Abstract It has recently been shown that relativistic quantum theory leads to a local interpretation of quantum mechanics wherein the universal wavefunction in configuration space is entirely replaced with an ensemble of local fluid equations in spacetime. For want of a fully relativistic quantum fluid treatment, we develop a model using the nonrelativistic Madelung equations, and obtain conditions for them to be local in spacetime. Every particle in the Madelung fluid is equally real, and has a definite position, momentum, kinetic energy, and potential energy. These are obtained by defining quantum momentum and kinetic energy densities for the fluid and separating the momentum into average and symmetric parts, and kinetic energy into classical kinetic and quantum potential parts. The two types of momentum naturally give rise to a single classical kinetic energy density, which contains the expected kinetic energy, even for stationary states, and we define the reduced quantum potential as the remaining part of the quantum kinetic energy density. We treat the quantum potential as a novel mode of internal energy storage within the fluid particles, which explains most of the nonclassical behavior of the Madelung fluid. For example, we show that in tunneling phenomena, the quantum potential negates the barrier so that nothing prevents the fluid from flowing through. We show how energy flows and transforms in this model, and that enabling local conservation of energy requires defining a quantum potential energy current that flows through the fluid rather than only flowing with it. The nonrelativistic treatment generally contains singularities in the velocity field, which undermines the goal of local dynamics, but we expect a proper relativistic treatment will bound the fluid particle velocities at c.
Abstract It has recently been shown that relativistic quantum theory leads to a local interpretation of quantum mechanics wherein the universal wavefunction in configuration space is entirely replaced with an ensemble of local fluid equations in spacetime. For want of a fully relativistic quantum fluid treatment, we develop a model using the nonrelativistic Madelung equations, and obtain conditions for them to be local in spacetime. Every particle in the Madelung fluid is equally real, and has a definite position, momentum, kinetic energy, and potential energy. These are obtained by defining quantum momentum and kinetic energy densities for the fluid and separating the momentum into average and symmetric parts, and kinetic energy into classical kinetic and quantum potential parts. The two types of momentum naturally give rise to a single classical kinetic energy density, which contains the expected kinetic energy, even for stationary states, and we define the reduced quantum potential as the remaining part of the quantum kinetic energy density. We treat the quantum potential as a novel mode of internal energy storage within the fluid particles, which explains most of the nonclassical behavior of the Madelung fluid. For example, we show that in tunneling phenomena, the quantum potential negates the barrier so that nothing prevents the fluid from flowing through. We show how energy flows and transforms in this model, and that enabling local conservation of energy requires defining a quantum potential energy current that flows through the fluid rather than only flowing with it. The nonrelativistic treatment generally contains singularities in the velocity field, which undermines the goal of local dynamics, but we expect a proper relativistic treatment will bound the fluid particle velocities at c.
Abstract In single-particle Madelung mechanics, the single-particle quantum state Ψ ( x → , t ) = R ( x → , t ) e i S ( x → , t ) / ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R ( x → , t ) 2 and local momentum ∇ → S ( x → , t ) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q ( x → , t ) = − ℏ 2 2 m ∇ → R ( x → , t ) R ( x → , t ) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q ( x → , t ) < 0 . For energy superposition states with band-limit
Starting from the Pauli current we obtain the decomposition of the non-relativistic local velocity in two parts: one parallel and the other orthogonal to the momentum. The former is recognized to be the ``classical'' part, that is the velocity of the center-of-mass, and the latter the ``quantum'' one, that is the velocity of the motion in the center-of-mass frame (namely, the internal ``spin motion'' or {\em Zitterbewegung}). Inserting the complete expression of the velocity into the kinetic energy term of the classical non-relativistic (i.e., Newtonian) Lagrangian, we straightforwardly derive the so-called ``quantum potential'' associated to the Madelung fluid. In such a way, the quantum mechanical behaviour of particles appears to be strictly correlated to the existence of spin and Zitterbewegung.
Starting from the Pauli current we obtain the decomposition of the nonrelativistic local velocity in two parts: one parallel and the other orthogonal to the impulse. The former is recognized as the “classical” part, that is, the velocity of the center-of-mass, and the latter the “quantum” one, that is, the velocity of the motion in the center-of-mass frame (namely, the internal “spin motion” or zitterbewegung), Inserting the complete expression of the velocity into the kinetic energy term of the classical nonrelativistic (i.e. Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is known, to the Madelung fluid. In such a way, the quantum mechanical behavior of particles appears to be strictly correlated to the existence of spin and zitterbewegung.
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.
Inverted Relativistic Phase-Fluid Mechanics on Curved Spacetime introduces a covariant matter-phase model in which the phase field defines the local four-momentum, written in readable form as: p_mu = hbar * nabla_mu phi or equivalently: four-momentum = Planck constant × phase gradient The density n weights the conserved phase current and determines how strongly the phase flow contributes to the stress-energy tensor and therefore to spacetime curvature. In the minimal model, n has no independent gradient term. It acts as an algebraic density variable enforcing a relativistic Hamilton-Jacobi or mass-shell relation. This gives an inverted relativistic phase fluid: the phase determines motion, while the density weights its physical and gravitational effect. The model is then extended through the Madelung decomposition: psi = sqrt(n) * exp(i phi) This adds the amplitude-gradient term and produces the relativistic quantum-pressure contribution: Q_rel = - hbar^2 * Box(sqrt(n)) / sqrt(n) where Box is the covariant d’Alembert operator on curved spacetime. In this way, the minimal phase-fluid picture becomes connected to the full Klein-Gordon/Madelung quantum-fluid structure on curved spacetime. The paper provides a covariant bridge between: proper-time matter phase, de Broglie momentum-phase relations, relativistic fluid mechanics, Madelung quantum hydrodynamics, quantum pressure, stress-energy generation, and gravitational curvature. The model does not replace general relativity and does not claim to be a completed theory of quantum gravity. Instead, it formulates a compact theoretical framework in which matter phase, density, quantum amplitude structure and spacetime curvature are coupled without introducing a preferred reference frame. A compact summary of the construction is: The phase carries the motion, the density carries the weighting, the quantum pressure carries the wave nature, and the metric carries gravity. This preprint is intended as a theory note and starting point for further analytical and numerical work, including fixed-background simulations, weak gravitational backreaction, and full Einstein-Klein-Gordon/Madelung evolution.
In 1948, Schwinger developed a local Lorentz-covariant formulation of relativistic quantum electrodynamics in space-time which is fundamentally inconsistent with any delocalized interpretation of quantum mechanics. An interpretation compatible with Schwinger’s theory is presented, which reproduces all of the standard empirical predictions of conventional delocalized quantum theory in configuration space. This is an explicit, unambiguous, and Lorentz-covariant “local hidden variable theory” in space-time, whose existence proves definitively that such theories are possible. This does not conflict with Bell’s theorem because it is a local many-worlds theory. Each physical system is characterized by a wave-field, which is a set of indexed piece-wise single-particle wavefunctions in space-time, each with its own coefficient, along with a memory which contains the separate local Hilbert-space quantum state at each event in space-time. Each single-particle wavefunction of a fundamental system describes the motion of a portion of a conserved fluid in space-time, with the fluid decomposing into many classical point particles, each following a world-line and recording a local memory. Local interactions between two systems take the form of local boundary conditions between the differently indexed pieces of those systems’ wave-fields, with new indexes encoding each orthogonal outcome of the interaction. The general machinery is introduced, including the local mechanisms for entanglement and interference. The experience of collapse, Born rule probability, and environmental decoherence are discussed, and a number of illustrative examples are given.