Conformal Invariance of Clifford Monogenic Functions in the Indefinite Signature Case

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions - called monogenic functions - are defined by means of the Dirac operators that factor a certain wave operator. One of the fundamental features of quaternionic analysis is the invariance of quaternionic analogues of holomorphic function under conformal (or Mobius) transformations. A similar invariance property is known to hold in the context of Clifford algebras associated to positive definite quadratic forms. We generalize these results to the case of Clifford algebras associated to all non-degenerate quadratic forms. This result puts the indefinite signature case on the same footing as the classical positive definite case.


Introduction
Many results of complex analysis have analogues in quaternionic analysis.In particular, there are analogues of complex holomorphic functions called (left and right) regular functions.Some of the most fundamental features of quaternionic analysis are the quaternionic analogue of Cauchy's integral formula (usually referred to as Cauchy-Fueter formula) and the invariance of quaternionic regular functions under conformal (or Möbius) transformations.For modern introductions to quaternionic analysis see, for example, [Sud79;Col+04].
Complex numbers C and quaternions H are special cases of Clifford algebras.(For elementary introductions to Clifford algebras see, for example, [Che54;Gar11].)There is a further extension of complex and quaternionic analysis called Clifford analysis.For Clifford algebras associated to positive definite quadratic forms on real vector spaces, Clifford analysis is very similar to complex and quaternionic analysis (see, for example, [BDS82; GM91; DSS92] and references therein).Furthermore, Ryan has initiated the study of Clifford analysis in the setting of complex Clifford algebras [Rya82].
Let Cl(V ) be the universal Clifford algebra associated to a real vector space V with nondegenerate quadratic form Q. The Dirac operator D on V is introduced by means of factoring the wave operator associated to Q. Then the Clifford monogenic functions f : V → Cl(V ) are defined as those satisfying Df = 0.An analogue of Cauchy's integral formula for such functions was established in [LS21].In this paper we focus on another important feature of Clifford analysis -conformal invariance of monogenic functions.If A = a b c d is a Vahlen matrix producing a conformal (or Möbius) transformation on V and f : V → Cl(V ) is a monogenic function, then it is known in the case of positive definite quadratic forms Q that is also monogenic [Rya84;Boj89].We extend this result to all non-degenerate quadratic forms Q having arbitrary signatures.While the formula in the mixed signature case visually appears the same, its meaning is slightly different.First of all, the transformation (1) takes place on the conformal closure of the vector space V , and it is different from the one-point compactification of V .Secondly, one needs to revisit the definition of the monogenic functions in this context and choose the "right" Dirac operator D. The choice of signs in our definition of the operator D is compatible with [BS86; BT88; LS21].Furthermore, we argue that this is the most natural choice of the Dirac operator that makes the results valid, while other choices would not work.Interestingly, Bureš-Souček [BS86] arrive at the same Dirac operator from a different perspective -invariance under the action of the group Spin(p, q).These two features of Clifford analysis with indefinite signature -the analogue of Cauchy's integral formula and the conformal invariance of monogenic functions put the indefinite signature case on the same footing as the classical positive definite case.
The paper is organized as follows.Section 2 is a review of relevant topics.We start with the conformal transformations and the conformal compactification of V .Then we discuss Clifford algebras and associated groups -such as Lipschitz, pin and spin groups -and their actions on V by orthogonal linear transformations.After this we introduce the basics of Clifford analysis: the Dirac operator D and the left/right monogenic functions.We show that our definition of D is independent of the choice of basis of V (Lemma 12) and give an alternative basis-free way of defining D (Proposition 12).In Section 3 we introduce Vahlen matrices and explain the relation between Vahlen matrices and conformal transformations on V .The results of this section are well known, especially in the positive definite case (see, for example, [Ahl85; Vah02; Mak92; Cno96] and references therein), and we mostly follow [Mak92].In Section 4 we prove our main result (Theorem 36) that the function (2) is left monogenic.We also state a similar formula for right monogenic functions.We emphasize that our proof is independent of the signature of the quadratic form Q. Finally, in Appendix A we discuss what would happen if the Dirac operator were defined differently and show that a different choice of signs in the definition of the operator D would result in the loss of the conformal invariance.In [LS21] the same choice of signs was motivated by completely different reasons.
