(Ω, Cl(V^p,q))satisfying

∂F =−S, (15)

with S ∈ C_c^∞(Eⁿ, Cl(V^p,q)). Similarly, within CA^* over Eⁿ, we want to derive an integral representation for functionsF^∗ ∈C^∞(Ω, Cl(V^∗p,q))satisfying

∂^∗F^∗ =−S^∗, (16)

withS^∗ ∈C_c^∞(Eⁿ, Cl(V^∗p,q)).

From a mathematical point of view, it is not necessary to relate the space V^p,q to E^r,s (or V^∗p,qtoE^∗r,s) any further. Their only connection so far is through the functions that we study, which: (i) have as domain a subset of the affine spaceEⁿunderlyingE^r,sandE^∗r,sand (ii) have as codomain the Clifford algebra generated by eitherV^p,qorV^∗p,q.

For further convenience, we denote by{²_i}a general basis forV^p,q and by© θ^jª

the canon- ically associated cobasis forV^∗p,q with respect toh. Similarly, let{ε_µ}be a general basis for E^r,sand{ϑ^ν}the canonically associated cobasis forE^∗r,swith respect tog.

3.2 Solderings

Sometimes a physics application might require that an additional relation between the spaces V^p,q and E^r,s is to be imposed and that m = n. Such a relation is commonly defined in the literature by a smooth bijective map E^r,s → V^p,q such that u 7→ v , hχ, ui with either (a) χ ∈V^p,q⊗E^∗r,sor (b)χ∈ E^∗r,s⊗V^p,q. With respect to the above bases,hχ, uiis defined in terms of the bilinear binary function (5), relative tog, as for (a) and (b) respectively,

hχ, ui = ­

χⁱ_ν(²_i⊗ϑ^ν), u^µε_µ®

,χⁱ_νu^µ²_i⊗ hϑ^ν,ε_µi=χⁱ_κu^κ²_i. (17) hχ, ui = ­

χ_ν^j(ϑ^ν ⊗²_j), u^µε_µ®

,u^µχ_ν^jhϑ^ν,ε_µi ⊗²_j =u^κχ_κ^j²_j. (18) In the physics literature, this map is usually called a soldering as it “solders” V^p,q to Eⁿ by establishing a bijective relation between the linear space V^p,q and the common tangent space E^r,s. We assume here thatχis independent ofx∈Eⁿ.

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^∗p,q 3 E^∗p,q →V^p,q χ∈V^p,q⊗E^p,q χ∈E^p,q⊗V^p,q 4 E^∗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,qtoEⁿ.

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^r,s or (b)χ⁻¹ ∈E^r,s⊗V^∗p,q. With respect to the above bases,hχ⁻¹, viis defined in terms of the bilinear binary function (5) now relative toh, as for (a) and (b) respectively,

­χ⁻¹, v®

= D¡

χ⁻¹¢ _ν

j

¡θ^j ⊗εν

¢, vⁱ²i

E ,vⁱ¡

χ⁻¹¢ _ν

j

­θ^j,²i

®⊗εν =v^k¡ χ⁻¹¢ _ν

k εν(19),

­χ⁻¹, v®

= D¡

χ⁻¹¢_µ

j

¡ε_µ⊗θ^j¢ , vⁱ²_i

E ,¡

χ⁻¹¢_µ

jvⁱε_µ⊗­ θ^j,²_i®

=¡ χ⁻¹¢_µ

kv^kε_µ,(20) wherein

χⁱ_κ¡

χ⁻¹¢ _κ

j = δⁱ_j andχ^k_µ¡ χ⁻¹¢ _ν

k =δ^ν_µ, (21)

χ_κⁱ¡ χ⁻¹¢_κ

j = δⁱ_j andχ_µ^k¡ χ⁻¹¢_ν

k =δ^ν_µ. (22)

A solderingχalso relates higher tensor spaces in a natural way. In particular, the sesquilinear product structuresg andhbecome related by

g_µν =h_ijχⁱ_µχ^j_ν. (23)

Hence, if a solderingχis 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 andd_V^∗_E ∈ V^∗p,q⊗E^p,q be nondegenerate (i.e., maximal rank) tensors. The tensorsd_{V E}andd_V^∗_E have representatives,

dV E , d^iκ(²i⊗εκ), (24)

dV^∗E , d_j^κ¡

θ^j⊗εκ

¢. (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 εκ, [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_{V E} and d_V^∗_E can be identified as anisotropic Dirac operators, taking values inV^p,qandV^∗p,q, respectively.

