Does there exist the applicability limit of PDE to describe physical phenomena? -- A personal survey of Quantization, QED, Turbulence

TL;DR

This paper explores the applicability and limitations of partial differential equations (PDEs) in describing complex physical phenomena such as quantum mechanics, quantum electrodynamics, and turbulence, questioning whether these phenomena can be fully captured by PDEs.

Contribution

It provides a personal survey analyzing the extent to which PDEs can describe advanced physical phenomena like QED and turbulence, highlighting potential limitations and open questions.

Findings

QED's equations do not straightforwardly correspond to classical PDEs.

Turbulence remains a challenging phenomenon for PDE-based modeling.

Quantum phenomena may require frameworks beyond traditional PDEs.

Abstract

What does it mean to study PDE(=Partial Differential Equation)? How and what to do "to claim proudly that I'm studying a certain PDE"? Newton mechanic uses mainly ODE(=Ordinary Differential Equation) and describes nicely movements of Sun, Moon and Earth etc. Now, so-called quantum phenomenum is described by, say Schr\"odinger equation, PDE which explains both wave and particle characters after quantization of ODE. The coupled Maxwell-Dirac equation is also "quantized" and QED(=Quantum Electro-Dynamics) theory is invented by physicists. Though it is said this QED gives very good coincidence between theoretical and experimental observed quantities, but what is the equation corresponding to QED? Or, is it possible to describe QED by "equation" in naive sense?