Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper examines under which conditions a classical fluid model may reproduce the simplest formulations of both theories. By identifying the angular momentum of irrotational vortices in an inviscid, barotropic, isothermal fluid with sound speed c with the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity fields of a vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is equivalent to Schrödinger’s equation. The model is shown to yield classical analogues for de Broglie wavelength, the Einstein–Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wavefield of a vortex in convection yields the Lorentz transformation and the Klein–Gordon equation, with Schrödinger’s equation appearing as the low-Mach-number limit. These results demonstrate that, under explicit assumptions, a classical continuum can reproduce quantum and relativistic theory in their simplest form, without assuming the postulates principal to those theories.
Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper examines under which conditions a classical fluid model may reproduce the simplest formulations of both theories. By identifying the angular momentum of irrotational vortices in an inviscid, barotropic, isothermal fluid with sound speed c with the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity fields of a vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is equivalent to Schrödinger’s equation. The model is shown to yield classical analogues for de Broglie wavelength, the Einstein–Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wavefield of a vortex in convection yields the Lorentz transformation and the Klein–Gordon equation, with Schrödinger’s equation appearing as the low-Mach-number limit. These results demonstrate that, under explicit assumptions, a classical continuum can reproduce quantum and relativistic theory in their simplest form, without assuming the postulates principal to those theories.
Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper extends upon known analogies by showing that under specific assumptions, an Euler-Korteweg vortex model can be cast into equations that are mathematically equivalent to the Schrödinger and Klein-Gordon equations. By assuming that the angular momentum of an irrotational vortex in an inviscid, barotropic, isothermal fluid with sound speed c is equal in magnitude to the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity equations of an Euler-Korteweg vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is formally equivalent to Schrödinger's equation. The model is shown to yield analogues to de Broglie wavelength, the Einstein-Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wave field of a vortex in convection requires the Lorentz transformation and yields a wave equation mathematically equivalent to the Klein-Gordon equation, with Schrödinger's equation appearing as the low-Mach-number limit.
This article proposes a deterministic and geometrically grounded reformulation of quantum mechanics based on vortex dynamics in a structured, superfluid-like vacuum. Modeling the electron as a self-sustaining irrotational vortex with both internal rotation and external translation, the framework derives fundamental quantum phenomena—including de Broglie wavelength, Compton wavelength, spin, and quantization—directly from physical principles of motion and vacuum geometry. A generalized wave function is introduced, embedding both translational and rotational phase components, leading to a modified Schrödinger-like equation that naturally incorporates internal angular momentum. Interference, tunneling, and entanglement are reinterpreted as emergent behaviors of coherent vortex trajectories, eliminating the need for wavefunction collapse or intrinsic randomness. The Born rule is shown to arise from deterministic mechanisms such as phase averaging, internal oscillations, and ergodic dynamics. This vortex model aligns with classical, quantum, and relativistic principles, resolves key interpretational paradoxes, and offers a unified, causal framework for understanding quantum phenomena as structured motion within a physically real vacuum.
Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.
The Madelung equations offer a hydrodynamic description of quantum systems, from single particles to quantum fluids. In this formulation, the probability density is mapped onto the fluid density and the phase is treated as a scalar potential generating the velocity field. As examples of potential flows, quantum fluids described in this way are inherently irrotational, but quantum vortices may arise at discrete points where the phase is undefined. In this paper, starting from this irrotational description of a quantum fluid, a coarse-graining procedure is applied to arrive at a macroscopic description of the quantum fluid in terms of a hierarchy of moments in which the role of velocity is played by a Favre average of the microscopic velocity field. This hierarchy is truncated using an explicit closure derived from an expansion in a finite length scale. The resulting coarse-grained fields are shown to allow for finite vorticity at any point in the fluid. Furthermore, it is shown that this vorticity obeys a similar equation to the vorticity equation in classical hydrodynamics and includes a vortex-stretching term. The particular closure employed here also gives rise to a novel stress term in the fluid equations, which in the appropriate limit appears analogous to an artificial viscous stress from computational fluid dynamics.
A unified fluid and field theory containing all forces as special cases postulates that primordial fluid is formed by discrete, 3D-extended energy-like sagions obeying usual energy-momentum conservation laws. Contrary to Newtonian mechanics, mass and force are not primitive notions (hence, "Cartesian"), but the theory is atomistic (hence "neo"). Collective fluid behaviour obeys the homogeneous Klein-Gordon equation, or homogeneous 3D-wave equation, which besides usual harmonic periodic solutions has three families of quantized, Galilean-Lorentz isomorph, nonperiodic solutions discovered around 1995. Here, quantum mechanics is relativistic ab initio thus ensuring consistency with Einstein's general theory of relativity, which is extended to include effects upon gravity of chemical and nuclear composition, as indeed observed by Eötvös (1890s), and Majorana (1919). Planck constant is identified with the bit of minimum angular momentum in sagion-sagion interactions, thus introducing quantum features at the root of classical mechanics. Acceleration produced by successive pushes of a small projectile (say, a sagion, disagion, trisagion, 4-sagion) leads to a curve resembling Einstein's mass increase. This reinterprets one aspect of the restricted theory of relativity: rather than mass increase, Bertozzi experiment demonstrates inefficient transfer of linear momentum. The lower branch in our background universal acceleration curve is associated to static friction.
The propagation of fluctuations in the velocity potential of an inviscid, irrotational fluid is governed by a wave equation very similar to that governing the propagation of massless scalar fields in curved spacetime. This similarity allows certain fluid systems to be used as experimentally realisable models for objects that cannot be studied under controlled conditions, such as black holes and ultrarelativitsic rotating stars. The analogy applies only to the propagation of fields in a fixed background, not to the behaviour of the background itself. However, even this limited similarity has excited a great deal of interest from researchers, since it suggests that phenomena such as superradiance and Hawking radiation may have analogues in superfluids. In recent years, experimental advances in the creation and manipulation of Bose-Einstein condensates have provided researchers with very low temperature systems possessing a uniquely simple microscopic theory. Furthermore, under certain long-length-scale approximations, Bose-Einstein condensates can be described as classical irrotational inviscid fluids, and therefore can be treated as analogues of appropriate gravitational systems. We present a comprehensive and detailed critical review of the mean-field theory of Bose-Einstein condensates, the conditions under which Bose-Einstein condensates can be treated as an analogue gravity system, the construction of analogue black holes, and several derivations of Hawking radiation and analogue Hawking radiation. The interdisciplinary nature of analogue gravity means that the relevant results are widely scattered in the literature, and we consider this critical review and discussion of the relevant literature to be a central and noteworthy element of the project. We then analyse in detail a recent derivation of analogue Hawking radiation in Bose-Einstein condensates as a consequence of quantum depletion. We elucidate the methodology and the physical meaning of the results by comparison with the derivations of Hawking radiation considered previously, and discuss possible extensions and improvements. We also present an original investigation of analogue superradiance in simple vortices, with the vortex density profile chosen to mimic a realistic vortex in a Bose-Einstein condensate. We find that superradiant amplification occurs and is likely to be significant, but our long-length-scale approximation is of dubious validity near the scattering point.