Quantum Phenomena: From Accompanying Fields to Detection

01Why is matter localized yet wave-like?

ASD describes moving matter as a joint system of a localized topological core and its surrounding accompanying field. Matter waves are the long-wavelength, low-speed effective envelope of a selected extended gapped channel in that system. This connects localized particle exchange to an extended spatial phase that responds to apertures, barriers and detectors; it is the physical starting point of Chapter 5 in the final manuscript.

Electrons arriving one by one at a detector leave localized records. Repeating the same preparation can make these records accumulate into a spatial pattern of bright and dark fringes. The localization of matter in collisions and absorption, and its interference and diffraction during propagation, need to be connected within one physical picture. Starting from localized spatial structures of matter, ASD asks how a stable structure evolves together with surrounding space as it moves. How does this evolution establish a phase that boundaries can alter, and how does it enter an actual detection event?

In this picture, the localized core and the accompanying field outside it have different roles. The core carries the stable structure of matter and its concentrated energy exchange. The accompanying field describes the spatial response associated with the core and its preparation history, including phase outside the core, deformation, and energy and momentum exchange. A freely propagating wave can transport energy far away; the outgoing condition here specifies outward propagation in the far region, without an additional wave incident from infinity. These three objects may involve shared spatial degrees of freedom, but they cannot simply be treated as the same solution.

The system under study therefore includes the source, core, accompanying field, boundaries and detector. Masks, slits, phase plates and detecting materials are themselves material systems, whose effects must enter the equations or boundary conditions. The framework developed here concerns effective waves of single particles and the overall center-of-mass motion of composites whose internal structure is not significantly excited. The latter condition is described as approximately frozen internal states. Stable topological cores, phase locking and universal detection statistics are further dynamical problems connected by this framework.

02How motion becomes a source for spatial fields

Translating an entire static matter configuration to a position $\mathbf X(t)$ does not by itself produce a complete moving solution. Motion changes the time derivatives of the field. Substituting a rigidly translated static profile into the original dynamical equation generally leaves a residual. This residual drives core deformation and the response outside the core until core and field jointly satisfy the equation. The accompanying field is the spatial structure studied through this joint evolution, rather than a probability cloud specified in advance.

The accompanying-field page derives the source term step by step from this substitution, showing how acceleration, quadratic velocity terms and internal phase enter the calculation. It takes the complete static configuration and action as inputs, then decomposes small perturbations into overall translation and other responses. This mode projection distinguishes core displacement from the other field responses. Core motion and its accompanying field should thus be connected by the dynamics of the same system, rather than prescribed independently.

A source term alone does not determine the actual response: the initial field, preparation history and boundaries must also be specified. A co-moving near field, a decaying structure of finite extent and outgoing radiation have different boundary and energy-flux conditions. Uniform motion alone cannot establish that every driven system is nonradiating. What is needed is a complete configuration that can move stably with the core while satisfying total energy and momentum conservation.

03What slits change and what detectors record

Consider a prepared matter beam approaching a mask. Whether a finite core can pass through an opening depends on its interaction with the material boundary. The field outside the core must satisfy propagation conditions across the entire aperture. When the energy distribution is narrow, the propagation direction deviates only slightly from the beam axis and the mask can be described as a thin screen—the nearly monoenergetic and paraxial approximations—the aperture transmission function specifies how the incident envelope enters each opening. The propagation equation then determines the amplitude and phase at the screen. The phase difference between two slits produces interference fringes, while the finite width of each slit produces a diffraction envelope.

This calculation gives the spatial distribution of a continuous field. Predicting actual counts also requires coupling the field and localized core to the detector: which degrees of freedom absorb energy, how a record is triggered, how the detector resets, and how finite efficiency, absorption thickness and the time window affect the accumulated pattern. Continuous propagation and discrete recording are therefore connected stages of the physical process, each with its own equations and conditions.

04From rapid field oscillation to a slowly varying envelope

Turning this physical picture into a calculable matter wave first requires identifying the relevant propagation channel. Chapter 5 selects an extended gapped branch with a nonzero natural frequency $\omega_0$: its frequency remains nonzero at zero wave number, while it also supports spatial phase variation. An internal oscillator confined near the core does not become such an extended branch merely by being written as a complex number. The existence of the spectrum and its response coefficients must be established from the core–field system.

Under this spectral condition, select a positive-frequency field component using the convention $e^{-i\omega t}$, $\omega>0$, and write

Here $\mathbf r$ is spatial position and $t$ is the metrological parameter recording the evolution of the process. The complex envelope $\Psi$ varies slowly in space and time; the real field is the real part of the expression above. The complex number records two orthogonal phase components and does not yet define a detection probability. Substitution into the second-order field equation cancels the rapid carrier terms, leaving the first and second time derivatives of the envelope and its spatial-gradient terms.

