Generalizations of wave equations to multidimensional variational problems

TL;DR

This survey explores the extension of wave equations to multidimensional variational problems, focusing on generalized Schrödinger and Plato equations relevant to quantum field and string theories.

Contribution

It introduces a framework for defining and analyzing generalized wave equations in multidimensional variational contexts, connecting them to quantum field and string theories.

Findings

Proposes a mathematical formulation for generalized Schrödinger and Plato equations.

Links these equations to fundamental theories in physics.

Provides a basis for future rigorous analysis of multidimensional variational wave equations.

Abstract

This is a survey paper based on previous results of the author. In the paper, we define and discuss the generalizations of linear partial differential equations to multidimensional variational problems. We consider two examples of such equations: first, the generalized Schr\"odinger equation which is a natural candidate for the mathematical equation of quantum field theory, and second, the quantum Plato problem which is a natural candidate for a simplest mathematical equation of string theory and, more generally, theory of $D$-branes. We propose a way to give a mathematical sense to these equations.