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.
Let O be an octonionic Hilbert space. Let A:D(A)⊆O→O and B:D(B)⊆O→O be possibly unbounded self-adjoint linear operators. Then for all h∈D(AB)∩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 Δh(A):=∥Ah−⟨Ah,h⟩h∥. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.
Let X be a p-adic Hilbert space. Let A:D(A)⊆X→X and B:D(B)⊆X→X be possibly unbounded self-adjoint linear operators. For x∈D(A) with ⟨x,x⟩=1, define Δx(A):=∥Ax−⟨Ax,x⟩x∥. Then for all x∈D(AB)∩D(BA) with ⟨x,x⟩=1, we show that (1)max{Δx(A),Δx(B)}≥|⟨[A,B]x,x⟩2+(⟨{A,B}x,x⟩−2⟨Ax,x⟩⟨Bx,x⟩)2||2| and (2)max{Δx(A),Δx(B)}≥|⟨(A+B)x,y⟩|,∀y∈X satisfying ‖y‖≤1,⟨x,y⟩=0. We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.
Let X be a p-adic Hilbert space. Let A:D(A)⊆X→X and B:D(B)⊆X→X be possibly unbounded linear operators. For x∈D(A) with ⟨x,x⟩=1, define $ \Delta _x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all x∈D(AB)∩D(BA) with ⟨x,x⟩=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.
Let X be a non-Archimedean Hilbert space over a conjugated non-Archimedean valued field K with Char(K) ̸= 2. Let A : D(A) ⊆ X → X and B : D(B) ⊆ X → X be possibly unbounded self-adjoint linear operators. For x ∈ D(A) with ⟨x, x⟩ = 1, define Δx(A) := ∥Ax − ⟨Ax, x⟩x∥. Then for all x ∈ D(AB) ∩ D(BA) with ⟨x, x⟩ = 1, we show that (1) Δx(A)+Δx(B)≥max{Δx(A),Δx(B)}≥∣2∣⟨[A,B]x,x⟩2+(⟨{A,B}x,x⟩−2⟨Ax,x⟩⟨Bx,x⟩)2 and (2) Δx(A)+Δx(B)≥max{Δx(A),Δx(B)}≥∣⟨(A+B)x,y⟩∣,∀y∈X satisfying ∥y∥≤1,⟨x,y⟩=0. Inequality (1) may be regarded as a non-Archimedean analogue of the Heisenberg-Robertson-Schrodinger uncertainty principle, while Inequality (2) serves as a non-Archimedean analogue of the Maccone-Pati uncertainty principle.
Let X be a non-Archimedean Hilbert space over a conjugated non-Archimedean valued field K with Char(K)=2. Let A:D(A)⊆X→X and B:D(B)⊆X→X be possibly unbounded self-adjoint linear operators. For x∈D(A) with ⟨x,x⟩=1, define $ Δ_x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all x∈D(AB)∩D(BA) with ⟨x,x⟩=1, we show that \begin{align*} (1) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_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 Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_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*} Inequality (1) may be regarded as a non-Archimedean analogue of the Heisenberg-Robertson-Schrodinger uncertainty principle, while Inequality (2) serves as a non-Archimedean analogue of the Maccone-Pati uncertainty principle.
Let E be a Hilbert C*-module over a unital C*-algebra A. Let A:D(A)⊆E→E and B:D(B)⊆E→E be possibly unbounded self-adjoint morphisms. Then for all x∈D(AB)∩D(BA) with ⟨x,x⟩=1, we show that \begin{align*} (1) \quad \Delta _x(B)^2d_x(A)^2+\Delta _x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2}{2} \end{align*} and \begin{align*} (2) \quad \Delta _x(A)\Delta _x(B)\geq \frac{\sqrt{\|(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2\|}}{2}, \end{align*} where \begin{align*} &\Delta _x(A)\coloneqq \|Ax-\langle Ax, x \rangle x \|, \quad d_x(A)\coloneqq \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\\ &[A,B] \coloneqq AB-BA, \quad \{A,B\}\coloneqq AB+BA, \\ & \{\langle Ax, x \rangle,\langle Bx, x \rangle\}\coloneqq \langle Ax, x \rangle\langle Bx, x \rangle +\langle Bx, x \rangle\langle Ax, x \rangle, \\ & [\langle Ax, x \rangle,\langle Bx, x \rangle]\coloneqq \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle. \end{align*} We call Inequalities (1) and (2) as noncommutative Heisenberg-Robertson-Schrodinger uncertainty principles. They reduce to the Heisenberg-Robertson-Schrodinger uncertainty principle (derived by Schrodinger in 1930) whenever A=C.
Let E be a Hilbert C*-module over a unital C*-algebra A. Let A:D(A)⊆E→E and B:D(B)⊆E→E be possibly unbounded self-adjoint morphisms. Then for all x∈D(AB)∩D(BA) with ⟨x,x⟩=1, we show that \begin{align*} (1) \quad\quad\quad d_x(A)^2+d_x(B)^2&\geq \mp \langle \{A,B\}x, x \rangle +\langle (A\pm B)x, y \rangle\langle y, (A\pm B)x \rangle\pm \{\langle Ax, x \rangle, \langle Bx, x \rangle\}, \\ &\quad \forall y \in \mathcal{E} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle x,y \rangle =0 \end{align*} \begin{align*} (2) \quad \quad \quad d_x(A)^2+d_x(B)^2&\geq \mp i\langle [A,B]x, x \rangle +\langle (A\pm iB)x, y \rangle\langle y, (A\pm iB)x \rangle \mp i[\langle Ax, x \rangle, \langle Bx, x \rangle], \\ &\quad \forall y \in \mathcal{E} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle x,y \rangle =0. \end{align*} where \begin{align*} & d_x(A)\coloneqq \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\quad [A,B] \coloneqq AB-BA, \\ & [\langle Ax, x \rangle,\langle Bx, x \rangle]\coloneqq \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle. \end{align*} We call Inequalities (1) and (2) as noncommutative Maccone-Pati uncertainty principles. They reduce to uncertainty principles derived by Maccone and Pati [\textit{Phys. Rev. Lett., 2014}] whenever A=C.