How Can Continuous Fields Yield Discrete Detection Statistics?

01How the detector joins the matter–field process

An interference field has continuously varying amplitude and phase, while a screen leaves localized records. A physical system lies between those descriptions: matter in the detector is driven, absorbs energy and converts that energy into countable events. The detector's absorption response is a necessary part of connecting the continuous field distribution to discrete records.

In ASD, the localized matter core, extended accompanying field and detector exchange energy and momentum together. This page first develops the linear material response already established in the manuscript, then specifies the conditions under which it yields a squared-amplitude position distribution. $\Psi(\mathbf r,t)$ remains the slowly varying complex envelope of a selected propagation channel, and $|\Psi|^2$ its squared modulus. It is not initially defined as a particle's landing probability.

For a small detector element at $\mathbf r_D$, take the real driving field to be $\Phi_{\rm real}=\operatorname{Re}[\Psi e^{-i\omega_0t}]$. Here $\omega_0$ is the carrier frequency of a nearly monochromatic signal; the envelope must vary sufficiently slowly during a cycle average. This local model makes one material response coordinate explicit so that absorption can be calculated from work.

02From local material response to absorbed power

Let $q_D$ be the response coordinate, $M_D$ its mass parameter, $\gamma_D>0$ its damping rate and $\omega_D$ its natural frequency. The coupling $g_D\Phi_{\rm real}$ has units of force, while $\xi_D$ is a fluctuating material force. The manuscript uses the following driven damped model with the complex-amplitude convention $e^{-i\omega t}$:

$\chi_D(\omega)$ is the force-to-displacement response function. For harmonic driving at a specified frequency, the displacement amplitude is the force amplitude multiplied by $\chi_D$. Its imaginary part describes the absorptive response. Positive damping gives positive absorption at positive frequency. This model applies within the linear response range of the selected material coordinate; its parameters must be determined for the actual detector.

The instantaneous power delivered by the field is $g_D\Phi_{\rm real}\dot q_D$. Substituting the harmonic response, averaging over a carrier cycle and subtracting the background power due to independent noise gives the first relation below:

The mean power $\overline P_D$ has units of energy per time. Its coefficient contains the coupling, driving frequency and absorptive response, so the same incident envelope need not produce the same absorption in different materials. The squared modulus arises here from multiplying the linear drive by the linear response and taking the cycle average.

The second relation takes an additional step. $R_D$ is the long-term mean rate of recorded events and $\varepsilon_D$ is the fixed effective absorbed energy required for each record. Converting energy rate to event rate through $R_D\simeq\overline P_D/\varepsilon_D$ requires a nearly ideal integrating trigger with negligible leakage, dead time and saturation during accumulation. These are extra detector-model conditions. A noise-dominated, avalanche-gain or strongly thresholded device does not satisfy them merely because it has a threshold.

03From event rate to a conditional screen distribution

A screen contains many responding elements. Their density, local absorption coefficient, effective energy per record and efficiency can be collected into $\eta_D(\mathbf r,t)$ to define an event-rate density $\mathcal R$ per volume and time. A real screen has a finite absorbing thickness, so position counts must also be integrated through that thickness and accumulated over the selected time window:

Here $Y$ denotes a position on the screen, $z$ the thickness coordinate, $dA$ a screen area element and $\mathcal T$ the recording window. $\mathcal R_A$ is the event rate per area. The denominator is the total expected number of records across the active screen in that window and must be nonzero. Dividing the local count by this total yields the position distribution conditioned on detection, $p(Y\mid\mathrm{det})$.

This order follows the actual measurement: absorption, accumulation by position and time, then normalization. Only with uniform efficiency and the appropriate temporal structure and incident-velocity distribution does the result reduce to a normalized squared modulus on one cross-section. Measuring position at a fixed time, recording arrival times and accumulating a screen pattern over a long interval are distinct protocols.

For the adopted source-free phase-flow model in a region with nonzero amplitude, $j_n=v_n|\Psi|^2$ with $v_n=n_iB_{ij}\partial_jS/m_0$. A steady screen sampled by incoming flux therefore has $p(Y\mid\mathrm{det})\propto\eta_D(Y)v_n(Y)|\Psi(Y)|^2$, before normalization. Uniform efficiency alone does not remove the velocity weight: $v_n$ must also be constant across the compared positions, or cancel under the stated time-integration protocol, to obtain a normalized squared modulus. This makes explicit the extra condition needed in the paragraph after final equation (5.3.21). An absorbing boundary can also reflect or reshape a packet, so the source history and acceptance conditions remain part of the calculation.

