Conformal Invariance of Clifford Monogenic Functions in the Indefinite Signature Case

Abstract

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 Möbius) 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.

Appendix: 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:

$$\begin{aligned} D^* = e_1 \frac{\partial }{\partial {x_1}} + \cdots + e_p \frac{\partial }{\partial {x_p}} + e_{p+1} \frac{\partial }{\partial {x_{p+1}}} + \cdots + e_{p+q} \frac{\partial }{\partial {x_{p+q}}}. \end{aligned}$$

Note that this differential operator depends on the choice of basis—a different choice of orthogonal basis of $V \simeq {\mathbb {R}}^{p,q}$ 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 ${\mathbb {R}}^{2,1}$ with orthogonal generators of $\text {Cl}({\mathbb {R}}^{2,1})$ satisfying $e_1^2 = -1$, $e_2^2 = -1$ and $e_3^2 = 1$. The function $f(x) = x_1e_1 - x_2e_2$ is in the kernel of both $D^*$ and $D$, but if we consider a Vahlen matrix

$$\begin{aligned} A = \begin{pmatrix} \cosh \alpha + e_2 e_3 \sinh \alpha &{} 0 \\ 0 &{} \cosh \alpha + e_2 e_3 \sinh \alpha \end{pmatrix} \end{aligned}$$

producing a hyperbolic rotation in the $e_2e_3$-plane inside ${\mathbb {R}}^{2,1}$, we find that $J_A(x) f(Ax)$ is not in the kernel of $D^*$. Indeed, by direct computation,

$$\begin{aligned} J_A(x)= & {} \cosh \alpha - e_2 e_3 \sinh \alpha ,\\ A e_1 = e_1, \quad A e_2 = e_2 \cosh (2\alpha )\!+ & {} \! e_3 \sinh (2\alpha ), \quad A e_3 = e_2 \sinh (2 \alpha ) + e_3 \cosh (2\alpha ). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} J_A(x) f(Ax)&= (\cosh \alpha - e_2 e_3 \sinh \alpha ) \bigl ( x_1 e_1 - ( x_2 \cosh (2\alpha ) + x_3 \sinh (2\alpha )) e_2 \bigr ) \\&= x_1 e_1 (\cosh \alpha - e_2 e_3 \sinh \alpha ) \\&- ( x_2 \cosh (2\alpha ) + x_3 \sinh (2\alpha ) ) ( e_2 \cosh \alpha - e_3 \sinh \alpha ). \end{aligned} \end{aligned}$$

Then $D^*$ acting on the first term yields

$$\begin{aligned} D^* \bigl [ x_1 e_1 (\cosh \alpha - e_2 e_3 \sinh \alpha ) \bigr ] = -(\cosh \alpha - e_2 e_3 \sinh \alpha ), \end{aligned}$$

whereas the second term results in

$$\begin{aligned} D^* \bigl [ - ( x_2 \cosh (2\alpha ) + x_3 \sinh (2\alpha ) ) ( e_2 \cosh \alpha - e_3 \sinh \alpha ) \bigr ] = \cosh (3\alpha ) + e_2 e_3 \sinh (3\alpha ). \end{aligned}$$

They clearly do not cancel each other. To contrast, if we use the true Dirac operator,

$$\begin{aligned} \begin{aligned}&D \bigl [ x_1 e_1 (\cosh \alpha - e_2 e_3 \sinh \alpha ) \bigr ] = -(\cosh \alpha - e_2 e_3 \sinh \alpha ), \\&D \bigl [ - ( x_2 \cosh (2\alpha ) + x_3 \sinh (2\alpha ) ) ( e_2 \cosh \alpha - e_3 \sinh \alpha ) \bigr ] = \cosh \alpha - e_2 e_3 \sinh \alpha , \end{aligned} \end{aligned}$$

as desired.

In general, the true Dirac operator $D$ plays well with orthogonal linear transformations, while the ostensible Dirac operator $D^*$ does not. More precisely, restricting to the Vahlen matrices producing orthogonal linear transformations on $V$, that is, matrices of the form , where $a \in \Gamma (V)$, Eq. (23) becomes

$$\begin{aligned} J_A(x) f(Ax) = \frac{\hat{a}^{-1}}{|\tilde{a}a|^{n/2-1}} f(a x \hat{a}^{-1}) \sim \hat{a}^{-1} f(a x \hat{a}^{-1}). \end{aligned}$$

It was established in Theorem 36 that

$$\begin{aligned} Df=0 \qquad \text {implies} \qquad D \bigl ( \hat{a}^{-1} f(a x \hat{a}^{-1}) \bigr ) = 0. \end{aligned}$$

On the other hand, as the above example shows,

$$\begin{aligned} \text {when} \quad D^*f=0, \quad \text {typically} \quad D^* \bigl ( \hat{a}^{-1} f(a x \hat{a}^{-1}) \bigr ) \ne 0. \end{aligned}$$

The ostensible Dirac operator $D^*$ also does not behave well with respect to the inversion. The constant function $f(x)=1$ is annihilated by $D^*$ (and $D$). Applying to this function the inversion realized by results in $J_A(x)$, and $J_A(x)$ is not in the kernel of $D^*$. For concreteness, consider ${\mathbb {R}}^{1,1}$ with generators of $\text {Cl}({\mathbb {R}}^{1,1})$ satisfying $e_1^2 = -1$, $e_2^2 = 1$. By Eq. (33), we have

$$\begin{aligned} J_A(x) = {{\,\textrm{sgn}\,}}(x^2) \frac{x}{|x^2|} = \frac{x_1e_1 + x_2e_2}{(x_2)^2 - (x_1)^2}. \end{aligned}$$

Then $D^*[x_1e_1 + x_2e_2] = 0$, and

$$\begin{aligned} D^* J_A(x) = 2 \frac{(x_1e_1-x_2e_2)(x_1e_1 + x_2e_2)}{((x_2)^2 - (x_1)^2)^2} = 2 \frac{-(x_1)^2-(x_2)^2+2x_1x_2e_1e_2}{((x_2)^2 - (x_1)^2)^2}, \end{aligned}$$

which is clearly non-zero.