Abstract
From the macroscopic to the microscopic world, quantum mechanical effects in acoustics and elastic waves have become increasingly important. Observations on the quantum effects of acoustic and elastic waves using experimental methods have been reported in the literature. However, the conventional formulations of acoustic and elastic waves are still mainly governed by classical models. In this study, we investigated the quantization of acoustic and elastic waves using generalized Lorenz gauges. The potential variables of acoustic and elastic waves can be quantized in a manner similar to that of electrodynamics. The results include the Schrödinger equation with minimal coupling between the field and particles. The quantization of field variables is established as a consequence of the gauge symmetry property of the Schrödinger equation. Later, we explored the connections between the parallel formulations of mechanics and waves through an algebraic aspect. This highlights the isomorphism pattern from the theoretical characterization within the parallel formulations. To support the results, the derivations of potential formulations based on Lorenz gauges and functional mapping between field variables are presented.