This research was made possible by the Indiana University, Bloomington, Math REU (research experiences for undergraduates) program, funded by NSF Award #2051032.We would like to thank Professor Dylan Thurston for organizing and facilitating this program.We would also like to thank Ms. Mandie McCarthy and Ms. Amy Bland for their administrative work, the various professors for their talks, and the other REU students for their company.Finally, we would like to thank the reviewers for their feedback and suggesting additional references.

The Conformal Compactification of R p,q
We start with a review of the conformal compactification N (V ) of a vector space V ≃ R p,q and the action of the indefinite orthogonal group O(V ⊕ R 1,1 ) ≃ O(p + 1, q + 1) on N (V ) by conformal transformations.The reader may wish to refer to, for example, [Sch08;Kob] for more detailed expositions of the results.
Let V be an n-dimensional real vector space, and Q a quadratic form on V .Corresponding to Q, there is a symmetric bilinear form B on V .The form Q can be diagonalized: there exist an orthogonal basis {e 1 , . . ., e n } of (V, Q) and integers p, q such that By the Sylvester's Law of Inertia, the numbers p and q are independent of the basis chosen.
The ordered pair (p, q) is called the signature of (V, Q).From this point on we restrict our attention to non-degenerate quadratic forms Q, in which case p + q = n.If Q is such a form with q = 0 or p = 0, then it is positive or negative definite respectively.Associated with the quadratic form is the indefinite orthogonal group consisting of all invertible linear transformations of V that preserve Q: We frequently identify (V, Q) with the generalized Minkowski space R p,q , which is the real vector space R p+q equipped with indefinite quadratic form The symmetric bilinear form associated to Q is In this context, it is common to write O(p, q) for the indefinite orthogonal group O(R p,q ).By a conformal transformation of (V, Q), we mean a smooth mapping ϕ : U → V , where U ⊂ V is a non-empty connected open subset, such that the pull-back for some smooth function Ω : U → (0, ∞).(Note that we do not require conformal transformations to be orientation preserving.) The conformal compactification N (V ) of V is constructed using an ambient vector space V ⊕ R 1,1 .The 2-dimensional space R 1,1 has a standard basis {e − , e + }, and the quadratic form By definition, N (V ) is a quadric in the real projective space: Then N (V ) has a natural conformal structure inherited from V ⊕R 1,1 .More concretely, V ⊕R 1,1 contains the product of unit spheres S p × S q , and the quotient map ̟ : V ⊕ R 1,1 \ {0} ։ P (V ⊕ R 1,1 ) restricted to S p × S q induces a smooth 2-to-1 covering π : S p × S q → N (V ), which is a local diffeomorphism.The product S p × S q has a semi-Riemannian metric coming from the product of metrics -the first factor having the standard metric from the embedding S p ⊂ R p+1 as the unit sphere, and the second factor S q having the negative of the standard metric.Then the semi-Riemannian metric on N (V ) is defined by declaring the covering π : S p × S q → N (V ) to be a local isometry.We also see that N (V ) is compact, since it is the image of a compact set S p × S q under a continuous map π.
The embedding map ι : V ֒→ N (V ) can be described as follows.We embed V into the null cone (without the origin) and compose ι with the quotient map Proposition 1.The quadric N (V ) is indeed a conformal compactification of V .That is, the map ι : V → N (V ) is a conformal embedding, and the closure ι(V ) is all of N (V ).
The indefinite orthogonal group O(V ⊕ R 1,1 ) acts on V ⊕ R 1,1 by linear transformations.This action descends to the projective space P (V ⊕ R 1,1 ) and preserves N (V ).Thus, for every T ∈ O(V ⊕ R 1,1 ), we obtain a transformation on N (V ), which will be denoted by ψ T .
The following proposition essentially asserts that the restrictions of ψ T to V are compositions of parallel translations, rotations, dilations and inversions.
Proposition 3.For each T ∈ O(V ⊕ R 1,1 ), the conformal transformation ψ T can be written as a composition of ψ T j , for some When p + q > 2, these are all possible conformal transformations of N (V ).In fact, an even stronger result is true.
Theorem 4. Let p + q > 2. Every conformal transformation ϕ : U → V , where U ⊂ V is any non-empty connected open subset, can be uniquely extended to N (V ), i.e. there exists a unique conformal diffeomorphism φ : Moreover, every conformal diffeomorphism N (V ) → N (V ) must be of the form ψ T , for some T ∈ O(V ⊕ R 1,1 ).Thus, the group of all conformal transformations N ( We emphasize that by the group of conformal transformations of V we mean all conformal transformations N (V ) → N (V ), and these are not required to be orientation preserving.
