3-Heisenberg-Robertson-Schrodinger Uncertainty Principle
K. Mahesh Krishna
TL;DR
This work extends the Heisenberg–Robertson–Schrödinger uncertainty principle to three (potentially unbounded) 3-self-adjoint operators acting on a 3-product space. By introducing a 3-product, 3-self-adjointness, and the associated 3-uncertainty $\Delta_x(3,\cdot)$, the author derives a three-operator uncertainty bound: for all $x$ with $\langle x,x,x\rangle=1$ and appropriate domains, $\Delta_x(3,A)\Delta_x(3,B)\Delta_x(3,C) \ge |\langle (ABC-aBC-bAC-cAB)x,x,x\rangle+2abc|$, and an AM-GM-type bound $\frac{1}{27}(\Delta_x(3,A)+\Delta_x(3,B)+\Delta_x(3,C))^3 \ge \Delta_x(3,A)\Delta_x(3,B)\Delta_x(3,C)$. The framework leverages trilinear forms and accommodates unbounded operators, with concrete instantiation in $L^3(\Omega,\mu)$ guiding the construction. Overall, it generalizes Schrödinger’s refinement to a triple-operator setting, providing a new analytic tool for studying joint fluctuations in Banach-space contexts.
Abstract
Let $\mathcal{X}$ be a 3-product space. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$, $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ and $C: \mathcal{D}(C)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded 3-self-adjoint operators. Then for all \begin{align*} x \in \mathcal{D}(ABC)\cap\mathcal{D}(ACB) \cap \mathcal{D}(BAC)\cap\mathcal{D}(BCA) \cap \mathcal{D}(CAB)\cap\mathcal{D}(CBA) \end{align*} with $\langle x, x, x \rangle =1$, we show that \begin{align*} (1)\quad \quad Δ_x(3, A) Δ_x(3, B) Δ_x(3, C)\geq |\langle (ABC-a BC-b AC-c AB)x, x, x\rangle +2abc|, \end{align*} where \begin{align*} Δ_x(3, A):= \|Ax-\langle Ax, x, x \rangle x \|, \quad a:= \langle Ax, x, x \rangle, \quad b := \langle Bx, x, x \rangle, \quad c := \langle Cx, x, x \rangle. \end{align*} We call Inequality (1) as 3-Heisenberg-Robertson-Schrodinger uncertainty principle. Classical Heisenberg-Robertson-Schrodinger uncertainty principle (by Schrodinger in 1930) considers two operators whereas Inequality (1) considers three operators.