We derive the double-slit interference pattern within Homogeneous Compression Theory (HCT) starting from a discrete relational microstructure, without postulating the Born rule, a wavefunction, particle trajectories, or collapse. Three steps carry the derivation: (i) a positivity theorem for the bilateral correlation matrix, proved from microstructural definitions via the Schur product theorem and Bochner's theorem; (ii) linear bi-local propagation from the slits to the screen, which yields the interference law I=I1+I2+2|mu|sqrt(I1I2)cos as an algebraic consequence of a Hermitian correlator; (iii) detection modeled as local bilateral calorimetry, giving event rates proportional to the bi-local diagonal, R(X) proportional to B(X,X). Particle-like clicks and wave-like fringes emerge as the diagonal and off-diagonal aspects of a single Hermitian correlator. Definite trajectories are shown to do no explanatory work.
We derive the double-slit interference pattern within Homogeneous Compression Theory (HCT) starting from a discrete relational microstructure, without postulating the Born rule, a wavefunction, particle trajectories, or collapse. Three steps carry the derivation: (i) a positivity theorem for the bilateral correlation matrix, proved from microstructural definitions via the Schur product theorem and Bochner's theorem; (ii) linear bi-local propagation from the slits to the screen, which yields the interference law I=I1+I2+2|mu|sqrt(I1I2)cos as an algebraic consequence of a Hermitian correlator; (iii) detection modeled as local bilateral calorimetry, giving event rates proportional to the bi-local diagonal, R(X) proportional to B(X,X). Particle-like clicks and wave-like fringes emerge as the diagonal and off-diagonal aspects of a single Hermitian correlator. Definite trajectories are shown to do no explanatory work.
The double-slit experiment is typically presented as evidence for the non-classical nature of microscopic particles, especially when performed in the single-particle regime. However, the interference pattern that emerges on the detection screen is generated by wave propagation through two structured apertures, each acting as a narrow resonant cavity supporting longitudinal modes of a scalar field. In this work we derive the outgoing fields from the slits directly from boundary conditions and show that their superposition yields the standard far-field intensity profile of diffraction theory. The remaining aspect of the experiment — namely discrete, localized detection events — is accounted for by a minimal discrete-event model in which the continuous classical field drives quantized energy transfers at the detector. This approach is fully classical, requires no collapse postulate or nonlocality, and remains compatible with all known data. It clarifies which features of the experiment arise from field propagation and which arise from the discrete nature of detection.
In this work the author presents a fully explicit mathematical derivation of interference-like structures in a simulated double-slit configuration within the framework of the Schrödinger equation. The mathematical structure of the slits is directly incorporated into the initial wavefunction, avoiding simplified assumptions such as plane waves. The slit is modeled as a spatial constraint, and the resulting state is expanded over the eigenbasis of the system. The time evolution of the wavefunction naturally leads to the formation of structured spatial patterns. A multiscale potential based on the Livio function is introduced and treated through an explicit matrix formulation, allowing a direct comparison between the free case and a system with structured perturbations. In addition, a stochastic reconstruction of the probability density is performed by sampling individual detection events according to the Born rule, simulating the arrival of particles at a finite rate and showing the progressive emergence of the pattern over time. The results indicate that the interference fringes emerge as an intrinsic consequence of the spectral decomposition of a spatially structured quantum state. The close qualitative agreement between the simulated patterns and those observed in physical experiments therefore suggests that wave–particle-like behavior is intrinsically contained in the mathematical framework of quantum mechanics.
This volume establishes that the quantum propagator and double-slit interference pattern are derived consequences of the algebraic structure of H0(A3), not independent postulates. Starting from the presentation <a, b, c | a²=a, b²=b, c²=c, aba=bab, bcb=cbc, ac=ca>, we construct the transfer operator T = La + Lb + Lc as the adjacency operator of the Cayley graph, show that its spectrum {0, 1, 2, 3} with degeneracies (1, 11, 11, 1) is forced by idempotence, and derive the propagator K(d, s; t) = ⟨d|exp(−iTt)|s⟩ via spectral decomposition. Explicit numerical computation on the 24-element monoid verifies quantum interference: for certain detectors, the coherent probability |A1 + A2|² differs from the incoherent sum |A1|² + |A2|². We show that the commutation relation ac = ca in A3 suppresses interference at 20 of 24 detectors, while the fully braided algebra H0(Ã2) with C(3) = 0 produces maximal fringes. Measurement is not postulated but derived: the idempotent projection Ls² = Ls destroys coherence between paths, yielding the Born rule as a theorem. Keywords: 0-Hecke monoid, derived propagator, double-slit interference, transfer operator, idempotent projection, commutation deficit, Cayley graph Laplacian
Double-slit interference is one of the most fundamental quantum phenomena among photons and other particles. We propose a novel and realist quantum framework in which interference patterns in double-slit experiments arise not from wavefunction self-interference or collapse, but from causal non-local, quantized cavity-induced momentum exchange between the electron and the double-slit field potential. Unlike Copenhagen or Bohmian interpretations, our theory treats the electron as a deterministic, soliton-like entity governed by classical-like trajectories. The quantized potential of the slits—described via creation and annihilation operators—is responsible for discrete momentum transfer through coupling with cavity or barrier modes, modeled via octonionic algebra that differentiates internal particle structure from external field dynamics. The resulting interference pattern emerges from non-local, quantized interactions between the electron and field modes without invoking the probabilistic wavefunction. Our model aligns with experimental violations of Bell’s inequality and avoids conceptual paradoxes such as wavefunction collapse and self-interference. The framework makes several testable predictions, offering a promising new direction for understanding quantum interference with clearer physical realism.
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.
Quantum interference is traditionally interpreted as evidence of intrinsic wave–particle duality, invoking self-interference and wavefunction superposition. Here, we propose an alternative causal framework in which interference arises from interactions between a classical electron and a quantized electromagnetic field. In this picture, electrons follow definite, localized trajectories and produce discrete detection events, while the observed interference pattern emerges statistically from stochastic, nonlocal momentum transfers mediated by quantized field modes. Numerical simulations of the double-slit experiment show that dot-like interference fringes arise without invoking wavefunction collapse or intrinsic matter waves. Doubling the electron mass narrows the fringe spacing in agreement with de Broglie scaling, indicating that the effective wavelength emerges dynamically from particle–field coupling rather than representing an inherent property of the particle. The formulation is based on a rigorously defined Hamiltonian and derived equations describing electromagnetic coupling between particles and the slit environment. Although neutral particles such as neutrons or atoms carry no net charge monopoles, they possess electric or magnetic dipole and higher multipole moments that enable electromagnetic interactions with the slit apparatus, allowing discrete momentum transfer through the quantized field. Our simulations show that interference fringes arise from intermittent singleparticle detection events caused by stochastic momentum impulses from quantized electromagnetic modes. These results suggest a reinterpretation of wave–particle duality: the apparent wave-like behavior of matter may arise from interaction with the inherently wave-like quantized electromagnetic field—whose quanta are photons, the massless U(1)-gauge bosons—rather than from an intrinsic wave nature of the particles themselves. Our analysis is rigorous and is not speculative, because the proposed Hamiltonian for this system is physical. Within this nonlocal hidden-variable framework, the dot-like interference pattern observed in the double-slit experiment emerges from individual electrons traversing one slit at a time while receiving discrete momentum impulses from stochastic, spatially extended quantized electromagnetic fields.