[Paper Review] Geometric view on noneikonal waves

I. Y. Dodin|arXiv (Cornell University)|Oct 18, 2013

Quantum Mechanics and Applications105 references3 citations

TL;DR

This paper proposes an axiomatic geometric framework for classical nondissipative waves by modeling them as vectors in a Hilbert-like space with a Hermitian metric, deriving wave equations as Schrödinger-type equations. The key contribution is a unified invariant theory that generalizes action conservation, Whitham’s equations, and the Liouville equation under a single formalism, establishing a one-to-one correspondence with quantum mechanics and enabling the use of quantum-mechanical tools for classical waves.

ABSTRACT

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 $|ψ angle$ 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 Schrodinger 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, $|ψ angle\langleψ|$. 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 $|ψ angle\langleψ|$. 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.

Motivation & Objective

• To develop a universal, axiomatic framework for classical nondissipative waves independent of specific field equations.
• To establish a one-to-one correspondence between classical wave dynamics and quantum mechanics by modeling waves as abstract vectors in a Hilbert space with a Hermitian metric.
• To derive wave equations, action conservation, and kinetic equations from geometric principles without relying on Fourier or Wigner-Weyl transforms.
• To generalize existing theories—such as the Wigner-Weyl-Moyal formalism and action conservation theorems—into a single invariant theory.
• To recover ray tracing and fluid limits (e.g., Whitham’s equations) as projections of a fundamental von Neumann-type equation for the wave action density operator.

Proposed method

• Waves are defined as abstract vectors |ψ⟩ in a space Ψ equipped with a Hermitian metric, which may be indefinite in the presence of negative-energy waves.
• The generic wave equation is derived as a Schrödinger-like equation: i∂t|ψ⟩ = H|ψ⟩, where H is a Hermitian operator on Ψ.
• The classical wave action is defined as the density operator |ψ⟩⟨ψ|, which evolves according to a von Neumann-type equation: i∂tρ = [H, ρ].
• Phase space and coordinate/momentum representations emerge naturally from the geometry of Ψ, without assuming them a priori.
• The wedge product and symplectic structure on phase space are introduced via dual vector fields and a tensor ϖαβ, enabling derivation of Poisson brackets and Liouville dynamics.
• The Liouville equation is recovered as a special case of the von Neumann equation in a symplectic phase space, with the Liouvillian operator Ĥ = −i{·, H} acting as a Hamiltonian on the function space F.

Experimental results

Research questions

• RQ1 Can classical wave dynamics be formulated as an axiomatic geometric theory independent of specific field equations?
• RQ2 How can wave action conservation and kinetic equations be derived from fundamental geometric principles rather than ad hoc assumptions?
• RQ3 To what extent can the mathematical machinery of quantum mechanics be applied directly to classical waves?
• RQ4 How do Whitham’s equations and the Liouville equation emerge as limits within this unified framework?
• RQ5 What is the role of the Wigner-Weyl-Moyal formalism in this geometric theory, and how is it generalized?

Key findings

• The generic wave equation is derived as a Schrödinger-type equation, establishing a one-to-one correspondence between classical wave dynamics and quantum mechanics.
• The wave action density operator |ψ⟩⟨ψ| satisfies a von Neumann equation, and all kinetic equations arise as projections of this equation in phase space.
• The Liouville equation is recovered as a special case of the von Neumann equation in a symplectic phase space, with the Liouvillian operator Ĥ = −i{·, H} acting as the Hamiltonian.
• Whitham’s equations are obtained as the fluid limit of the wave equation in the geometrical-optics approximation, consistent with the geometric framework.
• The formalism naturally reproduces the Wigner-Weyl-Moyal formalism and generalizes action conservation theorems without requiring Fourier or Wigner transforms as foundational tools.
• The theory shows that ray tracing, especially nonlinear ray tracing, is more subtle than commonly assumed, as it arises from a nontrivial limit of the fundamental wave equation.