We conclude this subsection with a classification of points in the conformal compactification N (V ).The points in N (V ) are represented by the equivalence classes [v + x − e − + x + e + ] in the real projective space We can distinguish between the following three classes: ).These points can be identified with vectors in V through (6).
• Q(v) = 0, x − = − 1 2 and x + = 1 2 .Such points represent the image of the null vectors v ∈ V (i.e., vectors with Q(v) = 0) under the unique extension to N (V ) of the inversion map v → v/Q(v).
• Q(v) = x − = x + = 0 and v = 0. We can think of these points as the limiting points of the lines generated by the non-zero null vectors v ∈ V .

Clifford Algebras and Associated Groups
For elementary introductions to Clifford algebras see, for example, [Che54;Gar11].Let V be a real vector space together with a non-degenerate quadratic form Q. Recall the standard construction of the universal Clifford algebra associated to (V, Q) as a quotient of the tensor algebra.We start with the tensor algebra over V , consider a set of elements of the tensor algebra and let (S) denote the ideal of V generated by the elements of S. Then the universal Clifford algebra Cl(V ) associated to (V, Q) can be defined as a quotient We note the sign convention in (7) chosen for the generators of the ideal (S) is not standard in the literature, but it seems to be more prevalent in the context of Clifford analysis.We fix an orthogonal basis {e 1 , e 2 , . . ., e n } of V satisfying (3), and let e 0 ∈ Cl(V ) denote the multiplicative identity of the algebra.Thus Cl(V ) is a finite-dimensional algebra over R generated by e 0 , e 1 , . . ., e n .We consider subsets A ⊆ {1, . . ., n}.If A has k > 0 elements, or, more precisely, e A is the image of this tensor product in Cl(V ).If A is empty, we set e ∅ be the identity element e 0 .These elements form a vector space basis of Cl(V ) over R. In particular, V can be identified with the vector subspace of Cl(V ) spanned by e 1 , e 2 , . . ., e n , and we frequently do so.
If u and v are elements of V ⊂ Cl(V ), then In particular, if u and v are orthogonal in V , uv + vu = 0.
If A is a unital algebra, by a Clifford map we mean an injective linear mapping j : The algebra Cl(V ) is universal in the sense that each such Clifford map j : V → A has a unique extension to a homomorphism of algebras Cl(V ) → A.
There are three important involutions defined on the Clifford algebra Cl(V ).
• The grade involution ˆ: Thus, in the standard basis (8), the grade involution is given by where |A| is the cardinality of A.
• The reversal ˜: Cl(V ) → Cl(V ) is the transpose operation on the level of tensor algebra which descends to the Clifford algebra because the two sided ideal (S) generated by ( 7) is preserved.The transpose operation is defined on each summand V ⊗k of the tensor algebra Hence, for vectors v 1 , . . ., v k ∈ V , we have In the standard basis (8), the reversal is given by The reversal is an algebra anti-homomorphism: ab = bã.
• The Clifford conjugation is defined as the composition of the grade involution and the reversal (note that these two involutions commute).We denote the Clifford conjugation by ā = â.In the standard basis (8), the Clifford conjugation is given by e A = (−1) Like the reversal, the Clifford conjugation is an algebra anti-homomorphism: ab = bā.
We frequently use that the grade involution, the reversal, and the Clifford conjugation all commute with taking the multiplicative inverse in Cl(V ): Also note that a vector v is invertible as an element of Cl(V ) if and only if it is not null, i.e., Q(v) = 0, and An essential feature of Clifford algebras is that the orthogonal transformations on V can be realized via the so-called twisted adjoint action by elements of Cl(V ).Let v ∈ V be a vector that is not null, then every x ∈ V splits into x = x ⊥ + λv with x ⊥ ∈ V orthogonal to v. The map σ v : Cl(V ) → Cl(V ) defined as preserves the space V and produces a reflection in the direction of v. Indeed, Combining such reflections produces various orthogonal transformations on V .
Theorem 5 (Cartan-Dieudonné).Let V be a vector space with non-degenerate quadratic form, then every orthogonal linear transformation on V can be expressed as the composition of at most dim V reflections in the direction of vectors that are not null.
