p-adic Heisenberg-Robertson-Schrodinger and p-adic Maccone-Pati Uncertainty Principles
K. Mahesh Krishna
TL;DR
Problem addressed: develop p-adic analogues of the Heisenberg-Robertson-Schrodinger and Maccone-Pati uncertainty principles in p-adic Hilbert spaces. Method: define the p-adic uncertainty functional $Δ_x(A)=|| Ax- <Ax, x> x ||$ and prove both HR-S type and MP type inequalities for possibly unbounded operators, including self-adjoint and adjointable cases; the bounds feature the commutator [A,B], anticommutator {A,B}, and ultrametric refinements. Main results: (i) $Δ_x(A)Δ_x(B) \ge |\langle Ax, Bx\rangle-\langle Ax,x\rangle\langle Bx,x\rangle|$, and (ii) for self-adjoint A,B, $Δ_x(A)+Δ_x(B) \ge \max\{Δ_x(A),Δ_x(B)\} \ge \frac{ \sqrt{|\langle [A,B]x, x\rangle^2 + (\langle {A,B}x, x\rangle - 2\langle Ax, x\rangle\langle Bx, x\rangle)^2|}}{\sqrt{|2|}$, with a Maccone-Pati form $Δ_x(A)+Δ_x(B) \ge \max\{Δ_x(A),Δ_x(B)\} \ge |\langle (A±B)x, y\rangle|$ for all unit y orthogonal to x. Significance: extends foundational uncertainty relations to non-Archimedean, p-adic contexts, enabling theoretical investigations of p-adic quantum-like systems.
Abstract
Let $\mathcal{X}$ be a p-adic Hilbert space. Let $A:\mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded self-adjoint linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ Δ_x(A):= \|Ax- \langle Ax, x \rangle x \|.$ 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 \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 \quad \quad \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*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.