This paper defines an equation for causality. This equation is then combined with the postulates of quantum mechanics and mass-energy equivalence to produce a quantum mechanical telegrapher's equation and to reproduce the Schrodinger and Klein-Gordon equations. The incompressible Navier-Stokes equations and dynamic general equilibrium in economics (with an interpretation of a Nash equilibrium) are obtained when the equation of causality refers to itself, i.e. when the cause is its own effect. As it is shown that the Klein-Gordon equation is obtained by Wick rotating the cause vector with de Broglie angular frequency, this paper postulates an equation for Quantum Gravity, which relates the Navier-Stokes equations to the Einstein Field Equations of General Relativity.
This paper defines an equation for causality. This equation is then combined with the postulates of quantum mechanics and mass-energy equivalence to produce a quantum mechanical telegrapher's equation and to reproduce the Schrodinger and Klein-Gordon equations. The incompressible Navier-Stokes equations and dynamic general equilibrium in economics (with an interpretation of a Nash equilibrium) are obtained when the equation of causality refers to itself, i.e. when the cause is its own effect. As it is shown that the Klein-Gordon equation is obtained by Wick rotating the cause vector with de Broglie angular frequency, this paper postulates an equation for Quantum Gravity, which relates the Navier-Stokes equations to the Einstein Field Equations of General Relativity.
It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.
It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.
Starting from three fundamental duality relations (time-energy, space-momentum, angle-angular momentum), we establish an integral equation of quantum action and its dual counterpart. This integral equation directly links the evolution of the wave function to the accumulated action of spacetime and rotational degrees of freedom, forming a unified framework for quantum theory. Through semiclassical expansion, local approximation, and relativistic covariance, we rigorously derive the Schr\"odinger equation, Klein-Gordon equation, Dirac equation, Heisenberg equation, Schwinger-Dyson equation, Feynman path integral, Yang-Mills equations, Einstein field equations, and the Friedmann and Schwarzschild solutions. The framework also yields the kinematic effects of special relativity (time dilation, length contraction, mass-energy equivalence) as consequences of the duality structure. Moreover, the nonlocal kernel in the integral equation gives an exponential damping factor in momentum space, providing an intrinsic regularization mechanism for ultraviolet divergences. We extend the framework to quantum electrodynamics (QED), quantum chromodynamics (QCD), the electroweak theory, the full Standard Model, and quantum gravity, obtaining dual formulations for each. From the integral equation and its dual, we automatically derive quantization conditions for energy, momentum, angular momentum, and their duals, and propose experimental schemes to observe these discrete effects. All derivations are complete and rigorous, based on the same integral equation, providing a possible answer to Dirac's criticism that "quantum electrodynamics lacks a fundamental equation" and opening a new path toward a unified theory.
Derivation of Klein–Gordon equation Goals and prior knowledge The main wave equations of relativistic quantum mechanics are the Klein–Gordon, Dirac, Weyl , and Rarita–Schwinger equations. The principal aim of this chapter is to derive these, by means of the EPI principles (3.16) and (3.18). The derivations generally follow Frieden (1995), or Frieden and Soffer (1995), with numerous additions. In addition, the Schrödinger wave equation (SWE), which is non -relativistic, is also to be derived. Since rigorous use of EPI gives only relativistic wave equations (Chapter 3), EPI can only derive the SWE in some approximate sense. This is directly carried through in Appendix D, or by taking the well-known non-relativistic limit of the Klein–Gordon equation in Appendix G. Also, in Appendix D a simple EPI derivation of classical mechanics is obtained. In particular, in the limit h → 0 the EPI extremum condition (3.16) is found to give Newton's second law , and the zero-condition (3.18) gives the virial theorem , of classical mechanics. Other derived results are Eq. (4.17) – the equivalence of mass, momentum, and energy; the constancy of the Planck constant h (Sec. 4.1.14); the Compton wavelength as an ultimate resolution length (Sec. 4.1.17); and the Heisenberg uncertainty principle (Sec. 4.3). It is important to state what is assumed a priori . The relativistic properties of space and time coordinates were derived in Sec. 3.5. Hence, these are assumed here. Also, appropriate units for the concepts of mass and energy are assumed to be known, although the basic relation Eq. (4.17) between them is not presumed.
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 Schroedinger- and Klein-Gordon equations are directly derived from classical Lagrangians. The only inputs are given by the discreteness of energy (E=hbar.w) and momentum (p=hbar.k), respectively, as well as the assumed existence of a space-pervading field of "zero-point energy" (E_0=hbar.w/2 per spatial dimension) associated to each particle of energy E. The latter leads to an additional kinetic energy term in the classical Lagrangian, which alone suffices to pass from classical to quantum mechanics. Moreover, Heisenberg's uncertainty relations are also derived within this framework, i.e., without referring to quantum mechanical or other complex-numbered functions.