Chern--Simons Fluctuations and Information Geometry in Discrete Electromagnetism

Technical Summary: Chern–Simons Fluctuations and Information Geometry in Discrete Electromagnetism

Problem Statement
The paper addresses the formulation of statistical states for electromagnetic fields constrained by magnetic helicity within a discrete framework. While helicity-constrained statistical constructions exist in continuous magnetohydrodynamics (e.g., Frisch et al.) and are rooted in the maximum-entropy principle (Jaynes), this work aims to formulate the helicity constraint directly on a Whitney-discretized electromagnetic complex. A critical physical qualification is noted: magnetic helicity is not generally conserved under unconstrained Maxwell dynamics. Therefore, the goal is not to describe stationary dynamical equilibria of free Maxwell fields, but rather to construct instantaneous helicity-conditioned inference states that minimize relative entropy given a prescribed mean helicity.

Methodology
The author employs Whitney forms and finite element exterior calculus to discretize the electromagnetic field on a closed, oriented Riemannian three-manifold $M$ triangulated by a simplicial complex $K$.

1. Discrete Gauge Reduction:

   • The discrete magnetic potential is a 1-cochain $a \in C^1(K)$.
   • The simplicial de Rham complex preserves cohomological identities. The discrete gauge transformation is generated by 0-cochains $\chi$, mapping $a \mapsto a + \delta_0 \chi$.
   • Using the Whitney inner product, the space of 1-cochains is orthogonally decomposed into three sectors: gauge ($V_g = \text{im } \delta_0$), harmonic ($V_h = \mathcal{H}^1_K$), and transverse radiative ($V_T = \text{im } \delta_1^*$).
   • The author fixes the harmonic sectors (both potential and electric) to avoid non-normalizable Gaussian weights, restricting the analysis to the radiative phase space $P_T = V_T \oplus V_T$.

2. Discrete Chern–Simons Functional:

   • The spatial Abelian Chern–Simons functional is discretized as $CS_K(a) = \int_M W_1(a) \wedge dW_1(a)$, where $W_1$ is the Whitney map.
   • It is proven that this functional annihilates both exact (gauge) and harmonic components, depending solely on the coexact (radiative) sector $a_T$.
   • In a Whitney-orthonormal basis of $V_T$, $CS_K(a)$ is represented by a symmetric matrix $C$, such that $CS_K(a) = a^\top C a$.

3. Statistical Construction:

   • A Gaussian ensemble is defined on the reduced phase space with probability density:
     $$p_{\beta,\gamma}(a, e) = Z_K(\beta, \gamma)^{-1} \exp\left(-\beta E_K(a, e) - \gamma CS_K(a)\right)$$
     where $E_K$ is the reduced Maxwell energy, $\beta$ is the inverse temperature, and $\gamma$ is the Lagrange multiplier for the mean helicity constraint.
   • The partition function $Z_K$ and admissibility conditions are derived based on the positive definiteness of the matrix $Q_{\beta,\gamma} = \beta L + 2\gamma C$, where $L$ represents the discrete magnetic energy operator.

Key Contributions and Results

• Exact Gauge Invariance and Reduction: The paper demonstrates that the discrete Chern–Simons functional is exactly gauge invariant and depends only on the radiative sector, allowing for a rigorous reduction of the phase space without losing topological information (harmonic modes are fixed, not quotiented out as gauge).
• Explicit Partition Function and Admissibility: The partition function is derived explicitly as $Z_K(\beta, \gamma) = (2\pi)^n \beta^{-n/2} \det(Q_{\beta,\gamma})^{-1/2}$. The admissible region for parameters $(\beta, \gamma)$ is characterized by the condition $\beta L + 2\gamma C > 0$, which translates to constraints on the eigenvalues of the generalized helicity operator.
• Minimum Relative Entropy: It is proven that the distribution $p_{\beta,\gamma}$ uniquely minimizes the relative entropy $D(p \| p_{\beta,0})$ among all distributions satisfying a prescribed mean helicity constraint. This establishes the distribution as the optimal inference state given the constraint.
• Information Geometry and Susceptibility:
  • The Fisher information matrix is identified with the Hessian of the log-partition function.
  • A central result is the fluctuation-dissipation relation: the helicity susceptibility $\chi_H = -\partial_\gamma \langle CS_K \rangle$ is exactly equal to the variance of the discrete Chern–Simons functional, $\text{Var}_{\beta,\gamma}(CS_K)$, which corresponds to the Fisher information component $g_{\gamma\gamma}$.
  • This variance controls the local distinguishability of neighboring statistical states, quantified by the relative entropy expansion $D(p_{\beta,\gamma} \| p_{\beta,\gamma+\epsilon}) \approx \frac{\epsilon^2}{2} \text{Var}_{\beta,\gamma}(CS_K)$.
• Mesh-Level Computability: All quantities (partition function, mean helicity, susceptibility) are expressed in terms of computable simplicial data: incidence matrices ($D_0, D_1$), Whitney mass matrices ($M_1, M_2$), and the discrete Chern–Simons pairing matrix $S$.

Significance and Claims
The paper claims to provide an exact, mesh-level realization of information-geometric fluctuations for helicity-constrained electromagnetic fields. Unlike previous spectral truncation approaches, this method operates directly on the discrete complex, preserving the exact discrete gauge symmetry and cohomological structure.

The author emphasizes that these states represent conditioned inference rather than dynamical equilibrium, as magnetic helicity is not conserved under free Maxwell evolution. The construction is specific to the topologically trivial Abelian sector on a closed manifold. The paper does not claim mesh-refinement invariance, a continuum limit, or stationarity under free dynamics; these are identified as requiring further analysis. The work serves as a rigorous foundation for computing statistical properties of discrete electromagnetic fields constrained by helicity, linking Chern–Simons fluctuations directly to statistical distinguishability via Fisher information.