Geometric view on noneikonal waves

An axiomatic theory of classical nondissipative waves is proposed that is constructed based on the definition of a wave as a multidimensional oscillator. Waves are represented as abstract vectors |ψ〉|ψ〉 in the appropriately defined space Ψ with a Hermitian metric. The metric is usually positive-definite but can be more general in the presence of negative-energy waves (which are typically unstable and must not be confused with negative-frequency waves). The very form of wave equations is derived from properties of Ψ  . The generic wave equation is shown to be a quantumlike Schrödinger equation; hence one-to-one correspondence with the mathematical framework of quantum mechanics is established, and the quantum-mechanical machinery becomes applicable to classical waves “as is”. The classical wave action is defined as the density operator, |ψ〉〈ψ||ψ〉〈ψ|. The coordinate and momentum spaces, not necessarily Euclidean, need not be postulated but rather emerge   when applicable. Various kinetic equations flow as projections of the von Neumann equation for |ψ〉〈ψ||ψ〉〈ψ|. The previously known action conservation theorems for noneikonal waves and the conventional Wigner–Weyl–Moyal formalism are generalized and subsumed under a unifying invariant theory. Whitham's equations are recovered as the corresponding fluid limit in the geometrical-optics approximation. The Liouville equation is also yielded as a special case, yet in a somewhat different limit; thus ray tracing, and especially nonlinear ray tracing, is found to be more subtle than commonly assumed. Applications of this axiomatization are also discussed, briefly, for some characteristic equations.