2026-07-30
Diffraction of particles by a slit as a manifestation of the spatial phase of the operator J
Part 2. Passing through two slits
A single slit already creates a diffraction envelope. But two identical slits produce more than just the sum of two such patterns: frequent light and dark bands appear on the screen, which neither slit produces individually. Where does this additional structure come from if each particle is still registered by a single, indivisible point?
In the first part of the article, the spatial period of a massive particle was obtained directly from the internal phase of the operator.
\[ \tag{1} J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a}. \] For a moving state, this period is equal to
\[ \tag{2} \boxed{ \lambda_J = \frac{\lambda_0}{\gamma\beta} = \lambda_0\cot(\pi b) }, \] where \(\lambda_0\) is the intrinsic spatial scale of the internal period, \(\beta=v/c\), \(\gamma=1/\sqrt{1-\beta^2}\), and \(b=\arcsin(\beta)/\pi\) is the exponent of the external motion. Integrating the phases over the width of a single slit yielded a single-slit distribution \(\operatorname{sinc}^2\).
Now we need to take the next step: replace one continuous interval of admissible continuations with two separate intervals and determine what changes when they are opened together. It turns out that the new structure arises not within each slit, but between the two families of phases. Therefore, the main topic of the second part is not so much the passage through two holes, but the emergence of a mutual phase.
Introduction.
Why two slits are a new geometric problem
Why two slits are a new geometric problem
If you open the left slit and measure the distribution \(P_-(\theta)\), then close it, open the right one and measure \(P_+(\theta)\), then it is natural to expect that when both slits are opened simultaneously, the sum will be obtained.
\[ \tag{3} P_{12}(\theta) \stackrel{?}{=} P_-(\theta)+P_+(\theta). \] But experience shows otherwise. The double-slit pattern contains alternating maxima and minima, so in general
\[ \tag{4} \boxed{ P_{12}(\theta) \ne P_-(\theta)+P_+(\theta) }. \] The difference between the right and left sides of formula (4) is the essence of interference. If the new geometry claims to describe diffraction, it must not simply reproduce the known fringe formula, but also show from which element of the operator \(J\) the missing term arises.
In the first section, the slit was not considered as a device physically dividing the particle. It limited the set of permissible one-dimensional extensions of motion, and differences in the lengths of these extensions became differences in the internal phase. For two slits, this principle remains: each individual realization remains one-dimensional, and each registered object remains intact. However, the set of admissible continuations now splits into two spatially separated families.
Therefore, the central chain of the second part can be written as follows:
\[ \tag{5} \boxed{ \begin{aligned} \text{two slit regions} &\longrightarrow \mathcal A_-+\mathcal A_+, \\[2mm] \mathcal A_-+\mathcal A_+ &\longrightarrow 2\operatorname{Re} \left( \mathcal A_-\mathcal A_+^* \right) \longrightarrow \text{interference fringes}. \end{aligned} } \] As in the first part, the geometric addition of phases must be separated from the registration law.
\[ \tag{6} P\propto|\mathcal A|^2. \] The phase coefficients and their dependence on \(b\) will be obtained from geometry. The quadratic rule (6), and later the coherence loss coefficient, are for now accepted as additional physical conditions. This distinction will prevent the interpretation of the model from being presented as a complete measurement theory.
1. What has already been obtained for a single slit
Let a uniformly open slit of width \(d\) occupy an interval
\[ \tag{7} -\frac d2 \leq \xi \leq \frac d2. \] In the far field, the difference in the path from point \(\xi\) to the observation direction \(\theta\) is equal to
\[ \tag{8} \Delta\ell_\xi \simeq \xi\sin\theta. \] The spatial phase coefficient of a moving operator can conveniently be denoted by
\[ \tag{9} \boxed{ q(\theta) = \frac{2\pi}{\lambda_J}\sin\theta }. \] Then the phase difference of the extension passing through the coordinate \(\xi\) is equal to
\[ \tag{10} \Delta\phi_\xi = -q\xi. \] Integrating the phases over one slit yields the amplitude coefficient
\[ \tag{11} \mathcal A_0(\theta) = \em \int_{-d/2}^{d/2} e^{-iq\xi}\,d\xi = \em d\, \frac{\sin(qd/2)}{qd/2}. \] After normalization and applying formula (6), we obtain a single-slit envelope.
\[ \tag{12} \boxed{ \frac{P_0(\theta)}{P_0(0)} = \left[ \frac{ \sin \left( \dfrac{\pi d\sin\theta}{\lambda_J} \right) }{ \dfrac{\pi d\sin\theta}{\lambda_J} } \right]^2 }. \] In the second part, each of the two slits will create exactly this factor. A new picture will emerge from phase shifts associated not with the width \(d\), but with the distance between the slit centers.
