UCM Paper 36 (Theoretical). Derives the Schrödinger equation from the UCM substrate action in the non-relativistic limit (Theorem 1). The Feynman path integral is identified with the sum over membrane phase paths (Proposition 2). Double-slit interference follows from instantaneous membrane phase propagation (κ_mem >> κ_cell, Paper 31): both slits simultaneously acquire the global phase pattern without hidden variables (Proposition 3). The de Broglie relation follows from the dispersion relation; confirmed against the Davisson-Germer experiment at 0.1%. The substrate quantum mass m₀ = E_Λ/c² is the U-cell rest mass; composite particle masses arise from substrate binding energy. UCM preprint series on Zenodo, ORCID 0009-0005-5088-4339.
UCM Paper 36 (Theoretical). Derives the Schrödinger equation from the UCM substrate action in the non-relativistic limit (Theorem 1). The Feynman path integral is identified with the sum over membrane phase paths (Proposition 2). Double-slit interference follows from instantaneous membrane phase propagation (κ_mem >> κ_cell, Paper 31): both slits simultaneously acquire the global phase pattern without hidden variables (Proposition 3). The de Broglie relation follows from the dispersion relation; confirmed against the Davisson-Germer experiment at 0.1%. The substrate quantum mass m₀ = E_Λ/c² is the U-cell rest mass; composite particle masses arise from substrate binding energy. UCM preprint series on Zenodo, ORCID 0009-0005-5088-4339.
The Schrödinger equation is not a postulate. It is the non-relativistic limitof the Klein-Gordon equation that governs Hopf winding modes in the Jobsoncell medium. The derivation chain has zero free parameters: 1. Jobson medium: K = 1/eps_0, rho = mu_0, c = sqrt(K/rho) [PROVEN] 2. Free pressure wave: omega = c*k (massless) [QM1] 3. Hopf winding rest energy = m_p*c^2 -> Compton frequency omega_C = m_p*c^2/hbar [QM2; the Compton wavelength IS lambda_p = hbar_c/m_p = Zone 1 boundary] 4. Winding dispersion: omega^2 = c^2*k^2 + omega_C^2 [Klein-Gordon, QM3] 5. Non-relativistic limit (v << c): omega_KG -> omega_C + hbar*k^2/(2*m_p) Remove rest-mass oscillation phi = psi * exp(-i*m_p*c^2*t/hbar): i*hbar * dpsi/dt = -(hbar^2/2*m_p) * nabla^2 * psi [SCHRODINGER, QM5] 6. Born rule: P(x) = |psi(x)|^2 from medium wave energy density [QM6] The Schrödinger equation is quantum mechanics. It is now derived from thesame medium (Jobson cell lattice) that gives alpha, G, the Higgs mass, andquantum entanglement. No additional postulates or free parameters. Consequences: - Double-slit: Hopf winding fields span both slits simultaneously. Interference = constructive/destructive Huygens superposition in medium. - Which-path: Zone 3 field decoherence resolves winding at r < r_lock(T). r_lock(300K) = 340 fm; coherence preserved for macroscopic slit separations. - Delayed choice: no paradox. Medium configuration never had a single-slit address. "Measurement" = local perturbation resolving global winding. Timing is irrelevant; topology is not retroactively changed. NEW PREDICTIONS (not in standard QM): - Minimum electron slit width for coherent diffraction: 2*lambda_bar_e = 772 fm (below this, slit-edge Zone 3 fields decohere the electron winding) - Minimum proton slit width: 2*lambda_bar_p = 0.421 fm = r_grind (nuclear hard core) - Fringe visibility vs temperature: V(T) ~ exp(-r_sep / r_lock(T)) where r_lock(T) = (alpha*hbar_c*r_p^2 / k_B*T)^(1/3)
The double-slit experiment has stood for a century as the paradigmatic demonstration of quantum superposition: the standard interpretation holds that a particle passes through both slits simultaneously, existing in a superposition of two spatially separated states. This paper proposes a physically distinct alternative: the particle's matter wave — a real physical wave propagating through space in accordance with de Broglie's relation λ = h/p — passes through both slits, while the particle itself (matter) remains in an undetermined state until a constructive interference maximum is formed at the detection screen. The particle does not simultaneously occupy two locations; the wave does. This distinction eliminates the need for superposition as an ontological commitment while fully reproducing the observed interference pattern. We further show that the disappearance of the interference pattern upon which-path measurement is explained without invoking wavefunction collapse: the measurement interaction redirects the matter wave to a single-slit geometry, eliminating the two-wave interference condition. Three experimentally testable predictions distinguish this framework from the standard interpretation: (1) weak-measurement trajectory reconstruction should reveal single particle paths inconsistent with simultaneous two-slit passage; (2) diffraction fringe spacing should depend on the mass density of the slit material, because denser slit edges generate a steeper refractive-index gradient for the matter wave; (3) the fringe pattern should exhibit a density-dependent phase shift measurable with current matter-wave interferometry technology. This reinterpretation requires no new mathematical formalism — the Schrödinger equation and Born's rule are retained — but replaces the ontological claim of simultaneous multi-location existence with the physically consistent picture of matter-wave propagation through an undetermined-state particle.
In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.
Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
The traditional path integral formulation of quantum mechanics relies on complex probability amplitudes and the principle of least action, which has conceptual ambiguities in interpreting microphysical mechanisms — serving merely as effective mathematical statistical tools. Based on the Unified Cosmic Mechanics Evolution Theory — where the momentum unit (m₀·c) is the only physical reality, the field is an interaction protocol, and the particle is a momentum encapsulation structure — this paper reconstructs the mathematical description of multi-slit experiments. The new formulation replaces the complex path integral with a dual-integral system: the self-integral describes momentum unit self-evolution along their intrinsic direction, endowing linear motion; the interaction integral describes momentum flow interaction with the slit wall's electromagnetic field, realizing multi-path distribution and direction deflection. These two integrals together constitute the momentum flow distribution integral, interpreting "interference" as the superposition of real momentum flows dominated by the interaction integral. Key insight: Just as a stone falling from a height depends on the perceptual cross-section 1/(4πr²) for its integral result, all forces should be similarly understood — their underlying dynamics must be reduced to the perceptual cross-section mechanism. The interaction integral follows this logic: the relativistic mass-energy equation is a space-time state shaping equation, where the emergent space-time integral reflects Pythagorean conservation (Lorentz covariance); general relativistic integrals involve variable space-time redundancy under field strength; special relativistic integrals arise from perceptual space-time window compression during motion. For micro-particles in multi-slits, this paper primarily adopts the interaction integral mechanisms of special relativity, electromagnetism, and photoelectricity. This framework remains consistent with existing experimental phenomena and proposes testable predictions including slit thickness-dependent deflection and multi-slit interference fringe modulation.