By the Cartan-Dieudonné theorem, all orthogonal linear transformations on V can be expressed using a twisted adjoint action where a ∈ Cl(V ) is a product of vectors in V that are not null.Definition 6.Let Cl × (V ) denote the set of invertible elements in Cl(V ).The Lipschitz group 1 Γ(V ) consists of elements in Cl × (V ) that preserve vector space V under the twisted adjoint action: The following facts about the Lipschitz group Γ(V ) are well known.For each a ∈ Γ(V ), σ a restricts to a linear transformation of V that preserves the quadratic form Q(x).Moreover, we have an exact sequence 1 Definition 7. The pin group Pin(V ) and the spin group Spin(V ) are defined as The pin group and the spin group are double covers of the orthogonal group and special orthogonal group respectively.This can be summarized by the exact sequences 1 Sometimes this group is called Clifford-Lipschitz group or Clifford group.
Proposition 8.The Lipschitz group Γ(V ) is the subgroup of Cl × (V ) generated by R × and the invertible vectors in V .Moreover, every element in Γ(V ) can be expressed as a product of R × and at most n = dim V invertible vectors.

Monogenic Functions
In this subsection we review the definitions of the Dirac operator and monogenic functions in the context of Clifford analysis.Some comprehensive works on Clifford analysis include [BDS82; GM91; DSS92] as well as references therein.
We identify the vector space V with R p,q and introduce the Dirac operator D on R p,q This operator can be applied to functions on the left and on the right.The choice of signs in (13) is compatible with [BS86; BT88; LS21], and it is discussed further in Appendix A.
If f is a twice-differentiable function on R p,q with values in Cl(R p,q ) or a left Cl(R p,q )-module and g is a twice-differentiable function on R p,q with values in Cl(R p,q ) or a right Cl(R p,q )-module, Thus, we can think of the Dirac operator D as a factor of the wave operator p,q on R p,q .
Definition 11.Let U ⊆ R p,q be an open set, and M a left Cl(R p,q )-module.A differentiable function f : at all points in U .
Similarly, let M ′ be a right Cl(R p,q )-module, then a differentiable function g : at all points in U .
We often regard Cl(R p,q ) itself as a Cl(R p,q )-module and speak of left or right monogenic functions with values in Cl(R p,q ).Modules over Cl(R p,q ) are well understood since the seminal paper by Atiyah, Bott and Shapiro [ABS64]; they treated modules over complex and real Clifford algebras.The result is saying that such modules are completely reducible and that there is a classification of the irreducible ones.More detailed exposition of this topic can be found in, for example, [BS86; BT88].Hence, without loss of generality one can consider functions with values in one of these irreducible Cl(R p,q )-modules.
For the remainder of this subsection we discuss some properties of the Dirac operator D.
Lemma 12.The Dirac operator D is independent of the choice of orthogonal basis of V ≃ R p,q satisfying (3).
Proof.Let {e ′ 1 , . . ., e ′ n } be another basis of V satisfying (3), and let D ′ be the Dirac operator associated to this basis.There exists a matrix T ∈ O(p, q) such that Furthermore, we can define the Dirac operator in a basis independent fashion.Let V * denote the dual space of V , and treat the non-degenerate symmetric bilinear form B on V as an element of Denote by B * the element in V ⊗ V corresponding to B. Also, let V ′ denote the space of all translation-invariant first order differential operators on V .Naturally, V and where ∂ v denotes the directional derivative in the direction of vector v.This isomorphism induces a map d : , where D(V ) denotes the space of all V -valued translation-invariant first order differential operators on V : Proposition 13.The image d(B * ) ∈ D(V ) is precisely the Dirac operator D defined by (13).
Proof.Let {e 1 , . . ., e n } be any basis of V , and let {e * 1 , . . ., e * n } be the dual basis of V * , then The matrix with entries B * (e * i , e * j ) is the inverse of the matrix with entries B(e i , e j ).If {e 1 , . . ., e n } is an orthogonal basis of V satisfying (3), it is clear that d(B * ) equals the Dirac operator D defined by (13) relative to the basis {e 1 , . . ., e n }.
Corollary 14.For any basis {e 1 , . . ., e n } of V , let B −1 be the inverse of the matrix with entries B(e i , e j ), then Expression ( 14) essentially serves as the definition of the Dirac operator in [BT88]; this definition is standard in physics literature.

