Abstract In the Planck aether hypothesis it is assumed that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying a nonrelativistic law of motion. If described by a quantum mechanical two-component nonrelativistic nonlinear operator field equation, this model has a spectrum of particles greatly resembling the particles of the standard model, with Lorentz invariance as a derived dynamical symmetry valid in the limit of energies small com pared to the Planck energy. Here we show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of the Boltzmann equation for the positive and negative Planck masses, with departures from quantum mechanics suppressed by the Planck length.
Abstract In the Planck aether hypothesis it is assumed that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying a nonrelativistic law of motion. If described by a quantum mechanical two-component nonrelativistic nonlinear operator field equation, this model has a spectrum of particles greatly resembling the particles of the standard model, with Lorentz invariance as a derived dynamical symmetry valid in the limit of energies small com pared to the Planck energy. Here we show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of the Boltzmann equation for the positive and negative Planck masses, with departures from quantum mechanics suppressed by the Planck length.
To bypass Lehmann’s theorem against Heisenberg’s “Unified Field Theory of Elementary Particles,” requiring a Hilbert space with an indefinite metric, which is in conflict with the quantum mechanical probability interpretation, it was proposed by the author that Lorentz invariance, on of the fundamental assumptions made by Heisenberg, is a dynamic symmetry, approximately only valid for energies small compared to the Planck energy of ~10^19 GeV, with the fundamental symmetry of nature the Galilei group, in agreement with Mach’s principle. There then Heisenberg’s theory can be reformulated as an exactly non-relativistic quantum field theory with a positive definite metric in Hilbert space. With the Hamiltonian operator in such a theory commuting with the particle number operator, Heisenberg’s ground state of the vacuum is permitted to be a zero temperature plasma made up of positive and negative Planck mass particles which are one Planck mass per Planck length volume, interacting with the Planck force over a Planck length. Making for this Planck mass plasma the Hartree-Foch approximation, one obtains the Landau-Ginzburg equation of a super-fluid, and from the Boltzmann equation the quantum potential of the Madelung transformed Schrodinger equation. Quantum mechanics is thereby explained as a completely deterministic theory, as required by Kant’s law of causality. This paper was inspired by a remarkable paper recently published by I. Licata [1], who compared the work by the author for a deterministic interpretation of quantum mechanics to the work of ‘t Hooft with the same goal.
In this work we analyze the relation between the stochastic representation of nonrelativistic quantum mechanics and the Newtonian stochastic mechanics. In order to understand how the nodes of a wave function are accounted for in stochastic quantum mechanics, we solve the Fokker-Planck equation associated with the quantum-mechanical density ${\ensuremath{\rho}}_{0}$=x2 and current j0=0, and find that the stochastically perturbed particle does not cross the nodal surface x=0 of the density ${\ensuremath{\rho}}_{0}$. Further we consider the classical motion of a particle subject to a stochastic perturbation, in the absence of conventional external forces and friction, and moreover we consider the classical stochastic motion of a harmonically bound particle in the presence of a friction force proportional and opposite to the velocity of the particle, to show that although the quantum-mechanical equations for the probability density and current can be put in a form resembling Newton's second law, the actual significance of those equations is not related to Newtonian stochastic mechanics. From the fact that, in the stochastic representation of quantum mechanics, the particles have continuous trajectories, we infer that the interactions would possess, in this theory, a global character. We finally show that the property of reversibility in stochastic quantum mechanics depends on the existence of suitably chosen compensating drifts in the mean forward velocity or acceleration of the particle under consideration.
Quantum mechanics is defined here as the theoretical framework crystallised at the fifth Solvay Conference of 1927, comprising five postulates: the wavefunction, the canonical commutation relation [x, p] = iℏ, the Born rule, the Schrödinger equation, and the transition probability rule. We present a complete derivation of all five Solvay postulates from purely classical foundations, starting from the directly observable Coulomb field of the electron. The single empirical input is ℏ—measured from pre-Solvay experiments (Planck 1900–1911, Einstein 1905, Casimir 1948/Lamoreaux 1997) without quantum mechanical interpretation, in the same way G was measured by Cavendish (1798) without general relativity. By Einstein’s photoelectric theory (1905) generalized to all electromagnetic fields, the Coulomb field energy is carried in electromagnetic quanta, each carrying both electric and magnetic fields with energy density ϵ0E^2. Lorentz invariance uniquely determines the ZPF spectral energy density u(ω) ∝ ω^3 (Boyer 1969, 1975). The Casimir force measurement gives the total ZPF energy per mode B = ℏ; classical E/B equipartition from Maxwell’s equations gives the electric energy per mode ℏω/2. The central original contribution is the physical justification of the Compton frequency cutoff ωc = mc^2/ℏ: an electron cannot emit an electromagnetic quantum with energy exceeding mc^2, because the residual electron would require negative mass, never observed in nature. By equating two independent expressions for the momentum variance of the ZPF-driven harmonic oscillator—one from the Abraham–Lorentz equation, one from Nelson’s osmotic velocity formula—D = ℏ/2m is derived without any quantum assumption. The ZPF-driven motion is then proved to be Brownian motion in the precise sense of Wiener (1923): all four Wiener conditions are verified from the properties of the ZPF established in Sections 3–4. This answers the question Nelson (1966) explicitly left open: “We do not attempt to give a physical explanation of why the electron should undergo Brownian motion.” The physical explanation is that the electron cannot escape its own Coulomb field. By Itô’s stochastic calculus and Nelson’s theorem, this gives the Schrödinger equation as an effective description valid for timescales Δt ≫ ℏ/mc^2. The transformation ψ = √ρe^(iS/ℏ) linearises the nonlinear Fokker–Planck equation, explaining why superposition holds. All five Solvay postulates emerge as consequences of classical physics plus the single measured constant ℏ. No Solvay postulate is assumed anywhere.