Within the long-wavelength range where the continuum description applies, the envelope frequency bandwidth must also be much smaller than $\omega_0$, with $ck/\omega_0\ll1$ and $|V|/(\hbar\omega_0)\ll1$, where $k=|\mathbf k|$. These slow-variation and weak-modulation conditions allow the smaller second envelope time derivative to be neglected relative to the term coupling the carrier frequency to the first envelope time derivative. Adopting the rest-energy matching condition $\hbar\omega_0=m_0c^2$ then gives the familiar gradient coefficient $\hbar^2/(2m_0)$. Here $m_0$ is the matched rest mass, $c$ is the characteristic speed of the field channel, and $\hbar=h/(2\pi)$ is the reduced action constant. In the isotropic limit with no explicit source, the result is

where $V$ is a weak scalar background modulation with units of energy in this model. General directional response, explicit sources and the discarded time term remain visible in the full effective derivation. The Schrödinger-type equation is thus a low-energy envelope result for a specified spatial-field channel. Rapid preparation and signal fronts require a return to the original second-order equation. Rest-energy matching is also an explicit input; the mathematical agreement of the equation does not establish that input as an independently obtained microscopic result.

05How phase connects momentum to observable patterns

The envelope provides a wave vector $\mathbf k$, while upstream ASD dynamics provides the mechanical momentum $\mathbf P$ of moving matter. Interpreting a propagation pattern as a matter wave requires connecting these quantities within the same core–field structure. For the monochromatic periodic mode used in the manuscript, define its action scale $\mathcal I$ as mode energy divided by angular frequency. Its momentum can be written as $\mathbf P_{\rm mode}=\mathcal I\mathbf k$. Only after matching the action of one matter excitation to $\hbar$, and matching the phase generator to the mechanical momentum of the complete structure, does one obtain $\mathbf P=\hbar\mathbf k$ and $\lambda=h/P$, where $P=|\mathbf P|$. The matter-wave page develops this matching and the relation between core velocity and group velocity.

Once phase and apparatus conditions are specified, concrete calculations become possible. Interference and diffraction determines aperture propagation and fringe geometry. For normalized wave packets with finite second moments and suitable decay conditions, the uncertainty relation follows from Fourier conjugacy as $\Delta x\,\Delta k\geq1/2$; the corresponding momentum-width bound additionally requires momentum matching. Tunneling determines transmitted flux by matching boundary conditions on both sides of a barrier. Connecting an evanescent field, which decays with distance inside the barrier, to actual core transport still requires the joint dynamics of the core, field and barrier.

06How the effective framework is tested

Chapter 5 uses linear material response to obtain mean absorbed power proportional to the squared modulus of the field envelope. Relating that power to a mean event rate additionally requires a fixed effective absorbed energy per record, with negligible leakage, recovery intervals after recording (dead time), saturation and similar effects. Under the appropriate detection-efficiency and time-integration conditions, the position distribution can reduce to a normalized squared modulus. This conditional route obtains a squared-modulus position law when its response and flux conditions hold, while retaining the dynamical tasks of explaining why one excitation produces only one record and establishing a more general Born rule.

Electron diffraction, interference patterns accumulated particle by particle, interference of neutral composites and barrier transmission specify relations that this framework must recover. Comparison with experiment must use the actual preparation, propagation boundaries and detection protocol. Recovering the mathematical form of established wave theory is a consistency requirement for the effective framework, not independent confirmation of ASD's microscopic interpretation. The quantum phenomenological requirements list these comparison tasks and their current discriminatory power.

The domain of the present calculations follows from the physical conditions at each step: a stable core is an upstream input; the gapped branch and energy and action matching are conditions of the effective construction; linear, long-wavelength and slowly varying approximations control propagation; and a detector model connects the field to statistics. For composites, overall center-of-mass waves can be studied when internal states are approximately frozen, external fields vary slowly and momentum transfer does not resolve internal structure. This does not require the center-of-mass wavelength to exceed the object's size.

The adopted action matching also yields $\mathbf v_g=c^2\mathbf P/E=\mathbf v$ in the isotropic free branch. For a stationary rectangular barrier the hyperbolic and exponential rational transmission formulas are exactly equivalent, while $T_{\rm WKB}$ names a first-order approximation for a slowly varying barrier. These established results and their assumptions are set out on the linked pages; microscopic locking, nonradial core stability and the exclusive single-record mechanism remain distinct closure tasks.

The next dynamical task is to connect these effective stages to the same source equations and conserved quantities. The core source equation, phase–momentum locking and universal Born statistics connect localized matter, continuous waves and discrete records, respectively. They specify how this physical picture can be tested and developed further.