04How material nonuniformity enters the counts

Even after the field is determined, spatially varying efficiency changes the raw screen pattern. In the conditional model this first appears as $p_{\rm obs}\propto\eta_D|\Psi|^2$. Saturation under stronger driving, crosstalk between elements and finite spatial resolution can be represented through a response expansion and a spatial kernel:

Equation (5.3.22) defines the selected quartic-plus-nonlocal response functional. Its equals sign applies within this constitutive model, rather than to every detector at arbitrary intensity.

$\eta_1$ and $\eta_2$ are response coefficients at the indicated orders. $K_D(\mathbf r,\mathbf r')$ describes how input at one position affects records at another; the integration variable $\mathbf r'$ must be distinguished from the output position $\mathbf r$. The coefficients and kernel require material and instrument calibration. A truncated expansion applies only within its response range and must keep the physical event rate nonnegative.

A raw pattern departing from an uncalibrated squared-amplitude curve therefore first raises a question about the detector model. Testing a fundamental statistical law requires comparison of data with predictions that include efficiency, resolution and the measurement protocol. Detector nonlinearity by itself is not a departure from the Born rule.

05What phase flow can establish

Alongside detector response, the manuscript examines a possible connection between envelope phase and core motion. Decompose a source-free envelope into amplitude and phase. In regions with $n>0$, the imaginary part of its effective equation gives a continuity equation:

Here $n=|\Psi|^2$ is an envelope weight that need not yet be normalized, $S$ a phase function with units of action, $m_0$ the effective mass parameter and $B_{ij}$ the directional response tensor of the effective channel. Components are written in fixed Cartesian coordinates. $v_i$ is the flow velocity identified from the envelope continuity equation. Equating the velocity of the localized core centre $\mathbf X(t)$ to this flow is the explicitly marked candidate motion law.

If all cores share the same given envelope and follow that law, their position probability density $\rho_c$ obeys the same continuity equation. An initial distribution $\rho_c=n/\int n\,d^3r$ is then preserved with source-free evolution and compatible boundaries. This preservation is called equivariance: it transports an already specified distribution and does not explain convergence from arbitrary initial conditions. If each core also drives its own field history self-consistently, the statistical object must instead include the joint core–field distribution.

The real part of the same decomposition contains the amplitude-curvature term

This is a mathematical rewriting of the effective envelope equation. A separate local overlap model uses $V_{\rm ov}(\mathbf X)=-\alpha_c\int w_c(\mathbf r-\mathbf X)U(\mathbf r)\,d^3r$, where $w_c$ is a normalized core weight and $U$ the field energy density; under this convention $\alpha_c$ has units of volume. The two potentials are generally different. Deflecting a core through an overlap force and prescribing its velocity through phase flow are different candidate connections, not two trajectory mechanisms already proved by one action.

On closed loops avoiding nodes, a single-valued complex envelope also requires $\oint\nabla S\cdot d\boldsymbol\ell=2\pi N\hbar$, with integer $N$. This follows from the single-valued scalar envelope used here. Its microscopic mapping to a particle's topological core still requires a specific calculation.

06From a conditional result to individual records

Along the linear-response route, the manuscript has connected field amplitude, mean absorbed power, mean event rate and conditional position distribution. It provides one conditional origin of a Born-type position distribution. The probability amplitude here is a normalized effective envelope used to predict statistics after preparation, propagation and detection protocols are specified. Its range is set jointly by the energy conversion and detection assumptions above.

The remaining closure concerns why one matter excitation triggers exactly one recording element, how topological charge and energy are transferred locally, how competing elements select a record, and why more general observables obey their corresponding statistics. A localized topological core offers a candidate mechanism for concentrated exchange. One record per excitation must be checked through joint equations for the core, field and material. The core and extended accompanying field belong to one complete structure; each energy contribution must be counted only once.

These tasks are collected in OP-QM-004: Born statistics. The quantum phenomenological requirements matrix further distinguishes squared-amplitude absorption, accumulated position patterns and individual-particle event statistics, preserving the separate requirements to be tested.