Constants of Motion for Non-Differentiable Quantum Variational Problems

TL;DR

This paper extends classical variational principles and Noether's theorem to non-differentiable functions within scale relativity, deriving constants of motion for quantum systems like the Schrödinger equation.

Contribution

It introduces a generalized framework for variational calculus and symmetry analysis applicable to non-differentiable quantum problems, expanding the theoretical foundation.

Findings

Derived constants of motion for linear Schrödinger variants

Extended Noether's theorem to non-differentiable settings

Applied results to specific quantum equations

Abstract

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schrodinger equation.