Noncommutative Heisenberg-Robertson-Schrodinger Uncertainty Principles

Abstract

Let $\mathcal{E}$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$. Let $A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}$ and $B: \mathcal{D}(B)\subseteq \mathcal{E}\to \mathcal{E}$ be possibly unbounded self-adjoint morphisms. 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 \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 \quad \quad \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 $\Delta _x(A):= \|Ax-\langle Ax, x \rangle x \|$, $d_x(A):= \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2}$, $[A,B] := AB-BA$, $\{A,B\}:= AB+BA$, $\{\langle Ax, x \rangle,\langle Bx, x \rangle\}:= \langle Ax, x \rangle\langle Bx, x \rangle +\langle Bx, x \rangle\langle Ax, x \rangle$, $[\langle Ax, x \rangle,\langle Bx, x \rangle]:= \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle$. 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 $\mathcal{A}=\mathbb{C}$.