We consider a particular model of a Source of independent particles and a macroscopic Detector that are both tuned to the same resonance frequency ν0≡1/P. Particles are emitted by the Source at exact multiples of the resonance period P, and the Detector absorbs them with a certain probability at any one of its points. The Detector may also announce the detection of the absorbed particle. Any particle that is not absorbed at a certain point passes through to a deeper layer in the interior of the Detector. Eventually, all particles will be absorbed, i.e., detected. We calculate the probability of detection for two particle time series generated by the same Source reaching the Detector with a time delay of δt and show that it manifests the illusion of collective (wave-like) interference with particle number conservation. We conclude that wave phenomena may reflect the nature of detectors rather than fundamental laws of physics.
We consider a particular model of a Source of independent particles and a macroscopic Detector that are both tuned to the same resonance frequency ν0≡1/P. Particles are emitted by the Source at exact multiples of the resonance period P, and the Detector absorbs them with a certain probability at any one of its points. The Detector may also announce the detection of the absorbed particle. Any particle that is not absorbed at a certain point passes through to a deeper layer in the interior of the Detector. Eventually, all particles will be absorbed, i.e., detected. We calculate the probability of detection for two particle time series generated by the same Source reaching the Detector with a time delay of δt and show that it manifests the illusion of collective (wave-like) interference with particle number conservation. We conclude that wave phenomena may reflect the nature of detectors rather than fundamental laws of physics.
We present a numerical implementation of the proposed Source–Detector Resonance (SDR) as a conceptual analog of a Double-Slit Interference Experiment with discrete particles. Two periodic streams of particles are emitted from two point sources at random integer multiples of a fundamental period P and corresponding frequency ω=2π/P and fly out towards a detection screen. The screen consists of a deep set of identical oscillators with eigenfrequency ω0=2π/P0. In the SDR scenario, ω≈ω0. When the particles reach the screen, they implement a periodic forcing of its oscillators at the stream’s fundamental frequency ω0. As a result, an oscillating pattern develops along the screen. The amplitude of oscillation of each oscillator saturates at a value that is determined by the balance between the periodic particle forcing and the damping of each oscillator. This is clearly proportional to the number of particles that reach a certain oscillator per unit time times the fraction of particles that reach it at its resonant frequency. The latter fraction is equal to the ratio of the Power Spectral Density (PSD) of the time series of the particles that reach the oscillator at its resonance frequency PSD(ω0) over the PSD at zero frequency PSD(0). If we further assume that each oscillator absorbs a particle and announces a detection with a probability that is proportional to the square of the ratio PSD(ω0)/PSD(0); the pattern of particle detections that develops over a thick layer of oscillators is shown to be the same as that of a Double-Slit Interference Experiment. Our result shows that when macroscopic resonant detectors interact with and detect periodic streams of discrete particles, they may create the illusion of an interference measurement, as if each discrete particle manifests a phase of its own.
In a single-particle detection experiment, a wavefront impinges on a detector but observers only see a point response. The extent of the wavefront becomes evident only in statistical accumulation of many independent detections, with probability given by the Born rule. Drawing on concepts from quantum optics, we analyze a simple model to reverse-engineer how this behavior can come about in terms of wave mechanics alone without a measurement axiom. The model detector consists of many molecules, each of which can be resonantly excited by the incoming particle and then emit a detection signature (e.g., localized flash of light). Different molecules have different resonant energies because local conditions (proximity of other molecules, Doppler shifts, etc.) vary. The detector is thus a quasi-continuum, and the incoming particle preferentially excites the molecule that it matches most closely in energy. (In actuality, molecules can be so numerous that many could closely match the incoming particle in energy; but in that case only one will be the first, and there will be nothing left for the others by the time the first match resonates and then emits. The model does not explicitly take into account the temporal advance of the particle wave packet through the detector medium.) The excited molecule can emit a detection signature, but that process competes with fluctuation-driven dephasing. We estimate the probability that a given molecule is resonantly excited, and we also estimate the probability that a detection signature is produced before being overwhelmed by fluctuations in the detector medium. The product of these two probabilities is proportional to the absolute-square of the incoming wavefunction at the molecule in question, i.e. the Born rule. We discuss ways to probe these mechanisms experimentally, and check the model with numbers from a neutron interference experiment.
