Geometric theory of wave kinetics and the ray-tracing controversy*

An invariant, geometric formulation of linear wave kinetics is proposed that allows casting \textit{any} wave equation (WE) in a quantumlike form. The wave amplitude is described by the Schr\"odinger equation, which, absent dissipation, has a Lagrangian form. Any approximations made to the Lagrangian preserve the conservative form of WE, automatically preventing standard errors (e.g., at guessing a WE from a dispersion relation or at approximating the dielectric tensor with its local value). The wave action is naturally introduced as a Hermitian operator (density matrix). The associated kinetic equation (KE) accounts for both diffraction and mode coupling and conserves wave quanta. Contrary to a popular misconception, taking the geometrical-optics limit does \textit{not} necessarily lead to what is known as the ``wave kinetic equation.'' This undermines the applicability of ray tracing for common practical applications; e.g., the correct KE manifestly prohibits stochasticity at $t \to \infty$.