Why Would Moving Matter Generate an Accompanying Field?

01The localized core moves with its surrounding field

In ASD's picture of matter, a particle is studied as a localized spatial structure with finite energy. Localization does not make the surrounding space irrelevant: translation, deformation and internal motion of the core can change the field outside it, which in turn acts back on the core through energy and momentum exchange. The accompanying field concerns this joint evolution, rather than assigning a “wave function” in advance as a probabilistic entity around the particle.

Three objects need to be distinguished. The localized core carries the stable structure and concentrated exchange associated with matter. The accompanying field is the spatial response associated with the core and its preparation history, and can include co-moving near fields, deformation and phase. A freely propagating wave is a disturbance evolving in its propagation channel under specified initial and boundary conditions, and can transport energy to the far region. An actual solution may contain all of these components. They must be distinguished through the allowed perturbation frequencies and spatial forms—the field spectrum—as well as boundaries and energy flux.

This page starts from a complete static configuration whose existence is assumed. It explains how motion enters the field equation and then how a specified source produces a spatial response. The static configuration here includes the core and its existing surrounding field; a stable particle solution still has to be established from the complete nonlinear dynamics.

02From a translated trial configuration to a mismatch source

Let $\mathcal Q_*(\mathbf r)$ denote the complete static configuration, $\mathbf X(t)$ the core center, and $\eta(\mathbf r,t)$ the remaining perturbation relative to the translated configuration. First consider an illustrative equation with a normalized time-derivative term and no velocity cross terms,

The variable $\mathcal Q$ may collect several spatial fields, while $\mathcal F$ denotes the selected nonlinear spatial-dynamics operator. Initially translate only the static profile and set $\boldsymbol\xi=\mathbf r-\mathbf X(t)$. Define velocity and acceleration by $\mathbf v=\dot{\mathbf X}$ and $\mathbf a=\ddot{\mathbf X}$. Taking two successive time derivatives at a fixed observation position, the chain rule gives

The spatial derivatives here act on the translated core profile. Substitute $\mathcal Q_*+\eta$ into the original equation and retain terms linear in $\eta$. The static equation cancels the original spatial-balance terms, yielding

The operator $\mathcal H_*(t)$ describes the linear restoring response to a small deformation of the static structure. Mathematically, it is the second variation of static energy normalized using the kinetic inner product. The source $S_{\rm tr}$ is the residual of rigid translation; the complete $S$ may also include contributions from internal phase, orientation, velocity-dependent deformation and external backgrounds. Nonlinear terms of order $O(\eta^2)$ are omitted when defining this first-order equation. This step turns the picture of motion affecting the surrounding field into an equation that can be calculated term by term.

Even when $\mathbf a=0$, rigid translation can leave a quadratic velocity term. It expresses a mismatch between the trial configuration and the original equation, not a claim that all motion must radiate. An actual moving solution allows the core profile and surrounding field to adjust together, changing or even cancelling some residuals. If the core itself is a periodic solution, the relevant question is how a perturbation changes over one background period. This requires linearization about the periodic background and Floquet analysis; a static perturbation spectrum cannot simply replace it.

03Separating core position from field deformation

Changing the core position produces a perturbation of the form $\partial_i\mathcal Q_*$. Including the same change again in $\eta$ would count one translational degree of freedom twice. A perturbation that shifts the center while preserving the internal shape is called a translation zero mode. Once assigned to $\mathbf X(t)$, that component must be excluded from the remaining perturbation by requiring orthogonality in the kinetic inner product:

In $\langle\cdot,\cdot\rangle_Z$, the weight $Z$ is determined by the kinetic terms of the action. Different field coefficients require the corresponding weights. Projection onto translation zero modes determines core motion; projection onto the orthogonal complement determines the other deformations, accompanying responses and radiation. The core trajectory and source strength must therefore be determined from the same action and solution, rather than chosen independently.

A further dynamical connection is also needed between internal oscillations and matter waves outside the core. A localized normal mode is a small perturbation oscillating with a fixed spatial shape near the core; its amplitude initially depends only on time. Promoting it to an amplitude that depends on spatial position requires additional extended propagation structure. The gapped branch used by the subsequent effective wave equation is a propagating branch with nonzero frequency at zero wave number that also supports spatial phase variation. Its existence is a condition to be established from the complete core–field spectrum.

04Finding the actual response to a specified source

Where the background and coefficients are approximately uniform and a single-channel approximation applies, the linear propagation operator may be taken as $\partial_t^2-c^2B_{ij}\partial_i\partial_j+\Omega_a^2$. Here $B_{ij}$ is a real, symmetric, positive-definite dimensionless response matrix, $c$ is the characteristic speed, and $\Omega_a$ is the channel's gap frequency. The subscript $a$ labels a channel and differs from the acceleration vector $\mathbf a$ above. The retarded Green function $G_a^{\rm ret}$ describes the response of subsequent spatial positions and times to a localized disturbance. The initial field and source history jointly determine the actual solution:

Here $t_0$ is the initial time, $\eta_{a,\rm in}$ is the homogeneous solution satisfying the specified initial data, and $S_a$ inside the integral is the source projected onto the selected channel. The variables $\mathbf r',t'$ label the source position and source time being integrated over, as distinct from the observation point $\mathbf r,t$. Primes mark integration variables here, not derivatives. Retaining the initial-data term means that the same instantaneous core position need not correspond to the same field: preparation history also matters. Coupled channels require a matrix Green function. If the projection basis varies with position, the mixing produced by derivatives of its basis vectors must also be retained. A single-channel adiabatic approximation applies only when the background and basis vary sufficiently slowly and interchannel conversion can be neglected in a controlled way.

