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Uncertainty principles as forcing mechanisms
Abstract
In this note we first consider the Heisenberg uncertainty principle. We show that it acts as a {\bf forcing mechanism} in the following sense. Real functions that approach its limit are close to Gaussian distributions, with proximity both in the sense of total variation distance and Wasserstein-1 distance. We next consider the uncertainty principle for functions defined on finite Abelian groups, and show that it acts as a forcing mechanism in the following sense. Functions that approach its limit are close (after translation and modulation by a complex exponential) to indicator functions of some subgroup, with proximity measured in 𝕃1-norm.
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BibTeX
@misc{massoulie2026uncertainty,
title = {{Uncertainty principles as forcing mechanisms}},
author = {Laurent Massoulié},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38909},
eprint = {2610.38909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38909},
}
RIS
TY - UNPB
TI - Uncertainty principles as forcing mechanisms
AU - Laurent Massoulié
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38909
UR - https://arxiv.org/abs/2610.38909
AB - In this note we first consider the Heisenberg uncertainty principle. We show that it acts as a {\bf forcing mechanism} in the following sense. Real functions that approach its limit are close to Gaussian distributions, with proximity both in the sense of total variation distance and Wasserstein-1 distance. We next consider the uncertainty principle for functions defined on finite Abelian groups, and show that it acts as a forcing mechanism in the following sense. Functions that approach its limit are close (after translation and modulation by a complex exponential) to indicator functions of some subgroup, with proximity measured in 𝕃^1-norm.
ER -