how does matrix theory yield the Laplace equation from the Cauchy-Riemann equations

Question

In her paper "How I became a torchbearer for matrix theory" Taussky writes:

In the late thirties Isuddenly realized that the Cauchy-Riemann equations and the fact that they imply the Laplace equation can be expressed via matrix theory and can be connected with the fact that the field of complex numbers has no zero divisors.

how can matrix theory yield the Laplace equation from the Cauchy-Riemann equations?

Answer 1

Consider the Wirtinger derivatives

$$\partial_z = \frac{1}{2} \left( \partial_x - i \partial_y \right)$$ $$\partial_{\bar{z}} = \frac{1}{2} \left( \partial_x + i \partial_y \right)$$

acting on smooth functions $f : \mathbb{C} \to \mathbb{C}$, where we tacitly identify $\mathbb{C} \cong \mathbb{R}^2$ via $z = x + iy \mapsto (x, y)$. Write $z = x + iy, \bar{z} = x - iy$, regarded as smooth functions $\mathbb{C} \to \mathbb{C}$. It is a pleasant exercise to check the following:

1. $\partial_z z = 1, \partial_z \bar{z} = 0$.
2. $\partial_{\bar{z}} z = 0, \partial_{\bar{z}} \bar{z} = 1$.
3. $f$ is holomorphic iff $\partial_{\bar{z}} f = 0$ (so this condition is equivalent to the Cauchy-Riemann equations) and anti-holomorphic iff $\partial_z f = 0$.
4. The Laplacian $\Delta = \partial_x^2 + \partial_y^2$ can be written $\Delta = 4 \partial_z \partial_{\bar{z}} = 4 \partial_{\bar{z}} \partial_z$. Hence it immediately follows that any holomorphic or anti-holomorphic function is harmonic.

Presumably this is what Taussky is referring to. The conceptual innovation here is the idea of treating differential operators as algebraic objects in their own right; they form an infinite-dimensional ring of linear operators generalizing the familiar concept of finite-dimensional matrices, which maybe is why Taussky calls this "matrix theory" (this is not exactly a standard term so I don't know what Taussky is using it to mean).

To get the above factorization we need to know that 1) the partial derivatives commute, and 2) that over $\mathbb{C}$ we can factor a sum of two squares. However, I don't believe this argument really uses that $\mathbb{C}$ is a field, so I don't know what Taussky has in mind there. Similar factorizations are available involving Clifford algebras in higher dimensions, and Clifford algebras aren't division algebras in general.