We present a statistical model of non-interacting individual classical particles that may lead to a microscopic implementation of quantum mechanics. The model requires the action of a special type of detector that detects and records individual particles. According to our model, microscopic particles are classical entities that follow classical trajectories without nonlocal interactions between them. As long as they remain undetected, particles carry the information of a phase equal to an action integral along their trajectory. At the point of their detection, our special type of detector collects the phases from all particles reaching it, adds them up over time as complex numbers, and divides them by the square root of their number. The detector records a detection each time the square of the magnitude of the ensuing vector crosses an integer value. We test our model with numerical simulations of a double-slit experiment, and obtain an interference pattern analogous to the one observed in an actual physical experiment.
The Observer Effect and Wave–Particle Duality in the S-field Framework --- 1. The traditional wave–particle problem In standard quantum mechanics, physical entities appear to behave: • as waves in interference experiments; • as particles when measured or detected. This has led to the idea that observation itself changes reality, often formulated as: “measurement collapses the wavefunction”. However, this formulation leaves unresolved questions: • What qualifies as an observer? • Why should information acquisition alter physical law? • Where exactly does the transition occur? --- 2. S-field perspective: no wave, no particle as primitives In the S-field framework, neither “wave” nor “particle” is fundamental. At the base level, there exist only: • oscillatory S-field states; • phase, frequency, and orientation; • dynamic resonance patterns. Wave and particle are emergent regimes, not ontological categories. 3. Wave behavior: coherent phase evolution A system exhibits wave-like behavior when: • S-oscillations evolve coherently; • multiple phase paths remain phase-correlated; • no dominant resonance locking occurs. In this regime: • interference patterns arise naturally; • delocalization reflects phase superposition; • the system evolves as a distributed resonance structure. “Wave” behavior is therefore a manifestation of extended phase coherence. --- 4. Particle behavior: resonance locking Particle-like behavior emerges when: • the S-oscillatory ensemble undergoes strong phase locking; • coherence collapses into a dominant resonant configuration; • the system enters a stable 4D projection. Localization is not imposed externally — it is a self-organized resonance outcome. A “particle” is a stable, self-locked oscillatory ensemble. 5. What observation actually does Observation does not create reality and does not collapse a wavefunction. Instead, observation means: • physical interaction with an environment; • introduction of new S-field couplings; • modification of phase relations. This interaction typically: • reduces global phase coherence; • enforces resonance selection; • stabilizes a particular outcome. Thus, observation is not epistemic — it is dynamical. --- 6. Why which-path information destroys interference Interference disappears because: • phase correlations are disrupted; • alternative phase paths decohere; • the system is forced into locally stable resonance modes. Which-path information is not special — any interaction that breaks coherence removes interference. No observer consciousness is required. 7. Delayed-choice experiments In delayed-choice scenarios: • the system remains in a coherent oscillatory regime; • no definitive resonance locking occurs until interaction; • temporal ordering appears paradoxical only classically. In the S-field framework: • time itself emerges from phase ordering; • coherence preservation allows late-mode selection; • no retrocausality is involved. The outcome reflects final coherence conditions, not past knowledge. --- 8. No collapse, only regime transition There is no physical “collapse”. Instead, there is: • a transition from distributed coherence to localized resonance stability; • a change of dynamical regime; • emergence of macroscopic definiteness. Wave → particle transition is micro → macro coherence restructuring. 