2. Two slits as the union of two intervals
Let us denote the width of each slit by \(d\), and the distance between their centers by \(D\). The origin of the coordinateWe place the inat in the middle between the slits. Then the left open region has the form
\[ \tag{13} \Omega_- = \left[ -\frac D2-\frac d2,\, -\frac D2+\frac d2 \right], \] and the right one has the form
\[ \tag{14} \Omega_+ = \left[ \frac D2-\frac d2,\, \frac D2+\frac d2 \right]. \] The complete region through which the setup allows continued movement is the union of two disconnected intervals:
\[ \tag{15} \boxed{ \Omega = \Omega_-\cup\Omega_+ }. \] Formula (15) is the main geometric difference from a single slit. The operator \(J\) is not decomposed into two halves and does not lose its unit norm. Only the outer region of admissible coordinates \(\xi\), over which phase summation is performed, is split into two parts.
Each point of both slits corresponds to its own admissible one-dimensional continuation.
\[ \tag{16} \ell_\xi: \qquad \text{source} \longrightarrow \xi \longrightarrow \text{screen point}, \qquad \xi\in\Omega. \] The one-dimensionality of an individual motion therefore does not imply the uniqueness of a geometrically admissible continuation. It means that in each individual event, not a branched particle is realized, but a single, integral connection between the source and the detection point.
3. Spatial phase via the external motion index b
From the definition of the external motion index
\[ \tag{17} b = \frac{\arcsin\beta}{\pi} \] it follows
\[ \tag{18} \sin(\pi b)=\beta, \qquad \cos(\pi b)=\frac{1}{\gamma}. \] Therefore
\[ \tag{19} \gamma\beta = \tan(\pi b). \] Using formula (2), the spatial phase coefficient (9) can be completely expressed in terms of \(b\):
\[ \tag{20} \boxed{ q(\theta) = \frac{2\pi}{\lambda_0} \tan(\pi b)\sin\theta }. \] Thus, the velocity change is not introduced into the double-slit formula as an external correction. It is already contained in the phase step between adjacent extensions. The larger \(b\), the faster the phase changes during the transition along the slit plane and the more densely the fringes are spaced on the screen.
4. Left slit amplitude
The phase-sensitive contribution of the left slit is equal to the integral over the domain \(\Omega_-\):
\[ \tag{21} \mathcal A_-(\theta) = \em \int_{-D/2-d/2}^{-D/2+d/2} e^{-iq\xi}\,d\xi. \] To separate the internal structure of the slit from the position of its center, we introduce a local coordinate
\[ \tag{22} \xi = -\frac D2+\eta, \qquad -\frac d2 \leq \eta \leq \frac d2. \] Then the exponential factor is factored into two terms:
\[ \tag{23} e^{-iq\xi} = e^{\,iqD/2}e^{-iq\eta}. \] The first term is the same for the entire left slit and is determined by the position of its center. The second term varies across the slit width and creates the familiar single-slit envelope. After integration, we obtain
\[ \tag{24} \boxed{ \mathcal A_-(\theta) = \em d\, \frac{\sin(qd/2)}{qd/2} e^{\,iqD/2} }. \] Therefore, moving the slit from the origin to the point \(-D/2\) does not change the absolute value of its own amplitude, but adds a total phase shift of \(+qD/2\).
