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Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

K. Mahesh Krishna

TL;DR

The paper extends the Heisenberg-Robertson-Schrodinger uncertainty principle to a nonlinear Banach-space setting by introducing a nonlinear framework for Lipschitz maps. It defines two uncertainties, Δ and ∇, via a Lipschitz functional and proves a chain of inequalities that bound nonlinear deviations, thereby generalizing the classical operator-based HR-S relation. A key result is that this nonlinear uncertainty principle reduces to the classical HR-S principle when the maps are linear operators on a Hilbert space, illustrating consistency with the standard quantum-mechanical uncertainty framework. The work broadens the scope of uncertainty relations to nonlinear contexts and Lipschitz maps on Banach spaces, potentially impacting analyses where linearity cannot be assumed.

Abstract

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.

Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

TL;DR

The paper extends the Heisenberg-Robertson-Schrodinger uncertainty principle to a nonlinear Banach-space setting by introducing a nonlinear framework for Lipschitz maps. It defines two uncertainties, Δ and ∇, via a Lipschitz functional and proves a chain of inequalities that bound nonlinear deviations, thereby generalizing the classical operator-based HR-S relation. A key result is that this nonlinear uncertainty principle reduces to the classical HR-S principle when the maps are linear operators on a Hilbert space, illustrating consistency with the standard quantum-mechanical uncertainty framework. The work broadens the scope of uncertainty relations to nonlinear contexts and Lipschitz maps on Banach spaces, potentially impacting analyses where linearity cannot be assumed.

Abstract

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.
Paper Structure (3 sections, 9 theorems, 18 equations)

This paper contains 3 sections, 9 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle
  3. Acknowledgments

Key Result

Theorem 1.1

ROBERTSONCASSIDYHEISENBERGVONNEUMANNBOOKDEBNATHMIKUSINSKIOZAWA (Heisenberg-Robertson Uncertainty Principle) Let $A: \mathcal{D}(A)\to \mathcal{H}$ and $B: \mathcal{D}(B)\to \mathcal{H}$ be self-adjoint operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$, we have

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • ...and 3 more