3-Heisenberg-Robertson-Schrodinger Uncertainty Principle

2024/12/02 by K. Mahesh Krishna, Krishna, K. Mahesh
Decision Sciences · #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2412.10396

Abstract

Let X be a 3-product space. Let A: D(A)⊆ X→ X, B: D(B)⊆ X→ X and C: D(C)⊆ X→ X be possibly unbounded 3-self-adjoint operators. Then for all x ∈ D(ABC)\capD(ACB) ∩ D(BAC)\capD(BCA) ∩ D(CAB)\capD(CBA) with ⟨ x, x, x ⟩ =1, we show that (1) Δ_x(3, A) Δ_x(3, B) Δ_x(3, C)≥ |⟨ (ABC-a BC-b AC-c AB)x, x, x⟩ +2abc|, where Δ_x(3, A):= ‖Ax-⟨ Ax, x, x ⟩ x ‖, a:= ⟨ Ax, x, x ⟩, b := ⟨ Bx, x, x ⟩, c := ⟨ Cx, x, x ⟩. We call Inequality (1) as 3-Heisenberg-Robertson-Schrodinger uncertainty principle. Classical Heisenberg-Robertson-Schrodinger uncertainty principle (by Schrodinger in 1930) considers two operators whereas Inequality (1) considers three operators.

Related

• p-adic Heisenberg-Robertson-Schrodinger and p-adic Maccone-Pati Uncertainty Principles
• Noncommutative Heisenberg-Robertson-Schrodinger Uncertainty Principles
• Uncertainty relations for two observables coupled with the third one
• Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle
• General and Stronger Uncertainty Relation