5. Amplitude of the Right Slit
Similarly, the contribution of the right slit is
\[ \tag{25} \mathcal A_+(\theta) = \em \int_{D/2-d/2}^{D/2+d/2} e^{-iq\xi}\,d\xi. \] Put
\[ \tag{26} \xi = \frac D2+\eta, \qquad -\frac d2 \leq \eta \leq \frac d2. \] Now
\[ \tag{27} e^{-iq\xi} = e^{-iqD/2}e^{-iq\eta}, \] and therefore
\[ \tag{28} \boxed{ \mathcal A_+(\theta) = \em d\, \frac{\sin(qd/2)}{qd/2} e^{-iqD/2} }. \] Both slits produce the same function of the width \(d\), but opposite phase shifts from the positions of their centers. It is this pair of factors \(e^{\,iqD/2}\) and \(e^{-iqD/2}\) that contains the future interference structure.
6. Geometric sum of two phase families
When both slits are open and the extensions through them remain indistinguishable, the integral over the combined region (15) is equal to the sum of the integrals:
\[ \tag{29} \begin{aligned} \mathcal A_{12}(\theta) &= \em \int_{\Omega_-\cup\Omega_+} e^{-iq\xi}\,d\xi \\[1mm] &= \mathcal A_-(\theta) + \mathcal A_+(\theta). \end{aligned} \] Substitute formulas (24) and (28):
\[ \tag{30} \mathcal A_{12} = \em d\, \frac{\sin(qd/2)}{qd/2} \left( e^{\,iqD/2} + e^{-iqD/2} \right). \] The sum of opposite complex phases is equal to twice the cosine:
\[ \tag{31} e^{\,iqD/2} + e^{-iqD/2} = 2\cos\left(\frac{qD}{2}\right). \] Therefore, the total amplitude of the two slits takes the form
\[ \tag{32} \boxed{ \mathcal A_{12}(\theta) = 2\em d\, \frac{\sin(qd/2)}{qd/2} \cos\left(\frac{qD}{2}\right) }. \] Formula (32) clearly separates the two scales of the setup. The width of each slit \(d\) determines the slowly varying function \(\operatorname{sinc}\), while the distance between the centers \(D\) creates a more frequent cosine factor.
7. Distribution of Registrations for Two Slits
At the center of the pattern \(q=0\), so the amplitude is
\[ \tag{33} \mathcal A_{12}(0) = 2\em d. \] Applying the quadratic rule (6) and normalizing the distribution to the central value, we obtain the main observed result:
\[ \tag{34} \boxed{ \frac{P_{12}(\theta)}{P_{12}(0)} = \left[ \frac{ \sin \left( \dfrac{\pi d\sin\theta}{\lambda_J} \right) }{ \dfrac{\pi d\sin\theta}{\lambda_J} } \right]^2 \cos^2 \left( \frac{\pi D\sin\theta}{\lambda_J} \right) }. \] The first factor is the same as the result of the first part:
\[ \tag{35} E_d(\theta) = \operatorname{sinc}^2 \left( \frac{\pi d\sin\theta}{\lambda_J} \right). \] This is the general single-slit envelope, within which a double-slit structure can be located. The second factor
\[ \tag{36} I_D(\theta) = \cos^2 \left( \frac{\pi D\sin\theta}{\lambda_J} \right) \] is the interference between the centers of the two slits. Therefore, formula (34) can be written briefly as: \[ \tag{37} \boxed{ \frac{P_{12}(\theta)}{P_{12}(0)} = \underbrace{E_d(\theta)}_{\text{envelope of one slit}} \, \underbrace{I_D(\theta)}_{\text{mutual phase of two slits}} }. \]
One slit already forms an angular region in which noticeable detection is possible. The second slit does not cancel this envelope, but marks it with narrower bands of mutual amplification and suppression.
8. Central formula directly through b
Substitute the geometric period into formula (34)
\[ \tag{38} \frac{1}{\lambda_J} = \frac{\tan(\pi b)}{\lambda_0}. \] Then the entire distribution is expressed in terms of the operator's external motion index:
\[ \tag{39} \boxed{ \begin{aligned} \frac{P_{12}(\theta)}{P_{12}(0)} &= \left[ \frac{ \sin \left( \dfrac{\pi d}{\lambda_0} \tan(\pi b)\sin\theta \right) }{ \dfrac{\pi d}{\lambda_0} \tan(\pi b)\sin\theta } \right]^2 \\[1mm] &\quad\times \cos^2 \left( \frac{\pi D}{\lambda_0} \tan(\pi b)\sin\theta \right). \end{aligned} } \] Formula (39) is the main geometric result of the second part. In it, the width and position of the fringes are determined not by a separately introduced wave, but by the spatial manifestation of the internal periodicity of the moving \(J\). The parameters \(d\), \(D\), and \(\theta\) belong to the external geometry of the setup, and the quantity \(\lambda_J=\lambda_0\cot(\pi b)\) defines the phase scale of the state itself.
