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Quaternionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

Primarily human-written textHuman understanding: all partsmath.MP — Mathematical Physicsmath.FA — Functional Analysis

Contributed by K. MAHESH KRISHNA ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

Let Q\mathcal{Q} be a quaternionic Hilbert space. Let A:D(A)⊆Q→QA: \mathcal{D}(A)\subseteq \mathcal{Q}\to \mathcal{Q} and B:D(B)⊆Q→QB:\mathcal{D}(B)\subseteq \mathcal{Q}\to \mathcal{Q} be possibly unbounded self-adjoint linear operators. Then for all h∈D(AB)∩D(BA)h \in \mathcal{D}(AB)\cap \mathcal{D}(BA) with ∥h∥=1\|h\|=1, we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{Q} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where Δh(A)≔∥Ah−⟨Ah,h⟩h∥ \Delta _h(A)\coloneqq \|Ah-\langle Ah, h \rangle h \|. We call Inequality (1) as quaternionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as quaternionic Maccone-Pati uncertainty principle.

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Version history

  1. v1 · 7 pagesSubmitted by K. MAHESH KRISHNAInitial depositCurrentOct 20, 2026