The Geometry of Radiative Transfer

Radiative transfer describes the transport of electromagnetic energy in macroscopic environments when polarization effects are neglected [38]. The theory originates in work by Bouguer [6, 7] and Lambert [23] in the 18th century where the intensity of light and its measurement were first studied systematically, cf. Fig. 1, and in the 19th and early 20th century the theory was extended to include transport and scattering effects [27, 9, 43, 44]. To this day, however, radiative transfer is a phenomenological theory without connection to the fundamental theories of light and with a mathematical formulation that still employs the concepts introduced by Lambert in the 18th century—and this despite the importance of the theory in a multitude of fields, such as medical imaging, remote sensing, computer graphics, atmospheric science, climate modelling, and astrophysics. In the following, we will explain the physical foundations of radiative transfer in media with varying refractive index and we study the geometry of the theory and its symmetries; an overview is provided in Fig. 2. Exploiting recent advances in applied mathematics, we employ semi-classical analysis to lift Maxwell’s equations from configuration space Q⊆ R3 to phase space T ∗Q. By restricting the dynamics on T ∗Q to a non-zero energy level and considering the short wavelength limit one obtains a transport equation for polarized light, and further neglecting polarization and only considering the energy density that is transported on phase space leads to radiative transfer theory in a Hamiltonian formulation. Our derivation shows that the central quantity of radiative transfer theory is the phase space light energy density