On the Unification of Deterministic and Stochastic Electromagnetic Information Theory via Symplectic Geometry
For researchers in electromagnetic information theory, this work provides a geometric unification of previously separate deterministic and stochastic frameworks, revealing structural necessities rather than coincidences.
This paper unifies deterministic and stochastic electromagnetic information theory via symplectic geometry, showing that both formulations yield identical eigenvalues and spatial degrees of freedom (NDF) for spatially incoherent sources. The NDF is proven to be a symplectic invariant, with Liouville's theorem ensuring conservation under lossless propagation and Gromov's Non-Squeezing Theorem setting a fundamental bound on resolving power.
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 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; 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.