Information Loss under Gauge Reduction of Discrete Electromagnetic Fields
Abstract
We study relative entropy under gauge reduction in Whitney-discretized electromagnetism. The discrete Hodge decomposition separates exact gauge directions from harmonic and coexact physical degrees of freedom. For Gaussian distributions on the auxiliary potential space, relative entropy splits exactly into the relative entropy of the physical marginals and a nonnegative conditional gauge contribution. We express this contribution through Schur complements, conditional means, and physical--gauge correlations, and characterize equality between auxiliary and physical relative entropies. The physical divergence is also shown to be the minimum over all Gaussian auxiliary extensions. At the infinitesimal level, Fisher information admits an analogous decomposition. A two-variable model illustrates how statistically distinct auxiliary potentials may represent identical physical statistics. The construction provides an information-geometric interpretation of discrete gauge reduction without assigning physical significance to the eliminated gauge variables.
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BibTeX
@misc{magnot2026information,
title = {{Information Loss under Gauge Reduction of Discrete Electromagnetic Fields}},
author = {Jean-Pierre Magnot},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05473},
eprint = {2609.05473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05473},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Information Loss under Gauge Reduction of Discrete Electromagnetic Fields
AU - Jean-Pierre Magnot
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.05473
UR - https://arxiv.org/abs/2609.05473
AB - We study relative entropy under gauge reduction in Whitney-discretized electromagnetism. The discrete Hodge decomposition separates exact gauge directions from harmonic and coexact physical degrees of freedom. For Gaussian distributions on the auxiliary potential space, relative entropy splits exactly into the relative entropy of the physical marginals and a nonnegative conditional gauge contribution. We express this contribution through Schur complements, conditional means, and physical--gauge correlations, and characterize equality between auxiliary and physical relative entropies. The physical divergence is also shown to be the minimum over all Gaussian auxiliary extensions. At the infinitesimal level, Fisher information admits an analogous decomposition. A two-variable model illustrates how statistically distinct auxiliary potentials may represent identical physical statistics. The construction provides an information-geometric interpretation of discrete gauge reduction without assigning physical significance to the eliminated gauge variables.
ER -