2026-07-30
Diffraction of particles by a slit as a manifestation of the spatial phase of the operator J
Part 1. Passage through one slit
A particle passes through the setup one event at a time, leaving a single local point on the screen. But after many such events, the points align into an extended diffraction pattern [1]. How does an indivisible object obtain information about the entire width of the slit, and why does this pattern narrow as the speed of a massive particle increases?
Classical quantum mechanics answers this question using the wave function and the de Broglie wavelength. This paper poses a more specific but fundamental problem: to test whether the same spatial scale can emerge directly from the geometry of the state operator.
\[ \tag{1} J(a,b)=\j^a(-\j)^b, \] solely for geometric reasons. First, the spatial phase of the moving operator will be obtained, then the diffraction distribution for a single slit, and only then will the result be compared with the standard wavelength and real experiments.
Introduction.
Why the solution must be obtained solely from geometry.
Why the solution must be obtained solely from geometry.
For the new model, agreement with a known formula in itself proves nothing. If we start our reasoning with the de Broglie wavelength,
\[ \tag{2} \lambda_{\mathrm{dB}}=\frac{h}{p}, \] and then rewrite it in the notations \(J\), \(a\), and \(b\), then the geometric construction will not explain diffraction. It will only give a different name to an already known result. Such a coincidence will be a consequence of pre-installed physics, not an independent consequence of the model.
Therefore, the reverse approach has been chosen here. The initial principles are only two geometric motions of the operator, its unit norm, the relationship of the external exponent \(b\) with velocity, and the relativistic geometry of spacetime. From these assumptions, it is necessary to obtain a quantity possessing all the properties of the observed wavelength: it must define the phase difference between the paths, decrease with increasing velocity of the massive particle, and determine the position of the minima on the screen.
This is precisely the interest of the problem. If the quantity \(h/p\) appears at the end, although it was not used at the beginning, then the geometry will not simply repeat a known formula. It will show what internal motion of the particle is converted into a measurable spatial period. Then the wavelength will not be a separate postulate, but a geometric consequence of the transfer of the internal periodicity of a moving object between different points in space.
However, it is necessary to immediately define the limits of such a conclusion. Geometry can provide the phase and the law of its addition, but the transition from the resulting amplitude to the number of registered events requires a rule.
\[ \tag{3} P\propto |\mathcal A|^2. \] In this article, this rule is adopted as the registration law. In the future, it should be related to the geometry of the particle-detector interaction. This distinction is important: the spatial period will be derived from the model, while the quadratic probability rule remains an additional physical condition for now.
1. Two Motions within a Single Operator
The model is based on two mutually complementary idempotents
\[ \tag{4} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Each idempotent distinguishes its own complex plane:
\[ \tag{5} \left\{\ep,i\ep\right\}, \qquad \left\{\em,i\em\right\}. \] The hyperbolic unit and its inverse unit have decompositions
\[ \tag{6} \j=\ep-\em, \qquad -\j=-\ep+\em. \] Their continuous powers act as two independent selective rotations:
\[ \tag{7} \j^a=\ep+\em e^{i\pi a}, \qquad (-\j)^b=\ep e^{i\pi b}+\em. \] Multiplication of these operators is particularly simple due to the orthogonality of \(\ep\em=0\):
\[ \tag{8} \boxed{ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \] The exponent \(b\) controls the phase of the \((\ep,i\ep)\) plane and describes the external motion. The exponent \(a\) controls the phase of the \((\em,i\em)\) plane and describes the internal periodic state. These components are independent but belong to the same operator.
Conjugation changes the signs of both phases. Therefore, the norm of the complete state remains unity:
\[ \tag{9} \begin{aligned} J\overline J &= \left( \ep e^{i\pi b}+\em e^{i\pi a} \right) \left( \ep e^{-i\pi b}+\em e^{-i\pi a} \right) \\[1mm] &= \ep+\em =1. \end{aligned} \] Thus, an external influence can change the phase distribution between two planes, but should not destroy the integrity of the state. An individual registered particle always corresponds to the whole \(J\), and not to one of its parts.