9. Where does the cross term come from?
To see the difference between two simultaneously open slits and two separate experiments, we expand the square of the sum of the amplitudes:
\[ \tag{40} P_{12} \propto \left| \mathcal A_-+\mathcal A_+ \right|^2. \] Multiplying by the complex conjugate gives
\[ \tag{41} \begin{aligned} \left| \mathcal A_-+\mathcal A_+ \right|^2 &= \left( \mathcal A_-+\mathcal A_+ \right) \left( \mathcal A_-^*+\mathcal A_+^* \right) \\[1mm] &= |\mathcal A_-|^2 + |\mathcal A_+|^2 + \mathcal A_-\mathcal A_+^* + \mathcal A_-^*\mathcal A_+. \end{aligned} \] The last two terms are conjugate, therefore
\[ \tag{42} \boxed{ P_{12} = P_-+P_+ + 2\operatorname{Re} \left( \mathcal A_-\mathcal A_+^* \right) }. \] It is the last term that is missing in the simple sum \(P_-+P_+\). Substituting the phase factors from formulas (24) and (28), we obtain
\[ \tag{43} \mathcal A_-\mathcal A_+^* \propto e^{\,iqD}, \] and therefore, the cross term changes as
\[ \tag{44} 2\operatorname{Re} \left( \mathcal A_-\mathcal A_+^* \right) \propto 2\cos(qD). \] Consequently, interference is not an additional force between the slits and does not require physical fragmentation of the particle. It appears algebraically upon quadratic registration of the sum of two mutually phase-preserving families.
10. Maxima, Minima, and Missing Orders
Interference maxima occur when
\[ \tag{45} \frac{\pi D\sin\theta_m}{\lambda_J} = \pi m, \qquad m=0,\pm1,\pm2,\ldots \] or
\[ \tag{46} \boxed{ D\sin\theta_m = m\lambda_J }. \] The minima of the interference factor are located between adjacent maxima:
\[ \tag{47} \boxed{ D\sin\theta_m^{\min} = \left( m+\frac12 \right)\lambda_J }. \] The single-slit envelope is invertedvanishes at
\[ \tag{48} \boxed{ d\sin\theta_n = n\lambda_J, \qquad n=\pm1,\pm2,\ldots }. \] If the maximum of double-slit interference coincides with the zero of the envelope, it is not observed. From formulas (46) and (48), the condition for such a coincidence is
\[ \tag{49} \frac{m}{D} = \frac{n}{d}, \] from where
\[ \tag{50} \boxed{ m = n\frac{D}{d} }. \] Such maxima are called missing orders. Their absence particularly clearly demonstrates that the interference fringes and the single-slit envelope are not two independent patterns: their product is observed.
11. Distance between Fringes on the Screen
Let the screen be located at a distance \(L\) from the plane of the slits. At small angles
\[ \tag{51} \sin\theta \simeq \theta \simeq \frac{y}{L}. \] From formula (46), the positions of the interference maxima are equal
\[ \tag{52} y_m \simeq \frac{mL\lambda_J}{D}. \] Therefore, the distance between adjacent bands is
\[ \tag{53} \boxed{ \Delta y_{\mathrm{fr}} \simeq \frac{L\lambda_J}{D} = \frac{L\lambda_0}{D\gamma\beta} = \frac{L\lambda_0}{D}\cot(\pi b) }. \] The width of the central single-slit envelope between the first zeros of \(n=-1\) and \(n=+1\) is
\[ \tag{54} \boxed{ \Delta y_{\mathrm{env}} \simeq \frac{2L\lambda_J}{d} }. \] The ratio of the width of the central envelope to the distance between the fringes does not depend on the particle velocity:
\[ \tag{55} \boxed{ \frac{\Delta y_{\mathrm{env}}}{\Delta y_{\mathrm{fr}}} \simeq \frac{2D}{d} }. \] Therefore, increasing the velocity compresses the entire pattern as a whole, but the approximate number of interference gaps within the central envelope is determined only by the ratio of the geometric dimensions \(D/d\).
