[Paper Review] A connection between the Uncertainty Principles on the real line and on the circle

Nils Byrial Andersen|arXiv (Cornell University)|Jul 18, 2013

Mathematical Analysis and Transform Methods6 references3 citations

TL;DR

This paper establishes a novel connection between the Heisenberg Uncertainty Principle on the real line and the Breitenberger Uncertainty Principle on the circle by analyzing the commutator of multiplication and backward difference operators on Bernstein spaces. The key result shows that the Heisenberg principle emerges as a limit when the difference step size δ→0, while the Breitenberger principle arises exactly at δ=1 with R=π, unifying both principles through operator-theoretic analysis on L2 functions with bandlimited Fourier transforms.

ABSTRACT

The purpose of this short note is to exhibit a new connection between the Heisenberg Uncertainty Principle on the line and the Breitenberger Uncertainty Principle on the circle, by considering the commutator of the multiplication and difference operators on Bernstein functions

Motivation & Objective

• To establish a new mathematical connection between the Heisenberg Uncertainty Principle on the real line and the Breitenberger Uncertainty Principle on the circle.
• To demonstrate that both uncertainty principles arise as limiting cases of a single operator-theoretic inequality involving backward difference and multiplication operators.
• To unify these two distinct uncertainty principles through the framework of Bernstein spaces and commutator analysis.
• To show that the same underlying structure—via the commutator of difference and multiplication operators—generates both principles depending on the parameter δ.

Proposed method

• The study employs Bernstein spaces $B^{2}_{R}$, defined as L2 functions on R whose Fourier transforms are supported in [-R, R], to model bandlimited functions.
• It introduces the normalized backward difference operator $A_{ u}f(z) = \frac{f(z) - f(z - \delta)}{\delta}$ on $B^{2}_{R}$, which is shown to be normal.
• The multiplication operator $Bf(x) = xf(x)$ is defined on the subspace $\dot{B}^{2}_{R}$ where $xf(x) \in L^2(\mathbb{R})$.
• The commutator $[A_{\delta}, B]$ is computed explicitly as $[A_{\delta}, B]f(x) = f(x - \delta)$, which is central to deriving the uncertainty inequality.
• The uncertainty inequality $\|(A_{\delta} - a)f\|_2 \|(B - b)f\|_2 \geq \frac{1}{2}|\langle f(\cdot - \delta), f \rangle|$ is derived from a general theorem on symmetric and normal operators.
• The limiting cases δ→0 and δ=1 (with R=π) are analyzed to recover the Heisenberg and Breitenberger uncertainty principles, respectively.

Experimental results

Research questions

• RQ1 How are the Heisenberg and Breitenberger uncertainty principles related through a common mathematical framework?
• RQ2 Can the Heisenberg uncertainty principle on the real line be derived as a limit of a discrete uncertainty principle on the circle?
• RQ3 What role does the commutator of the backward difference and multiplication operators play in unifying these two uncertainty principles?
• RQ4 Does the same operator-theoretic inequality generate both uncertainty principles depending on the choice of δ?
• RQ5 Can the central difference operator also be used to recover the Heisenberg uncertainty principle in the limit δ→0?

Key findings

• The uncertainty inequality $\|(A_{\delta} - a)f\|_2 \|(B - b)f\|_2 \geq \frac{1}{2}|\langle f(\cdot - \delta), f \rangle|$ holds for all $f \in \dot{B}^{2}_{R}$ and all $a, b \in \mathbb{C}$, with $A_{\delta}$ being the normalized backward difference operator.
• In the limit $\delta \to 0$, the inequality reduces to the Heisenberg Uncertainty Principle: $\|f' - a f\|_2 \|(x - b)f\|_2 \geq \frac{1}{2}\|f\|_2^2$.
• At $\delta = 1$ and $R = \pi$, the inequality yields the Breitenberger Uncertainty Principle on the circle: $\|(e^{i\theta} - a)\check{f}\|_2 \|\left(\frac{d}{d\theta} - b\right)\check{f}\|_2 \geq \frac{1}{2}|\langle e^{i\theta}\check{f}, \check{f} \rangle|$.
• The same framework applies to central difference operators $C_{\delta}$, yielding a new uncertainty principle on the circle at $\delta = 1$, $R = \pi$: $\|\left(\sin \theta - a\right)\check{f}\|_2 \|\left(\frac{d}{d\theta} - b\right)\check{f}\|_2 \geq \frac{1}{2}|\langle \cos \theta \check{f}, \check{f} \rangle|$.
• The result provides a continuous interpolation between the two uncertainty principles via the parameter $\delta$, showing that both are special cases of a single inequality on Bernstein spaces.
• The inequality supports the intuition that small R (frequency localization) leads to large position uncertainty, as shown by the derived bound $\|(x - a)f\|_2 \geq \frac{\|f\|_2}{2R}$.