Vahlen Matrices
We introduce Vahlen matrices, which are certain 2 × 2 matrices with entries in Cl(V ), and explain the relation between Vahlen matrices and conformal transformations on V .The results of this section are well known, especially in the positive definite case [Ahl85; Vah02; Mak92; Cno96] (and many other works).The last two references are particularly relevant, since they deal with the indefinite signature case.
Proof.First, we construct a Clifford map j : We need to check that j(x)j 9).This proves that j : By the universal property of Clifford algebras, this map extends to an algebra homomorphism  : Cl(V ⊕ R 1,1 ) → Mat(2, Cl(V )).
To show that  is an isomorphism, note that for a ∈ Cl(V ) ⊂ Cl(V ⊕ R 1,1 ), we have (a) = a 0 0 â , and, for an arbitrary element a + be − + ce + + de − e + ∈ Cl(V ⊕ R 1,1 ), where a, b, c, d ∈ Cl(V ), we have (a + be − + ce It is clear that  is a bijection and hence an algebra isomorphism. From now on we use isomorphism  to identify Cl(V ⊕ R 1,1 ) with Mat(2, Cl(V )).The following result can be deduced from equation (15) describing this isomorphism.Observe that, for all x ∈ V ⊕ R 1,1 , Q(x) = ∆(j(x)). (16)

Vahlen Matrices
We introduce another key ingredient of this paper -Vahlen matrices.
Proposition 19 Proof.By Corollary 9 and linearity, it is sufficient to show that this list of conditions is equiv- These expressions being in V ⊕ R 1,1 for all v ∈ V is equivalent to the first four criteria.From the calculation we see that A Ā ∈ R × is equivalent to the last two criteria.
Recall the definition of the pin group (11).Computation (17) implies the following description of Pin(V ⊕ R 1,1 ) and a property of the pseudo-determinants.At this point, we would like to mention that Cnops [Cno96] has a more refined set of criteria for a matrix A ∈ Mat(2, Cl(V )) to be a Vahlen matrix.His criteria reduce to Ahlfors' criteria given in [Ahl85] when V has positive definite signature.(However, Cnops uses the sign convention v 2 = Q(v) in his definition of a Clifford algebra.)

Conformal Space
In order to relate Vahlen matrices to conformal transformations, we observe that the embedding (6) can be rewritten as This particular presentation encourages us to reinterpret the twisted adjoint action σ of the Lipschitz group as an action on a spinor-like object with two components.We describe a construction of the conformal space due to Maks [Mak92] that makes this idea precise.
Definition 22.The pre-conformal space W pre is the set of products {Ae}, where e = 1 + e − e + 2 = 1 0 0 0 and A ∈ Mat(2, Cl(V )) ranges over all Vahlen matrices.The group of Vahlen matrices Γ(V ⊕ R 1,1 ) acts on W pre by multiplication on the left.
We see that any element in the pre-conformal space W pre must have a matrix realization with entries x, y ∈ Cl(V ).And since these entries form the first column of a Vahlen matrix, by Proposition 19, they satisfy xx, y ȳ ∈ R and xȳ ∈ V .Simplifying notations, we drop the right column and write (x, y) or ( x y ) for an element of W pre represented by ( 19).In order to map the pre-conformal space W pre into the null cone N of V ⊕R 1,1 (recall equation ( 5)) in a way that is compatible with (18), we introduce a map γ : Equivalently, we can use Lemma 16 and express (20) in matrix form From this expression and equation ( 16), we see that γ takes values in the null cone N of V ⊕R 1,1 .For convenience we state the following result (see, for example, [Lam05]).
Lemma 23 (Witt's Extension Theorem).Let V be a finite-dimensional real vector space together with a non-degenerate symmetric bilinear form.
) is the quotient map.
Lemma 24.The map ̟ • γ : W pre → N (V ), which by abuse of notation we also denote by γ, is surjective.
Proof.By the Witt's extension theorem, for each null vector x ∈ N , there exists an orthogonal transformation in O(V ⊕R 1,1 ) that takes n ∞ to x.Since the twisted adjoint action σ is surjective onto O(V ⊕ R 1,1 ), there exists a Vahlen matrix A x such that σ Ax (n ∞ ) = x.By equations ( 12) and (20), we have Therefore, it becomes an equality γ(A x e) = x in N (V ).This proves γ : W pre → N (V ) is surjective.