12. How b and velocity change the pattern
For a massive particle
\[ \tag{56} \lambda_J = \lambda_0 \frac{\sqrt{1-\beta^2}}{\beta}. \] As the speed increases, \(\gamma\beta=\tan(\pi b)\) increases, so the following decrease simultaneously:
1. spatial period \(\lambda_J\);
2. distance between adjacent fringes \(\Delta y_{\mathrm{fr}}\);
3. width of the single-slit envelope \(\Delta y_{\mathrm{env}}\);
4. angles of all maxima and minima of the same order.
2. distance between adjacent fringes \(\Delta y_{\mathrm{fr}}\);
3. width of the single-slit envelope \(\Delta y_{\mathrm{env}}\);
4. angles of all maxima and minima of the same order.
In the nonrelativistic limit
\[ \tag{57} \beta\ll1, \qquad \gamma\simeq1, \] therefore
\[ \tag{58} \lambda_J \simeq \frac{\lambda_0}{\beta}, \qquad \Delta y_{\mathrm{fr}} \simeq \frac{L\lambda_0}{D\beta}. \] A small increase in velocity reduces the scale of the picture approximately inversely. In the relativistic region, additional compression is created by increasing \(\gamma\). Geometrically, this means that with increasing \(b\), the phase changes more rapidly when moving from one valid slit coordinate to another.
13. Why P12 is not equal to P- + P+
The distributions \(P_-\) and \(P_+\) are obtained in two different setup configurations: in the first case, the right slit is closed, and in the second, the left slit is closed. Their sum describes the statistical mixing of the results of two separate experiments.
The distribution \(P_{12}\) refers to the third configuration, in which both slits are open simultaneously. In this configuration, the phase coefficients are added before applying the quadratic rule:
\[ \tag{59} \boxed{ \begin{aligned} \text{two separate configurations:}\quad& P_-+P_+, \\[1mm] \text{one configuration with two slits:}\quad& \left| \mathcal A_-+\mathcal A_+ \right|^2. \end{aligned} } \] These operations do not commute. Adding squares loses mutual phase, while squared sum preserves it:
\[ \tag{60} |\mathcal A_-|^2+|\mathcal A_+|^2 \ne \left| \mathcal A_-+\mathcal A_+ \right|^2. \] Therefore, opening the second slit does more than simply add new transmission points. It creates the opportunity to compare the phases of two spatially separated families. It is this opportunity that changes the entire distribution, including regions in which the detection probability decreases.
14. What happens when one slit is closed
If the right slit is closed, the region \(\Omega_+\) is excluded from the integration:
\[ \tag{61} \Omega \longrightarrow \Omega_-, \qquad \mathcal A_{12} \longrightarrow \mathcal A_-. \] Then both \(P_+\) and the cross term disappear in formula (42):
\[ \tag{62} P_{12} \longrightarrow P_-. \] The phase factor \(e^{\,iqD/2}\) in the amplitude of the single remaining slit has unity modulus, so it does not create fringes by itself. After taking the square, only the single-slit envelope remains:
\[ \tag{63} \left| \mathcal A_- \right|^2 \propto \left[ \frac{\sin(qd/2)}{qd/2} \right]^2. \] The same thing happens when the left slit is closed. Thus, the interference factor only exists when two phase-related contributions are simultaneously present.
