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

K. Mahesh Krishna

K. Mahesh Krishna

Lubomír Skála

K. Mahesh Krishna

Let $\mathcal{X}$ be a p-adic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ \Delta _x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.

K. MAHESH KRISHNA

Let $\mathcal{O}$ be an octonionic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{O}\to \mathcal{O}$ and $B:\mathcal{D}(B)\subseteq \mathcal{O}\to \mathcal{O}$ be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$, 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{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $ \Delta _h(A):= \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.

Charles H. Holbrow, James N. Lloyd, Joseph C. Amato +2 more