Lemma 25.Let A = a b c d be a Vahlen matrix.If γ(Ae) is proportional to n ∞ , then c = 0 and a, d ∈ Γ(V ).

Proof. By equation (12), we have
We conclude that c = 0 and a = λ d.From Proposition 19 we see that be two Vahlen matrices.Then γ(A 1 e) and γ(A 2 e) are proportional if and only if there exists an r ∈ Γ(V ) such that (a 1 , c 1 ) = (a 2 r, c 2 r).
Proof.Suppose first (a 1 , c 1 ) = (a 2 r, c 2 r) for some r ∈ Γ(V ), then Since rr ∈ R × , we conclude that γ(A 1 e) and γ(A 2 e) are proportional.Conversely, suppose γ(A 1 e) = λγ(A 2 e) for some λ ∈ R × .Then , Lemma 25 implies that c = 0 and a ∈ Γ(V ), and thus In other words, A 1 e = A 2 ea, or in matrix form This finishes the proof.
This lemma inspires the following definition of the conformal space.
Definition 27.The conformal space W of V is the pre-conformal space W pre modulo the relation that (x 1 , y 1 ) and (x 2 , y 2 ) are equivalent if and only if there exists an r ∈ Γ(V ) such that (x 1 , y 1 ) = (x 2 r, y 2 r).
Proof.By Lemma 26, γ is well defined and injective, and, by Lemma 24, γ is surjective.
For an arbitrary element in W represented by X ∈ W pre , by equation ( 12), we have This becomes an equality in N (V ) and proves that the two actions of Vahlen matrices commute with γ.

Geometry of the Conformal Space W
Recall that eq. ( 18) was our inspiration for the conformal space, so we wish to identify v ∈ V with (v, 1) ∈ W , but we first need to show that (v, 1) ∈ W . Indeed, A = ( v 1 1 0 ) satisfies the conditions of Proposition 19, thereby is a Vahlen matrix.Geometrically, A is the composition of the inversion ( 0 1 1 0 ) followed by the translation ( 1 v 0 1 ), and so γ(Ae) = An ∞ Ã can be obtained from n ∞ by applying the inversion and mapping n ∞ into n 0 , then translating the result by v.
Maks [Mak92] categorizes the points (x, y) in W into three classes.
Otherwise, we can think of (x, y) as the limiting point of the line generated by the non-zero null vector xȳ ∈ V .
By Proposition 3 and equation (10), every Vahlen matrix can be written as a finite product of matrices of these four types.Let U ⊂ V be a non-empty connected open subset.When p + q > 2, by Theorem 4, every conformal transformation U → R p,q can be described by a Vahlen matrix acting through where cx + d is invertible for all x ∈ U .Whenever we write a Vahlen matrix acting on x in this fashion, we always assume that cx + d is invertible.Before we go into the next section, we need to show that the product (xc + d)(cx + d) is a real number.Consequently, when cx+d is invertible, we have (xc+ d)(cx+d) ∈ R × .It is worth mentioning that Maks [Mak92] claims an even stronger result that if an entry in a Vahlen matrix a b c d is invertible, then that entry belongs to Γ(V ).

Conformal Invariance of Monogenic Functions
Monogenic function might not stay monogenic under translations by conformal transformations.That is, starting with a monogenic function f and Vahlen matrix A = a b c d , the composition function f (Ax) need not be monogenic, where Ax is defined by ( 22).On the other hand, it is well known that in the positive definite case (see [Rya84;Boj89]), the function is monogenic.We extend this result to the case of the quadratic form Q on the underlying vector space V ≃ R p,q having arbitrary signature.
Lemma 31.Let A = a b c d be a Vahlen matrix, then for all x, y ∈ V such that cx + d and cy + d are invertible.
Proof.A proof by direct calculation is possible (see [Ahl85;Rya95]).Here, we give a proof by induction based on the fact that every orthogonal transformation in O(V ⊕R 1,1 ) is a composition of translations, orthogonal transformations in O(V ), dilations, and the inversion (Proposition 3).
