What is pluripotential theory?

Question

My tutor for electromagnetism showed me a problem about point charges in a disk and their equilibria. He referred me to a subject called "pluripotential theory". I googled it and I did not find what I was looking for at all!

So my question is, what is pluripotential theory, what is it used for and are there any interesting and/or intuitive aspects/results of the field.

EDIT: I might have been unclear with the phrasing of the question. I am not looking for an account of potentials of more than one point charge. I am aware that pluripotential theory is a field in it's own and this is what I am interested in learning about.

Answer 1

It is probably the case that your tutor meant to say potential theory, which is absolutely related. To try to describe the relation, let us start by picturing a situation you might encounter in the subject you are working on: you have some distribution of charges in, say, 3-dimensional space $\mathbb{R}^3$. Maybe it's a collection of point charges, maybe the charge is evenly distributed on a sphere, whatever. You've probably learned that such a charge distribution gives rise to an, electrostatic potential, which is a scalar function $\phi$ on $\mathbb{R}^3$. But you can go backwards too: if I give you the potential function $\phi$ without telling you what the charge distribution is, you can figure out what the charge distribution is. This is essentially Gauss' Law. Specifically, if $\rho$ is the charge density function, then Gauss' Law tells you that $$\rho = -\varepsilon_0\Delta\phi,$$ where here $\Delta$ is the Laplacian operator $$\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}.$$ Roughly speaking, this says that there is some sort of correspondence between charge distributions $\rho$ and potential functions $\phi$. This correspondence is not bijective, in that a given charge distribution $\rho$ can have more than one potential function associated to it. Indeed if $\psi$ is a function such that $\Delta\psi = 0$, then $\phi + \psi$ will also be a potential function for $\rho$, since we will still have that $\rho = -\varepsilon_0\Delta(\phi + \psi)$. Such functions $\psi$ are called harmonic functions, and are one of the main objects studied in potential theory.

Potential theory is a mathematical field that puts what I just described on solid theoretical ground. The concept of a charge distribution $\rho$ is replaced with a mathematical object called a measure, and the potential functions $\phi$ are replaced with the mathematical objects called subharmonic functions. Thus a (positive) measure $\mu$ might induce a potential function $\phi$, which is subharmonic, and conversely, given a subharmonic function $\phi$, one can obtain a measure $\mu$ as $\mu = \Delta\phi$. Generally speaking, potential theory is the study of harmonic and subharmonic functions, and their relationships to the measures they encode.

Just a couple more comments:

1. While potential theory and pluripotential theory are different subjects, they are related and often have analogous aims. For instance, in pluripotential theory one defines the notion of a plurisubharmonic function $\phi$ (think of as an analogue of potential function), a differential operator called $dd^c$ (think of as an analogue of the Laplacian $\Delta$), and the notion of a positive closed current $T$ (think of as an analogue of a distribution of positive charges), and one is often interested in solutions of the equation $T = dd^c\phi$ (think of as an analogue of $\rho = -\varepsilon_0\Delta \phi$). There certainly is more to pluripotential theory than that, however, as is alluded to in user98130's answer.
2. I have to strongly disagree with user98130's comment "it's an acquired taste (acquired within a PhD program in mathematics)". These are beautiful subjects, and if you are interested in learning them, then by all means keep being curious and learning, even if you don't want to enter a math PhD program. Math learning is something that should be available to anyone. Sites like this can be a great resource if you have questions along the way.

Answer 2

Your tutor introduced some terminological confusion. Studying sets of multiple point charges is a fun and worthwhile activity, but it should not be called pluripotential theory. That name is used for a branch of analysis in several complex variables, one that deals with pluriharmonic and plurisubharmonic functions, complex Monge–Ampère equation, positive currents, and reaches into the geometry of Kähler manifolds. There is a lot of interesting stuff there, but it's an acquired taste (acquired within a PhD program in mathematics). Since what you are really after are systems of point charges, pluripotential theory does not have to be within your area of concern.