Geometric View On Noneikonal Waves
Authors: I.Y. Dodin
Abstract: An axiomatic theory of classical nondissipative waves is proposed that is constructed based on
the de nition of a wave as a multidimensional oscillator. Waves are represented as abstract vectors Ψ
in the appropriately de fined space Ψ with a Hermitian metric. The metric is usually positive-defi nite 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 Schrodinger equation; hence one-to-one
correspondence with the mathematical framework of quantum mechanics is established, and the quantummechanical
machinery becomes applicable to classical waves "as is". The classical wave action is defi ned
as the density operator, |Ψ) Ψ|. The coordinate and momentum spaces, not necessarily Euclidean,
need |Ψ) Ψ| not be postulated but rather emerge when applicable. Various kinetic equations flow as projections of the
von Neumann equation for |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 geometricaloptics
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.
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Submitted to: Physics Letters A (March 2014)
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