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 $⟨x,x⟩=1$, we show that \begin{align*} (1) \quad \quad \quad Δ_x(B)^2d_x(A)^2+Δ_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 Δ_x(A)Δ_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 $\hat{I}{”}_x(A):=\Vert Ax-⟨Ax,x⟩x\Vert$, $d_x(A):=\sqrt{⟨Ax,Ax⟩-⟨Ax,x{⟩}^2}$, $[A,B]:=AB-BA$, $\{A,B\}:=AB+BA$, $\{⟨Ax,x⟩,⟨Bx,x⟩\}:=⟨Ax,x⟩⟨Bx,x⟩+⟨Bx,x⟩⟨Ax,x⟩$, $[⟨Ax,x⟩,⟨Bx,x⟩]:=⟨Ax,x⟩⟨Bx,x⟩-⟨Bx,x⟩⟨Ax,x⟩$. 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 $\mathcal{A}=\mathbb{C}$.

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