The Geometry of Radiative Transfer

TL;DR

This paper explores the geometric structure of radiative transfer theory, revealing its Hamiltonian formulation, symmetries, and connection to geometrical optics, providing a new mathematical perspective on electromagnetic propagation.

Contribution

It introduces a geometric and Hamiltonian framework for radiative transfer, linking it to phase space methods and Lie-Poisson systems, which is a novel approach in the field.

Findings

Radiative transfer can be formulated as a Hamiltonian system.

Geometrical optics emerges as a special case within this framework.

The theory exhibits Lie-Poisson structure with symmetry groups.

Abstract

We present the geometry and symmetries of radiative transfer theory. Our geometrization exploits recent work in the literature that enables to obtain the Hamiltonian formulation of radiative transfer as the semiclassical limit of a phase space representation of electromagnetic theory. Cosphere bundle reduction yields the traditional description over the space of "positions and directions", and geometrical optics arises as a special case when energy is disregarded. It is also shown that, in idealized environments, radiative transfer is a Lie-Poisson system with the group of canonical transformations as configuration space and symmetry group.