Stochastic construction of a Feynman path integral representation for Green’s functions in radiative transfer

Abstract

A construction of a stochastic, Lagrangian path integral representation is presented for classical Green’s functions or propagators for radiative transfer in random scattering medias. As Fermat rays comprising the initial collimated pulse (e.g., from a laser) enter the medium, they evolve into a bundle of random paths or trajectories due to scattering. Stochastically, this can be interpreted as a bundle of randomly evolving Markov trajectories traced out by a gas or ensemble of “Brownian-type” classical point photons undergoing multiple scatterings. The path integral is structurally of the same form as a Wick rotated Euclidean quantum Feynman integral with direct optical analogs of the Hamiltonian and Lagrangian emerging. However, the optical stochastic integral is real and is defined in real time becoming exponentially damped rather than oscillatory. The calculation also constitutes an alternative mathematical derivation of the time dependent diffusion equation from the radiative transfer theory. The limits of the path integral representation give a maximally diffused Gaussian distribution of the heat kernal form (the large scatter limit) and a Beer’s law exponential decay corresponding to the extremal Fermat rays obeying Euler–Lagrange equations (the zero scatter limit). The approach also highlights the direct structural analogs between classical radiative transfer and optical diffusion in real time (t) and ordinary quantum mechanics in Euclidean time (√(−1)t=it).