A core has a finite size, and its source has an actual shape. Far from the core, the response can be expanded successively in total source strength, the source's directional distribution and higher spatial moments; this is the multipole expansion. For a mismatch source containing derivatives, the monopole integral—the total source strength—may vanish. Calculation must then begin with the first nonzero multipole instead of replacing every channel by the same point source. A finite-core form factor describes how the actual spatial source distribution changes the response at different wave numbers. The wave-number range of a candidate lattice instead comes from its underlying connections. These are different scale mechanisms and cannot substitute for one another.

For an isotropic gapless channel, for example, the retarded propagation time is determined by $t-t_r=R(t_r)/c$. Eliminating the source time $t_r$ produces the directional factor $1-\mathbf n\cdot\mathbf v/c$, where $R$ is the distance from the retarded source position to the observation point and $\mathbf n$ is the unit vector along that direction. A fixed propagation speed and the directional response of a moving source can therefore coexist. Quantitative matching to the LVE directional relation and to matter-wave phase still has to be established separately through dynamics.

With slits or material interfaces present, the propagation response must also satisfy their boundary conditions. The resulting field carries information about the aperture and material. Together with how the core passes through an opening and how the detector absorbs and records a matter event, these conditions define the experimental problem. The representation of the propagating field alone has not yet supplied a probability for the core's detection position.

05Distinguishing a co-moving response from free radiation

Moving with the core specifies a spatial and temporal relation between field and source, but does not by itself specify the far-field energy flux. For a uniformly moving source of the form $f(\mathbf r-\mathbf vt)e^{-i\omega_st}$, $f$ denotes the source profile and $\omega_s$ its prescribed internal driving frequency. Resolving this source into plane-wave components at different frequencies and wave numbers places its Fourier spectrum on $\omega=\mathbf k\cdot\mathbf v+\omega_s$. One must then check whether this source spectrum intersects the selected free-propagation spectrum:

Here $\mathbf k$ is the wave vector and $\omega_a(\mathbf k)$ the natural frequency of channel $a$. The second relation applies specifically to an isotropic gapped branch: $\omega_0$ is its zero-wave-number frequency, $\gamma=(1-v^2/c^2)^{-1/2}$, and $v<c$ is required. A spectral intersection allows a propagating mode to be excited. Actual far-field flux additionally requires the source to couple to that mode—a nonzero source projection—and outgoing boundary conditions that allow energy to propagate into the far region.

In an isotropic gapless channel, uniform translation of a static profile corresponds to $\omega_s=0$. For $v<c$, there is no radiation intersection at nonzero wave number. Maintaining an internal periodic drive changes the condition. For example, a moving external source constrained to keep $\omega_s=\omega_0$ can intersect the gapped branch above and continuously supply energy. A stable moving structure without external power or net radiation therefore requires a self-consistent relation among its frequency, profile and field feedback.

The threshold equality needs separate treatment. Equation (5.1.29) implies that $\omega_s<\omega_0/\gamma$ has no radiation intersection; $\omega_s=\omega_0/\gamma$ touches the spectrum at its minimum and is a critical case. The paragraph following that equation writes a non-strict inequality but calls it strictly below threshold. This page uses the strict subthreshold condition for the no-intersection conclusion and leaves the critical response to a source-projection and limiting-flux calculation.

The final manuscript also supplies an explicit two-dimensional response in (5.2.1)–(5.2.6). Take a core moving along $x$ with positive constant $b_x,b_y$, a frequency offset $\Omega_s$ relative to the carrier, and source amplitude $W_0$. With $\xi=x-vt$, $\eta=y$, set

The phase removes the first derivative in the co-moving equation; transverse rescaling makes the principal part isotropic. The delta-source Jacobian must be transformed too:

Define $R_b^2=\xi^2+(b_x/b_y)\eta^2$. The outgoing solution for $K_s^2>0$ and the decaying solution for $K_s^2=-\mu_s^2<0$ are respectively

Here $H_0^{(1)}$ is the outgoing Hankel function and $K_0$ the decaying modified Bessel function. The latter requires $\Omega_s<-m_0v^2/(2\hbar b_x)$. For its real radial amplitude, the current of the same effective equation satisfies $\mathbf j-\mathbf v|\Psi|^2=0$: it translates with the core without relative outward flow. Convolution with a suitably regular finite-core profile replaces the logarithmic point-source singularity. The outgoing solution is continuously driven; combining incoming and outgoing waves into a standing wave does not make the infinite-space energy finite. These calculated responses specify what the chosen equation allows, not fitted electron source parameters.

06From an accompanying response to a matter wave

The effective construction provides a definite calculation path: specify a complete configuration and action, calculate the translation residual and mode projections, find the response using initial and boundary conditions and source history, and check far-field flux and total energy. Establishing this response as a matter wave additionally requires phase locking between core and field within the same freely moving solution, together with the total momentum and action of the complete structure.

Mechanical momentum includes contributions from the core and its necessary near fields. Relating the phase wave vector to that total momentum and independently establishing single-excitation action matching are required before using $\mathbf P=\hbar\mathbf k$. Normalizing a field envelope does not itself generate Born statistics. The first task leads to matter waves, and the second to detection and statistics.

Interference of electrons and neutral composites places experimental constraints on an accompanying-field account: it must recover the correct phase, propagation and detection patterns while respecting actual apparatus boundaries. The quantum test matrix collects these requirements. Completing the source equation and co-moving solution is the task of OP-QM-001; connecting phase to momentum is the task of OP-QM-002.