Quantizing the Quantum Uncertainty

TL;DR

This paper explores the quantization of quantum uncertainty as an operator with a discrete spectrum, revealing its role in wave-packet evolution and posing questions about potential experimental signatures indicating a transition from quantum mechanics to quantum field theory.

Contribution

It introduces a novel operator for quantum uncertainty with a discrete spectrum derived from Lie algebra, linking it to wave-packet dynamics and effective potentials in quantum mechanics.

Findings

Quantum uncertainty operator has a discrete spectrum from $ extrm{sl}_{2}$ algebra.

Spectrum influences the coupling of the conformal potential in wave-packet evolution.

Open question on experimental detection of quantized quantum uncertainty.

Abstract

The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the $\textrm{sl}_{2}$ Lie algebra formed by the operators $\hat{x}^{2}$, $\hat{p}^{2}$ and $\widehat{xp}$. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schr\"odinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?