In the presence of a solderingχof type 1.a (Table 1), the tensorsdV EanddV^∗E are mapped to the respective tensorsd^S_EE ∈E^p,q⊗E^p,q andd^S_E∗E ∈E^∗p,q⊗E^p,q, having representatives

d^S_EE = ¡ χ⁻¹¢ _λ

i d^iκ(ε_λ⊗ε_κ), (26)

d_E^S∗_E = χ^j_νd_j^κ(ϑ^ν⊗εκ). (27) Eq. (26) shows thatd^S_EEis a second order differential operator acting onF^∞and hence does not qualify as a Dirac operator, taking values inE^p,q. Moreover, the tensord_{V E}itself defines a soldering of type 3.a, which corresponds to a type 1.a solderingχ=[gdV E, having representa- tiveχⁱ_ν =d^iκg_κν. Under its own soldering,d_{V E}is mapped tod^S_EE = (g⁻¹)^λκ(ε_λ⊗ε_κ), which is the Laplacian for scalar functions (since the connection coefficients are zero onEⁿ).

Eq. (27) shows that d^S_E∗E is a first order differential operator acting on F^∞. Hence, the covector operatord^S_E∗E qualifies as a natural anisotropic Dirac operator, taking values inE^∗p,q. Also, any covector Dirac operatordV^∗E defines itself a soldering of type 4.a, which corresponds to a type 1.a inverse soldering χ⁻¹ = d_V^∗_E , having representative(χ⁻¹)_i^ν = d_i^ν. Under its own soldering,d_V^∗_Eis thus mapped tod_E^S∗E =ϑ^κ⊗ε_κ, which is the standard isotropic covector Dirac operator. In this special case,

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 tensorδ ∈(E^p,q⊗E^∗p,q)∩(E^∗p,q⊗E^p,q).

I.b Let d_EV ∈ E^p,q⊗V^p,q andd_EV^∗ ∈ E^p,q⊗V^∗p,q be nondegenerate tensors, with repre- sentatives

d_EV , d^κi(ε_κ⊗²_i), (29)

d_EV^∗ , d^κ_j¡

ε_κ⊗θ^j¢

. (30)

Similarly, the tensorsd_EV andd_EV^∗can be identified as anisotropic Dirac type operators, taking values inV^p,qandV^∗p,q, respectively.

In the presence of a soldering χ of type 1.b, the tensors d_EV and d_EV^∗ are mapped to the respective tensorsd^S_EE ∈E^p,q⊗E^p,qandd^S_EE∗ ∈E^p,q⊗E^∗p,q, having representatives

d^S_EE = d^κi¡ χ⁻¹¢_λ

i(ε_κ⊗ε_λ), (31)

d_EE^S ∗ = d^κ_jχ_ν^j(ε_κ⊗ϑ^ν). (32) Again, the vector operator d^S_EE is not a Dirac operator, while the covariant operator d^S_EE∗

qualifies as an anisotropic Dirac operator. Under its own soldering, dEV^∗ 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^∗p,q be nondegenerate tensors, with representatives

d_{V E}^∗ , dⁱ_κ(²_i⊗ϑ^κ), (33) d_V^∗_E^∗ , d_jκ¡

θ^j ⊗ϑ^κ¢

. (34)

Since cotangent vectors at a point of a manifold, such asϑ^κ, are not derivations on F^∞, they are not partial differential operators [6, p. 135]. Hence the tensorsdV E^∗ anddV^∗E^∗ can not be identified as Dirac operators.