It is straightforward to show that the formula holds for the Vahlen matrices listed in Subsection 3.4 that produce translations, orthogonal transformations, dilations, and the inversion on V .Then we show that if the formula is true for Vahlen matrices By Corollary 21, it is sufficient to show These two identities are related to each other through the reversal operation, so it is sufficient to show This identity follows from (A 1 x)(c The second identity in (24) follows from the first by applying the reversal.
Corollary 32.The partial derivatives of the conformal transformation produced by a Vahlen matrix A = a b c d can be expressed as Proof.Using equation ( 24), we obtain This expression for the partial derivatives allows us to verify directly that the mapping x → Ax is a conformal transformation (compare with (4)).
Corollary 34.Write B for the matrix of the bilinear form with entries B(e i , e j ) and B −1 for its inverse.We have: Proof.Let ∂A denote the matrix of partial derivatives with ∂(Ax) i ∂x j in the (ij)-entry.Then equation (28) can be rewritten as which in turn can be rewritten as Hence, using (26), and the result follows.
To each Vahlen matrix A = a b c d we associate a function defined for all x ∈ V such that cx + d is invertible.Note that, by Corollary 30, the expression (xc + d)(cx + d) in the denominator is always real.
Lemma 35.Given Vahlen matrices A 1 and A 2 , we have wherever both sides are defined.
Proof.Let j A (x) = cx + d, then J A can be expressed as and equation (25) can be restated as Therefore, Theorem 36.Let U ⊆ V be an open set, M a left Cl(V )-module, and f : U → M a differentiable function.For each Vahlen matrix A = a b c d , Similarly, if M ′ a right Cl(V )-module, and g : U → M ′ a differentiable function, In particular, if g is right monogenic, so is g(Ax) J A (x).
Proof.Choose a basis {e 1 , . . ., e n } of V .By equations ( 14) and (29), Thus, it remains to prove that J A (x) is left monogenic wherever it is defined.
If A is one of the Vahlen matrix listed in Subsection 3.4 that produces a translation, orthogonal linear transformation, dilation on V , then J A is just a constant function, hence monogenic.If A = ( 0 1 1 0 ) represents the inversion on V , then By direct calculation, and J A (x) is monogenic too.
By equation (32), J A (x)f (Ax) is left monogenic whenever f and J A are left monogenic.Thus, by Lemma 35, J A 2 A 1 is left monogenic whenever J A 1 and J A 2 are left monogenic.Since each Vahlen matrix can be written as a finite product of Vahlen matrices realizing translations, orthogonal transformations, dilations and the inversion on V , this proves that J A is left monogenic for all Vahlen matrices A. This finishes the proof of (30).
The proof of (31) is similar.
A A Note on the Definition of the Dirac Operator We introduced the Dirac operator with plus and minus signs in (13), and we have shown that this is a natural construction from the point of view of basis independence.Since this phenomenon does not happen in the positive definite case, it is perhaps worthwhile to show that the classical formula eq. ( 23) fails if the Dirac operator were defined differently, for example, having all positive signs: Note that this differential operator depends on the choice of basis -a different choice of orthogonal basis of V ≃ R p,q satisfying (3) typically leads to a different operator.This can be seen by trying to adapt the proof of Lemma 12 for D * .As a first example, consider R 2,1 with orthogonal generators of Cl(R 2,1 ) satisfying e 2 1 = −1, e 2 2 = −1 and e 2 3 = 1.The function f (x) = x 1 e 1 − x 2 e 2 is in the kernel of both D * and D, but if we consider a Vahlen matrix A = cosh α + e 2 e 3 sinh α 0 0 cosh α + e 2 e 3 sinh α producing a hyperbolic rotation in the e 2 e 3 -plane inside R 2,1 , we find that J A (x)f (Ax) is not in the kernel of D * .Indeed, by direct computation, J A (x) = cosh α − e 2 e 3 sinh α, Ae 1 = e 1 , Ae 2 = e 2 cosh(2α) + e 3 sinh(2α), Ae 3 = e 2 sinh(2α) + e 3 cosh(2α).

Lemma 16 .Definition 17 .
The three involutions of Cl(V ⊕ R 1,1 ) in terms of Mat(2, Cl(V )) are given by the grade involution: For a matrix A = a b c d ∈ Mat(2, Cl(V )), define its pseudo-determinant as ∆(A) = a d − bc.
5. a b = bã, c d = dc, and 6. the pseudo-determinant ∆(A) = a d − bc is a nonzero real number.