15. Individual Registrations and Pattern Accumulation
Formula (34) describes not the distribution of parts of a single object across the screen, but the statistics of a large number of complete events. Each time the detector is triggered, one local point is recorded. \[ \tag{64} \mathcal R_n = \left( t_n,y_n \right), \]
where \(t_n\) is the moment, and \(y_n\) is the coordinate of the \(n\)th registration. Each such registration corresponds to an integer state with unit norm:
\[ \tag{65} J_n\overline J_n = 1. \] During the initial events, the pattern is almost invisible. As their number increases, the empirical density
\[ \tag{66} \rho_N(y) = \frac{1}{N} \sum_{n=1}^{N} \delta \left( y-y_n \right) \] approaches the distribution defined by formula (34). The locality of an individual point and the extent of the resulting pattern therefore relate to different levels of description: the former characterizes a single interaction with the detector, while the latter characterizes the frequency of such interactions in different coordinates.
In the proposed interpretation, it is not the physical parts of the electron that are added up before registration, but the phase coefficients of the admissible extensions. What is recorded on the screen is not the coefficient or the idempotent component, but the entire operator \(J\).
16. Path Information and Mutual Phase Loss
Until now, it has been assumed that the continuations through the left and right slits are indistinguishable. If the setup associates them with different states of the measuring device or the environment, the phase coupling weakens. Phenomenologically, this can be described by the complex mutual coherence coefficient \(\mu\):
\[ \tag{67} \boxed{ P = P_-+P_+ + 2\operatorname{Re} \left( \mu \mathcal A_-\mathcal A_+^* \right), \qquad 0\leq|\mu|\leq1 }. \] The value \(|\mu|=1\) corresponds to complete preservation of the mutual phase:
\[ \tag{68} |\mu|=1 \quad\Longrightarrow\quad \text{full interference pattern}. \] Intermediate values reduce the contrast of the fringes:
\[ \tag{69} 0 < |\mu| < 1 \quad\Longrightarrow\quad \text{partially blurred interference}. \] If the phase coupling is completely lost, the cross term disappears:
\[ \tag{70} \boxed{ |\mu|=0 \quad\Longrightarrow\quad P=P_-+P_+ }. \] It is important that the disappearance of the fringes cannot be reduced solely to the mechanical impact of the particle without further analysis. What's important is that the two families of continuations become physically distinguishable due to their correlation with different detector states.
If we denote these states by \(|D_-\rangle\) and \(|D_+\rangle\), then the coherence coefficient is standardly expressed by their overlap:
\[ \tag{71} \mu = \langle D_+|D_-\rangle. \] For identical detector states, information about the gap is absent and \(|\mu|=1\). For orthogonal states, the gap can, in principle, be determined and \(\mu=0\). Formulas (67) and (71) are used here as a general theory of coherence. Their direct origin from the interaction of the operator \(J\) with the detector operator has yet to be determined within the model.
17. Comparison with Bach's Controlled Experiment
A particularly clear comparison is provided by the controlled double-slit experiment of Roger Bach, Damian Pope, Si-Hwang Liu, and Herman Batelaan, published in 2013 [1]. The authors placed a movable mask in front of a real double slit. Its position allowed reversible opening of only the first slit, only the second slit, both slits, or blocking transmission.
The experiment used electrons with an energy of \(600\, \text{eV}\), for which the de Broglie wavelength was approximately \(50\, \text{pm}\). Each slit had a width of \(62\, \text{nm}\), and the distance between their centers was \(272\, \text{nm}\).
| Mask position | Mathematical notation | Observed pattern | Geometric meaning |
|---|---|---|---|
| Only slit 1 is open | \(P_-=|\mathcal A_-|^2\) | Single-slit diffraction | One interval of admissible extensions |
| Only slit 2 is open | \(P_+=|\mathcal A_+|^2\) | Same Single-slit diffraction | Second interval without reciprocal term |
| Both slits are open | \(P_{12}=|\mathcal A_-+\mathcal A_+|^2\) | InterferenceBands within the envelope | Two phase-coupled families |
At reduced intensity, registrations occurred at approximately a frequency of \(1\, \text{s}^{-1}\). The authors demonstrated sequential accumulation of the pattern for 2, 7, 209, 1004, and 6235 electrons. Initially, individual, seemingly random points are visible; after accumulation of a large number of events, stable bands appear.
This experiment almost literally realizes the difference between \(P_-\), \(P_+\), and \(P_{12}\), expressed by formulas (42) and (59). It confirms two observed facts that must be taken into account in the interpretation: each detection is local, but the distribution with two open slits is not equal to the sum of two separately measured single-slit distributions.