2. External velocity and geometric exponent b
We define external motion as a dimensionless velocity
\[ \tag{10} \beta=\frac{v}{c}, \qquad |\beta| < 1 \] for a massive particle. By the definition of the model, the exponent \(b\) is related to the velocity by the relation
\[ \tag{11} b=\frac{\arcsin\beta}{\pi}. \] Therefore, two orthogonal projections of the external state are equal
\[ \tag{12} \sin(\pi b)=\beta, \qquad \cos(\pi b)=\sqrt{1-\beta^2}=\frac{1}{\gamma}, \] where
\[ \tag{13} \gamma=\frac{1}{\sqrt{1-\beta^2}} \] is the external Lorentz factor of the particle. Formula (12) shows that \(\beta\) and \(1/\gamma\) are not two independent corrections. They represent the sine and cosine projections of the same angle \(\pi b\):
\[ \tag{14} \boxed{ \beta^2+\frac{1}{\gamma^2}=1 }. \] This is why it is impossible to separately "add" the factor \(1/\gamma\) to the diffraction result without first determining which spatiotemporal phase the experiment is comparing. The correct combination of \(\beta\) and \(\gamma\) must arise from the geometry of the comparison of the two events.
3. Internal Periodicity and Intrinsic Phase
In the rest frame of a massive particle, the external direction is not distinguished, but the internal process continues to change. We write its exponent in terms of the proper time \(\tau\):
\[ \tag{15} a=\varpi_0\tau, \qquad \omega_0=\pi\varpi_0. \] Then the internal component of the operator has the form
\[ \tag{16} J_{\mathrm{int}}(\tau) = \em e^{i\pi a} = \em e^{i\omega_0\tau}. \] The quantity \(\omega_0\) here is the natural angular frequency of the stable state. For the electron in the previously considered mass map, it corresponds to the Compton angular frequency at rest. At this stage, its connection with mass is not used: the existence of an internal period is sufficient for geometric derivation.
A complete rotation of the internal phase occurs when
\[ \tag{17} \Delta\phi_0 = \omega_0\Delta\tau = 2\pi. \] The corresponding time period in the rest frame is
\[ \tag{18} T_0=\frac{2\pi}{\omega_0} = \frac{2}{\varpi_0}. \] For now, this is only a time period. To obtain diffraction, it is necessary to understand how the same internal phase is compared at different points in the laboratory space.
4. Why the substitution t = l/v gives an incomplete result
At first glance, the time of motion can be replaced by the time it takes to travel the path \(\ell\):
\[ \tag{19} t=\frac{\ell}{v}. \] Then the phase along one path formally takes the form
\[ \tag{20} \phi_{\mathrm{path}} = \omega_0t = \frac{\omega_0}{v}\ell, \] and the corresponding spatial period is equal to
\[ \tag{21} \lambda_{\mathrm{path}} = \frac{2\pi v}{\omega_0}. \] But this operation answers another question: how much internal phase will a particle accumulate while moving along one pre-selected path. A diffraction experiment compares not successive points of a single trajectory, but the phases of several admissible paths arriving at the same observation front.
To observe fringes, it is necessary to compare states at the same laboratory time \(t\), but at different spatial coordinates or different path lengths. Therefore, the time portion of the common phase must be identical and canceled, while the spatial portion must remain. Simple substitution (19) does not perform such a comparison and therefore loses the required combination \(\gamma\beta\).
In other words, expression (21) describes the phase along the motion, but not the wavefront across the family of possible motions. Diffraction requires the full spatiotemporal geometry of the phase.
5. Spatial Phase of a Moving Operator
To extend the proper phase from the particle's worldline to laboratory spacetime, we denote the time coordinate of the particle's rest frame by \(\tau(t,x)\). On the worldline itself, this coordinate coincides with the proper time. For an arbitrary laboratory event, it is related to the \(t\) and \(x\) transformations.
\[ \tag{22} \boxed{ \tau = \gamma \left( t-\frac{vx}{c^2} \right) }. \] This expression is important not only as a formula of special relativity. It demonstrates the geometric tilt of the planes of simultaneity: events that are simultaneous in the laboratory frame have different values of the time coordinate of the rest frame. It is this difference that transforms the original temporal periodicity into a spatial phase. Here \(\tau(t,x)\) does not denote the proper time of several simultaneously moving particles; it is a single phase continuation of the internal process to different spacetime events.
