The Second-Order Schrödinger Field as a Quantum Aether: Lagrangian Formulation, Stability, Acoustic Dynamics and Parameter Selection

This paper investigates a theoretical model where a complex scalar wave field serves as the order parameter for a continuous physical medium termed a "quantum aether". The medium's microscopic scales are set by the Compton length and Compton time. Key highlights of the work include: Field and Energy Structure: The governing field equation is second-order in time and takes a Klein–Gordon form with a density-dependent rest energy. The theory defines a Hamiltonian with a conserved energy (bounded below for positive coupling) and a Noether charge whose density is sign-indefinite. Acoustic Dynamics and Spectrum: Excitations of uniform backgrounds generate real squared frequencies featuring a gapless acoustic branch (sound waves) and a gapped high-energy branch. The sound speed remains below the speed of light and approaches a fixed high-density limit. Stability Analysis: Uniform backgrounds are spectrally stable above a critical density threshold, whereas lower densities exhibit exponential growth characteristic of spinodal instability. Mathematical Rigor: On bounded domains and periodic boxes, the evolution problem is proved to be globally well-posed despite non-Lipschitz logarithmic terms. Fixed-charge energy minimizations yield confined nonuniform stationary states, and sharp existence thresholds are established for localized stationary lumps. Parameter Framework: The dynamics depend on the Lagrangian constants through a single invariant parameter, and various parameter-fixing postulates,including a critical fold point where the aether at rest vanishes and the sound speed drops to zero, are rigorously evaluated. The physical aether interpretation serves as a conceptual framework mapping mathematical outcomes to medium properties, without claiming direct identification with the vacuum of quantum field theory or empirical measurement rules.