Experiments with path detectors further show that increasing the correlation between the selected path and the detector reduces the visibility of interference. This corresponds to a transition from \(|\mu|\simeq1\) to smaller values in formula (67). However, such experiments confirm the law of coherence loss itself, not the uniqueness of the geometric interpretation proposed here.
18. What is derived geometrically, and what is adopted additionally?
The results of the second part must be divided into several levels.
The first level is the geometry of regions and phases. Two slits are represented by the union of two intervals \(\Omega_-\cup\Omega_+\). The spatial phase of each extension is determined by the coefficient.
\[ \tag{72} q(\theta) = \frac{2\pi}{\lambda_0} \tan(\pi b)\sin\theta. \] Integration over these intervals results in the sum of the amplitudes (32), which contains the single-slot function and the reciprocal cosine factor.
The second level is the accepted registration rule. The transition from the sum of the phase coefficients to the number of events is performed according to the law \(P\propto|\mathcal A|^2\). It is the square of the modulus that creates the cross term. In this paper, this rule is not derived from the \(J\) norm.
The third level is the description of the loss of coherence. The coefficient \(\mu\) allows us to correctly describe the gradual disappearance of the fringes as path information appears. However, the interaction between the particle, the slit, and the detector has not yet been directly constructed as a transformation of their operators \(J\).
The fourth level is interpretation. It is proposed that the slits separate not a material particle, but a set of admissible continuations of its phase; an individual motion remains one-dimensional, and an individual registration remains holistic. This interpretation is consistent with the observed picture, but the experiments themselves do not clearly distinguish it from other interpretations of quantum mechanics.
Therefore, the main result of the article is not a statement about a pre-selected slit, but a more cautious position:
\[ \tag{73} \boxed{ \text{the one-dimensionality of an individual realization does not imply the uniqueness of an admissible continuation} }. \] 19. Conclusions
In the first part, a single slit limited a continuous family of admissible extensions and created a diffraction envelope. In the second part, the integration domain split into two separate intervals. Each interval retained the same single-slit function, but the position of its center added an opposite phase factor.
Adding these factors resulted in the amplitude
\[ \tag{74} \mathcal A_{12}(\theta) = 2\em d\, \frac{\sin(qd/2)}{qd/2} \cos\left(\frac{qD}{2}\right), \] and the quadratic registration rule—to the product of the single-slit envelope and the double-slit interference multiplier:
\[ \tag{75} \boxed{ \frac{P_{12}(\theta)}{P_{12}(0)} = \operatorname{sinc}^2 \left( \frac{\pi d\sin\theta}{\lambda_J} \right) \cos^2 \left( \frac{\pi D\sin\theta}{\lambda_J} \right) }. \] Since \(\lambda_J=\lambda_0\cot(\pi b)\), the position of all the fringes is directly related to the external motion index. The distance between them is
\[ \tag{76} \boxed{ \Delta y_{\mathrm{fr}} \simeq \frac{L\lambda_0}{D}\cot(\pi b) }. \] When the slits are opened together, a cross term \(2\operatorname{Re}(\mathcal A_-\mathcal A_+^*)\) arises. When one slit is closed, it disappears along with the corresponding amplitude. When path information is obtained, it is suppressed by the loss of mutual coherence, even if both geometric slits remain open.
In the proposed picture, the particle is not required to physically divide and simultaneously move along two spatial trajectories. The two slits define two families of admissible one-dimensional extensions of a single state. The difference between their phases determines the statistical distribution, while each individual interaction with the screen remains a local registration of the whole \(J\).
The overall idea of the two parts can be expressed as follows: one slit detects the spatial phase of each family of extensions; two slits detect the mutual phase between the families. The particle remains intact, but the distribution of its registration sites preserves the geometry of both alternatives.
Materials used
- R. Bach, D. Pope, S.-H. Liou, H. Batelaan. Controlled double-slit electron diffraction, New Journal of Physics 15, 033018 (2013).
- E. Buks, R. Schuster, M. Heiblum, D. Mahalu, V. Umansky. Dephasing in electron interference by a which-path detector, Nature 391, 871–874 (1998).