离散电磁场规范约化下的信息损失

中文摘要 AI 辅助

我们研究了Whitney离散化电磁学中规范约化下的相对熵。离散Hodge分解将精确规范方向与调和及余精确物理自由度分离。对于辅助势空间上的高斯分布，相对熵精确分解为物理边际分布的相对熵与非负条件规范贡献之和。我们通过Schur补、条件均值和物理-规范相关性来表达这一贡献，并刻画了辅助相对熵与物理相对熵相等的条件。物理散度也被证明是所有高斯辅助扩展中的最小值。在无穷小层面，Fisher信息具有类似的分解。一个双变量模型说明了统计上不同的辅助势如何可能代表相同的物理统计。该构造为离散规范约化提供了信息几何解释，而不赋予被消除的规范变量以物理意义。

英文摘要

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.

发表机构

• SFR MATHSTIC, LAREMA, Université d’Angers（昂热大学 SFR MATHSTIC、LAREMA）
• Lepage Research Institute（莱佩奇研究所）

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