On the Unification of Deterministic and Stochastic Electromagnetic Information Theory via Symplectic Geometry

TL;DR

This paper unifies deterministic and stochastic electromagnetic information theories using symplectic geometry, revealing fundamental invariants and bounds on information resolution.

Contribution

It introduces a geometric framework linking electromagnetic information measures through symplectic invariants and establishes fundamental bounds on resolution.

Findings

Eigenvalues and NDF are identical for incoherent sources.

Symplectic invariants govern electromagnetic information measures.

Liouville's theorem and Gromov's Non-Squeezing set fundamental bounds.

Abstract

This paper unifies deterministic and stochastic Electromagnetic Information Theory (EIT) through symplectic geometry. For spatially incoherent sources, both formulations yield identical eigenvalues and spatial Degrees of Freedom (NDF). This equivalence is shown to be a structural necessity: the radiometric \'etendue, the Hamiltonian phase-space volume, and the NDF are the same symplectic invariant of the source--observer configuration. Liouville's theorem guarantees conservation of the NDF under lossless propagation; Gromov's Non-Squeezing Theorem establishes an irreducible minimum phase-space cell, setting a fundamental geometric bound on resolving power. The physical manifestation of this symplectic structure is the formation of \textit{Spatial Information Flows} (SIFs) -- level-set curves of high mutual information which, for convex sources with rotational symmetry, coincide with the optimal sampling curves of Bucci et al. Spatial information in electromagnetic fields is therefore governed by the geometry of the source--observer configuration, providing the foundation for a geometric theory of electromagnetic information.