Noncommutative Heisenberg-Robertson-Schrodinger Uncertainty Principles
K. Mahesh Krishna
TL;DR
This work extends the Heisenberg–Robertson–Schrödinger uncertainty framework to Hilbert C*-modules by introducing module-appropriate, operator-valued notions of uncertainty. It defines $\Delta_x(A)$ and $d_x(A)$ for possibly unbounded self-adjoint morphisms $A$ on a module and derives two noncommutative inequalities that bound combinations of these quantities in terms of $\langle \{A,B\}x, x\rangle$ and $\langle [A,B]x, x\rangle$, valid for $x$ with $\langle x,x\rangle=1$ in the joint domain $\mathcal{D}(AB)\cap\mathcal{D}(BA)$. The main contributions are the explicit inequalities (1) and (2) and the demonstration that they recover the classical Schrödinger inequality when the algebra is commutative or reduces to $\mathbb{C}$. The results provide a rigorous noncommutative generalization with potential implications for uncertainty quantification in noncommutative geometry and operator-algebraic quantum frameworks.
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 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 $Δ_x(A):= \|Ax-\langle Ax, x \rangle x \|$, $d_x(A):= \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2}$, $[A,B] := AB-BA$, $\{A,B\}:= AB+BA$, $\{\langle Ax, x \rangle,\langle Bx, x \rangle\}:= \langle Ax, x \rangle\langle Bx, x \rangle +\langle Bx, x \rangle\langle Ax, x \rangle$, $[\langle Ax, x \rangle,\langle Bx, x \rangle]:= \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle$. 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 $\mathcal{A}=\mathbb{C}$.