In the presence of a soldering χof type 1.a, the tensorsdV E^∗ anddV^∗E^∗ are mapped to the respective tensorsd^S_EE∗ ∈E^p,q⊗E^∗p,q andd^S_E∗E^∗ ∈E^∗p,q⊗E^∗p,q, having representatives

d^S_EE^∗ = ¡

χ⁻¹¢ _λ

i dⁱ_κ(ε_λ⊗ϑ^κ), (35)

d^S_E∗E^∗ = χ^j_νd_jκ(ϑ^ν ⊗ϑ^κ). (36) Eq. (35) shows that d^S_EE^∗ is a first order differential operator acting on F^∞. Hence, the covector operatord^S_EE∗qualifies as an anisotropic Dirac operator, taking values inE^∗p,q. Under its own soldering,d_{V E}^∗ is also mapped toε_κ⊗ϑ^κ =δ.

Eqs. (34) and (36) show that d^S_E^∗_E^∗ is not a differential operator. Any tensord_V^∗_E^∗ defines a soldering of type 2.a, which corresponds to a type 1.a solderingχ = ]_hd_V^∗_E^∗, having repre- sentativeχⁱ_ν = h^ikd_kν. Under its own soldering, d_V^∗_E^∗ is mapped tod^S_E∗E^∗ = g_λκ¡

ϑ^λ⊗ϑ^κ¢ , which is the “dual Laplacian” (with respect to a natural basis, d^S_E^∗_E^∗ = ds², the square of the infinitesimal line element inEⁿ).

II.b Let d_E^∗_V ∈ E^∗p,q ⊗V^p,q and d_E^∗_V^∗ ∈ E^∗p,q ⊗V^∗p,q be nondegenerate tensors, with representatives

d_E^∗_V , d_κⁱ(ϑ^κ⊗²_i), (37) d_E^∗_V^∗ , d_κj¡

ϑ^κ⊗θ^j¢

. (38)

In the presence of a soldering χof type 1.b, the tensorsdE^∗V anddE^∗V^∗ are mapped to the respective tensorsd^S_E∗E ∈E^∗p,q⊗E^p,q andd^S_E∗E^∗ ∈E^∗p,q⊗E^∗p,q, having representatives

d^S_E^∗_E = d_κⁱ¡ χ⁻¹¢_µ

i(ϑ^κ ⊗εµ), (39)

d_E^S∗E^∗ = d_κjχ_ν^j(ϑ^κ⊗ϑ^ν). (40) Again, the covector operatord^S_E^∗_E qualifies as an anisotropic Dirac operator, taking values inE^∗p,q. The tensord^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 solderingχ,

∂_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 solderingχ⁻¹, (a) of type 1.a,

∂_EE^(1a)∗ , ­

χ⁻¹, d_E^∗_V®

=­¡

χ⁻¹¢ _ν

i

¡θⁱ⊗ε_ν¢

, d_µ^j(ϑ^µ⊗²_j)® , ¡

χ⁻¹¢_ν

id_µ^j­ θⁱ,²j

®(εν ⊗ϑ^µ) = ¡ χ⁻¹¢_ν

kd_µ^k(εν ⊗ϑ^µ), (43)

∂_EE^(2a)∗ , ­

χ⁻¹, dV E^∗

®=­¡

χ⁻¹¢ _ν

i

¡θⁱ⊗εν

¢,(²j⊗ϑ^µ)d^j_µ® , ¡

χ⁻¹¢ _ν

i d^j_µ­ θⁱ,²_j®

(ε_ν⊗ϑ^µ) = ¡ χ⁻¹¢ _ν

k d^k_µ(ε_ν ⊗ϑ^µ) ; (44) (b) of type 1.b,

∂_EE^(1b)∗ , ­

χ⁻¹, dE^∗V

® =­¡

χ⁻¹¢_ν

i

¡εν⊗θⁱ¢

, d_µ^j(ϑ^µ⊗²j)® , ¡

χ⁻¹¢_ν

id_µ^j­ θⁱ,²_j®

(ε_ν⊗ϑ^µ) = ¡ χ⁻¹¢ _ν

k d_µ^k(ε_ν ⊗ϑ^µ), (45)