9. One-sentence synthesis > Wave–particle duality is not a duality of objects, but a duality of coherence regimes in the S-field. --- 10. Conceptual closure This interpretation: • removes observer-dependence; • preserves realism; • unifies wave and particle behavior dynamically; • aligns quantum phenomena with micro–macro transition theory. Si(t) = Ai · e^( i( ωi·t + φi ) ) C = | (1 / N) · Σ_{i=1}^N e^( i φi ) | Order parameter (complex): R·e^( iΨ ) = (1 / N) · Σ_{j=1}^N e^( i φ_j ) , C ≡ R Kuramoto-type phase dynamics (micro ensemble): dφ_i/dt = ω_i + (K / N) · Σ_{j=1}^N sin( φ_j − φ_i ) Self-consistent equation for stationary coherence R: R = ∫ dω g(ω) · √(1 − (ω/(K R))^2 ) (for |ω| ≤ K R) — standard Kuramoto integral form Mass as emergent resonant integral: m ∝ ∫ ρ_S(χ) · C(χ) · dV_S(χ) Micro→macro criterion: C ≥ C_crit ⇒ stable 4D projection (particle regime) C < C_crit ⇒ incoherent / wave-like regime Interference visibility and intensity (two-path): I(Δφ) = I_0 · [ 1 + V · cos( Δφ ) ] , V ∝ C Coherence decay under environment coupling (phenomenological): dC/dt = α(K,…) · C − Γ_env · C ⇒ C(t) = C(0) · e^{ (α − Γ_env) t } Decoherence rate (Fermi-golden-rule style): Γ_env(ω) = (2π / ħ) · |V_int(ω)|^2 · ρ_env(ω) Measurement interaction as coupling-induced locking (mean-field): dr/dt = (K/2) · r · (1 − r^2) − γ · r , fixed points r = 0 (wave) and r = r_* > 0 (particle) when K > K_c Which-path disruption condition: any interaction s.t. Γ_env ≳ α(K,…) ⇒ V → 0 (interference destroyed) Time as accumulated phase change (collective): t ≡ (1 / Ω) · Σ_k Δφ_k , Ω ≡ ⟨ω⟩_ensemble Spatial separation as phase-metric: d_{AB} ∝ | ⟨ e^( i( φ_A − φ_B ) ) ⟩ | Classical limit (macroscopic): lim_{N→∞} Var(φ) → 0 ⇒ C → 1 ⇒ particle-like, deterministic dynamics Compact formal statement: Classicality = lim_{C,N → large} ( Quantum S-ensemble )
We consider the time-delayed coincidence counting of two photons emitted in a cascade by a single particle (atom, molecule, nucleus, etc.). The time-dependence of the probability amplitude of the second photon in the cascade has a sharply rising leading edge due to the detection of the first photon, as results from causality. If a macroscopic ensemble of resonant two-level absorbers is placed in the path of the second photon between the radiation source and the detector, the photon absorption does not follow Beer's law due to the time-asymmetric shape of the photon. For very short delay times almost no absorption takes place, even in an optically dense medium. We analyze the propagation of such a second photon in a thick resonant three-level absorber if a narrow electromagnetically induced transparency (EIT) window is present at the center of the absorption line. It is shown that the EIT medium can change the asymmetric time dependence of the photon probability amplitude to a bell shape (EIT filtering). This bell-shaped photon interacts much more efficiently with another ensemble of two-level absorbers chosen, for example, to store this photon and the information it carries. Ideally a photon wave packet with a Gaussian time-envelope is most effectively stored.
This paper demonstrates how a classical detector that collects non-interacting individual classical massive free particles can generate a quantum interference pattern. The proposed classical picture requires that particles carry the information of a phase equal to an action integral along their trajectory. At the point of their detection, a special type of detector collects the phases from all individual particles reaching it, adds them up over time as complex numbers, and divides them by the square root of their number. The detector announces a number of detections equal to the square of the amplitude of the resulting complex number. An interference pattern is gradually built from the collection of particle phases in the detection bins of the detector after several repetitions of the experiment. We obtain perfect agreement with three solutions of the Schrödinger equation for free particles: a Gaussian wavepacket, two Gaussian wavepackets approaching each other, and a Gaussian wavepacket reflecting off a wall. The main conclusion of the present work is that the interference of quantum mechanics is basically due to the detectors that collect the particles when there are macroscopic detectors operating as proposed. Finally, a simple physical experiment with a single-photon detector is proposed that will be able to test our theory.