Information Loss under Gauge Reduction of Discrete Electromagnetic Fields

Technical Summary: Information Loss under Gauge Reduction of Discrete Electromagnetic Fields

Problem Statement
In classical and quantum field theories, gauge potentials contain redundant variables; in the Abelian setting, potentials related by a gauge transformation $A \mapsto A + d\chi$ describe the same physical electromagnetic field. From a statistical perspective, this redundancy raises a fundamental question: how much distinguishability between two auxiliary descriptions is lost when unobservable gauge variables are removed? While the classical chain rule for relative entropy is well-known, its explicit realization within the context of discrete gauge theories—specifically those utilizing Whitney forms and finite element exterior calculus—has not been fully characterized for Gaussian states. The paper addresses the quantification of information loss when projecting from the space of discrete potentials to the space of physical observables.

Methodology
The study operates within the framework of Whitney-discretized electromagnetism on a finite triangulation $K$ of a closed oriented Riemannian three-manifold.

1. Discrete Hodge Decomposition: The space of discrete 1-cochains (potentials) $C^1(K)$ is decomposed using the discrete Hodge decomposition:
   $$C^1(K) = V_p \oplus V_g$$
   where $V_g = \text{im } \delta^0$ represents exact gauge directions, and $V_p = \text{ker } \delta^1 \cap \text{ker } \delta^*_0 \oplus \text{im } \delta^*_1$ contains the harmonic and coexact physical degrees of freedom.
2. Statistical Framework: The analysis considers non-degenerate Gaussian probability measures $\mu$ and $\nu$ on the auxiliary potential space $V = V_p \oplus V_g$. The physical marginals $\mu_p$ and $\nu_p$ are defined via the pushforward under the physical projection $P_p$.
3. Information-Theoretic Tools: The paper utilizes relative entropy (Kullback-Leibler divergence) and Fisher information to analyze the relationship between the full auxiliary distributions and their physical projections. Explicit formulas are derived using Schur complements and conditional Gaussian distributions.

Key Contributions and Results

• Exact Entropy Decomposition: The central result is an exact decomposition of the relative entropy between two auxiliary Gaussian measures $\mu$ and $\nu$:
  $$D(\mu \parallel \nu) = D(\mu_p \parallel \nu_p) + I_g(\mu, \nu)$$
  Here, $D(\mu_p \parallel \nu_p)$ is the relative entropy of the physical marginals, and $I_g(\mu, \nu)$ is a non-negative conditional gauge contribution representing the information lost during reduction.
  $$I_g(\mu, \nu) = \int_{V_p} D(\mu_{g|x} \parallel \nu_{g|x}) \, d\mu_p(x)$$
  This term quantifies the distinguishability residing entirely in the unphysical gauge components.

• Explicit Gaussian Formula: For Gaussian measures defined by means $m$ and covariances $\Sigma$ (partitioned into physical $A$, gauge $C$, and cross-correlation $B$ blocks), the paper derives a closed-form expression for $I_g(\mu, \nu)$. This expression separates the auxiliary distinguishability into three distinct sources:

  1. Differences in conditional gauge covariances (Schur complements $S_\mu, S_\nu$).
  2. Differences in conditional gauge means.
  3. Differences in the physical–gauge regression operators ($R = B^\top A^{-1}$).
     The paper characterizes the condition for equality $D(\mu \parallel \nu) = D(\mu_p \parallel \nu_p)$: this occurs if and only if the conditional gauge distributions are identical ($\mu_{g|x} = \nu_{g|x}$) almost everywhere.

• Physical Relative Entropy as a Minimum: The paper establishes that the physical relative entropy is the minimum possible distinguishability among all Gaussian auxiliary extensions compatible with the physical states:
  $$D(\bar{\mu} \parallel \bar{\nu}) = \min_{\mu \in \text{Ext}_G(\bar{\mu}), \nu \in \text{Ext}_G(\bar{\nu})} D(\mu \parallel \nu)$$
  This minimum is attained when the auxiliary extensions share the same conditional gauge distribution (e.g., via a product structure $\mu = \bar{\mu} \otimes \tau$).

• Fisher Information Decomposition: At the infinitesimal level, the Fisher information matrix admits an analogous decomposition:
  $$g_{aux}(\theta) = g_{phys}(\theta) + g_{cond}(\theta)$$
  where the conditional term $g_{cond}$ is positive semidefinite, confirming that the auxiliary parameter space contains strictly more information than the physical space.

• Illustrative Model: A two-variable model ($V_p = \mathbb{R}, V_g = \mathbb{R}$) demonstrates that two auxiliary distributions can be statistically distinct ($D(\mu \parallel \nu) > 0$) while having identical physical marginals ($D(\mu_p \parallel \nu_p) = 0$). In this case, the distinguishability arises solely from different correlations between the physical coordinate and the gauge coordinate.

Significance and Claims
The paper claims to provide an information-geometric interpretation of discrete gauge reduction. Its primary significance lies in:

1. Quantifying Redundancy: It precisely quantifies the statistical information discarded when moving from auxiliary potentials to gauge-equivalence classes, without assigning independent physical significance to the eliminated gauge variables.
2. Clarifying Distinguishability: It clarifies that two auxiliary descriptions may be statistically distinguishable purely due to their gauge representation, even if they yield identical physical predictions.
3. Methodological Rigor: The construction is strictly limited to finite-dimensional Whitney complexes and classical Gaussian fields. The author explicitly notes that continuum limits, quantum density operators, non-Abelian theories, and dynamical evolution are outside the scope of this work and require separate analysis.

The work serves as a foundational step in understanding how gauge symmetry manifests in the information geometry of discrete field theories, offering exact formulas for entropy and Fisher information that separate physical content from gauge artifacts.