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

This paper unifies deterministic and stochastic Electromagnetic Information Theory (EIT) through symplectic geometry.

For spatially incoherent sources, both formulations yield identical eigenvalues and spatial Number of Degrees of Freedom (NDF).

In the asymptotic regime and in the absence of losses, this equivalence is shown to be a structural necessity: the radiometric étendue, 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, while Gromov's Non-Squeezing Theorem establishes a 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), defined operationally as the spatial loci along which the spatial coherence, equivalently the mutual information, decays at the minimum possible rate.

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.