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The problem is that "adjoining" a "number" $j$ to $\mathbb R$ such that $j^2=+1$ rather than $i^2=-1$ gives, by Sun-Ze's theorem, $$\mathbb R[j] \approx \mathbb R[x]/\langle x^2-1\rangle \approx \mathbb R[x]/\langle x-1\rangle \oplus \mathbb R[x]/\langle x+1\rangle \approx \mathbb R\oplus \mathbb R$$ In particular, there are $0$-divisors, such as $(0,1)\cdot (1,0)=(0,0)$. A corresponding change of coordinates gives a analogue of the Cauchy-Riemann operator: just $\frac{\partial}{\partial x}\pm \frac{\partial}{\partial y}$.

Notably, these two operators are the factors of the one (spatial) dimensional wave equation, whose analytic features/failings caused so much consternation/interest pre-1800, namely, any function of the form $F(x,y)=f(x-y)$ is apparently annihilated by $\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$, even when $f$ is not as smooth as one might think it ought to be.

This is in extreme contrast to the $i^2=-1$ situation, where the Cauchy-Riemann operator's vanishing gives a definite constraint (a.k.a. "ellipticity").

Edit: [thx for helpful edit-corrections...] If one uses the "usual" norm/length in the denominator, in the definition of "derivative", this does avoid zero-divisors, but changes the thing to being something else entirely, I think.

The fact that $\mathbb R$ with $j$ such that $j^2=+1$ adjoined is not a field is inescapable, and quite unlike the case of complex numbers. On another hand, it is certainly true that Clifford analysis is useful, if not quite in this fashion. Yes, "Dirac operators" have many roles, as factoring second-order differential operators into first-order. Yes, the Laplacian in $\mathbb R^2$ factors into Cauchy-Riemann and its conjugate, and the one-dimensional wave equation (as noted above) factors into two "real" linear operators. This is the goal, actually. For higher dimensions, these operators cannot factor into scalar differential operators, but Clifford-algebra-valued ones. Nevertheless, I don't think it's quite that the underlying "scalars" are made more exotic, but, rather, are enhanced by allowing operator-valued functions.

2 edited body

The problem is that "adjoining" a "number" $j$ to $\mathbb R$ such that $j^2=+1$ rather than $i^2=-1$ gives, by Sun-Ze's theorem, $$\mathbb R[j] \approx \mathbb R[x]/\langle x^2-1\rangle \approx \mathbb R[x]/\langle x-1\rangle \oplus \mathbb R[x]/\langle x+1\rangle \approx \mathbb R\oplus \mathbb R$$ In particular, there are $0$-divisors, such as $(0,1)\cdot (1,0)=(0,0)$. A corresponding change of coordinates gives a analogue of the Cauchy-Riemann operator: just $\frac{\partial}{\partial x}\pm \frac{\partial}{\partial y}$.

Notably, these two operators are the factors of the one (spatial) dimensional wave equation, whose analytic features/failings caused so much consternation/interest pre-1800, namely, any function of the form $F(x,y)=f(x-y)$ is apparently annihilated by $\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$, even when $f$ is not as smooth as one might think it ought to be.

This is in extreme contrast to the $i^2=+1$ i^2=-1$situation, where the Cauchy-Riemann operator's vanishing gives a definite constraint (a.k.a. "ellipticity"). 1 The problem is that "adjoining" a "number"$j$to$\mathbb R$such that$j^2=+1$rather than$i^2=-1$gives, by Sun-Ze's theorem, $$\mathbb R[j] \approx \mathbb R[x]/\langle x^2-1\rangle \approx \mathbb R[x]/\langle x-1\rangle \oplus \mathbb R[x]/\langle x+1\rangle \approx \mathbb R\oplus \mathbb R$$ In particular, there are$0$-divisors, such as$(0,1)\cdot (1,0)=(0,0)$. A corresponding change of coordinates gives a analogue of the Cauchy-Riemann operator: just$\frac{\partial}{\partial x}\pm \frac{\partial}{\partial y}$. Notably, these two operators are the factors of the one (spatial) dimensional wave equation, whose analytic features/failings caused so much consternation/interest pre-1800, namely, any function of the form$F(x,y)=f(x-y)$is apparently annihilated by$\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$, even when$f$is not as smooth as one might think it ought to be. This is in extreme contrast to the$i^2=+1\$ situation, where the Cauchy-Riemann operator's vanishing gives a definite constraint (a.k.a. "ellipticity").