math.MP — Mathematical Physics

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

Contributed by 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.