Traditionally, Newtonian mechanics is recognized as an approximate theory which is said to arise from quantum mechanics (apparently a more detailed theory) in the large energy level limit (correspondence principle). We argue that both quantum mechanics and Newtonian mechanics are both approximate and both complement each other at any energy level. In the Newtonian case, quantum effects are so small that they are not visible, but in the quantum case, Newtonian effects are visible, we argue. We first note that exp(-iEt+ipx), the free particle wavefunction (complex probability) does not exist without the Newtonian expression x=vt. x=vt implies it is possible for dx->0, dt->0, while hbar/p and hbar/E show that there are limitations with respect to interactions. The only way to ascertain x=vt is through measurement and if these measurements have dt=hbar/E and dx=hbar/p uncertainties, one cannot have dx->0, dt->0 resolution. Nevertheless, the quantum free particle does move as x=vt (as long as one respects the uncertainties of dt,dx).The quantum exp(-iEt+ipx) does not exist without the Newtonian x=vt. In a single particle quantum bound state (time-independent), we suggest there are significant approximations. First, a conservation of energy equation exists in Newtonian physics, namely pp/2m + V(x) = E ((1)). This equation does not contain time and does not explicitly show x(t), but that does not mean that such motion does not physically exist. The quantum bound state solution follows from ((1)) which is an equation which does not display full information, even in the classical case, and so this loss of information carries down to the quantum solution. In particular, in the quantum case one solves a time-independent problem and so uses a(p)exp(ipx) + a(-p)exp(-ipx) with a(p)=a(-p) or a(p)= -a(-p). These conditions are equivalent to insisting that the local momentum average at each x (-i W* grad W + i W grad W) is 0. The fact that one is solving a quantum bound state using only energy considerations means that one averages out time which means averaging momentum. Thus, W*(x)W(x) describes a resolution spatial pattern for interaction, but W = W(x) exp(-iEt+ip x) where p=0 at each x. In other words, just because one does not see forward/backward motion in a particular quantum solution (based on energy alone) does not mean that it does not exist and that it cannot be measured, we suggest. For high energy levels, one should be able to measure such motion. The factor exp(-iEt+ipx) is a further approximation of the bound state which may be used if one wishes to describe a particle of mo in a potential V(x) created by a particle of mass M as an M+m+E rest mass system. Thus, one may see that various levels of approximation are used in the same quantum bound state solution. Furthermore, classical and Newtonian mechanics are still linked at any energy level because the quantum average KE (in the nonrelativistic case) must equal the classical kinetic energy. Thus, the notion of x= vrms(x) t is present and if hbar/(m vrms) << dx (where dx <<L with L being the system size), then one again has exp(-ipx+i KE t) and x=vrms(x) within each tiny dx, as an approximation.
This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative approach to QFT for combining the quantum mechanics and special theory of relativity which keeps the concept of wave function (belonging to some representation of Lorentz group) through the whole theory. Scalar product has been redefined to take into the account the nonunitarity of representations of Lorentz group. Understanding parity symmetry turns out to be the key ingredient throughout the process. Instead of trying to guess an equation or a set of equations for some wave functions or fields (or equivalently trying to guess a Lagrangian for the same), one derives them based only on the superposition principle and properties of wave functions under Lorentz transformations and parity. The resulting model has striking similarities with the standard quantum field theory and yet has no negative energy states, no zitterbewegung effects, symmetric energy momentum tensor and angular momentum density tensor for \emph{all} representations of Lorentz group (unifying the theoretical description of all particles), as well as clear physical interpretation. It also offers a possible interpretation why particles and antiparticles have opposite quantum numbers.