Substitute (22) into the proper phase \(\phi_0=\omega_0\tau\):
\[ \tag{23} \begin{aligned} \phi(t,x) &= \omega_0\gamma \left( t-\frac{vx}{c^2} \right) \\[1mm] &= \omega_0\gamma t - \omega_0\gamma\frac{v}{c^2}x. \end{aligned} \] Now the phase naturally has two parts:
\[ \tag{24} \phi(t,x) = \Omega t-k_Jx, \] where
\[ \tag{25} \Omega=\gamma\omega_0, \qquad k_J=\frac{\omega_0}{c}\gamma\beta. \] The first quantity specifies the frequency of the phase change at a fixed laboratory coordinate. The second specifies the change in the same phase when passing between different points at the same instant of laboratory time.
Let two admissible paths differ in length by \(\Delta\ell\), and let their phases be compared at the same \(t\). The time terms in formula (23) are the same and cancel out:
\[ \tag{26} \boxed{ \Delta\phi = -\frac{\omega_0}{c} \gamma\beta\,\Delta\ell }. \] It is formula (26), and not the substitution \(t=\ell/v\), that determines the spatial phase difference on which the diffraction pattern depends.
6. Geometric Spatial Period
We define the spatial period \(\lambda_J\) as the change in path length that creates a complete phase revolution:
\[ \tag{27} |\,\Delta\phi\,| = \frac{2\pi}{\lambda_J}\Delta\ell. \] Comparing formulas (26) and (27) yields
\[ \tag{28} \boxed{ \lambda_J = \frac{2\pi c}{\omega_0\gamma\beta} }. \] We introduce the proper spatial scale of the internal period
\[ \tag{29} \lambda_0 = cT_0 = \frac{2\pi c}{\omega_0} = \frac{2c}{\varpi_0}. \] Then the central result of the geometric derivation takes a compact form:
\[ \tag{30} \boxed{ \lambda_J(\beta) = \frac{\lambda_0}{\gamma\beta} = \lambda_0 \frac{\sqrt{1-\beta^2}}{\beta} }. \] The resulting value has the required properties. As the velocity increases, the factor \(\gamma\beta\) increases, so the spatial period decreases. At low velocities \(\gamma\approx1\), and the period is approximately inversely proportional to the velocity. In the relativistic region, an additional reduction in scale appears due to the geometry of time.
It is important that formulas (22)–(30) did not use momentum, Planck's constant, or the de Broglie relation. The spatial period arose as a transformed natural periodicity of the internal component \(J\).
7. The same result directly through the exponent b
The connection with the state operator becomes especially clear if we exclude \(\beta\) and \(\gamma\) in favor of the geometric angle \(\pi b\). From formula (12) it follows
\[ \tag{31} \gamma\beta = \frac{\sin(\pi b)}{\cos(\pi b)} = \tan(\pi b). \] Therefore, the spatial period is
\[ \tag{32} \boxed{ \lambda_J(b) = \frac{\lambda_0}{\tan(\pi b)} = \lambda_0\cot(\pi b) }. \] In this notation, the entire external kinematic scale of diffraction is determined by a single exponent of the operator \((-\j)^b\). The sequence of causal relationships within the model can be represented as follows:
\[ \tag{33} \boxed{ \begin{aligned} b &\longrightarrow \tan(\pi b)=\gamma\beta \longrightarrow \lambda_J=\lambda_0\cot(\pi b), \\[2mm] \lambda_J &\longrightarrow \text{scale of the diffraction pattern}. \end{aligned} } \] As \(b\to0\), the particle approaches rest, \(\tan(\pi b)\to0\), and the spatial period tends to infinity. This does not mean that the physical extent of the particle at rest is infinite. This means the absence of a finite spatial frequency of translational motion: at \(v=0\), the internal process changes in time but does not form a running spatial phase along the laboratory axis.
At \(b\to1/2\), which corresponds to \(\beta\to1\), the factor \(\tan(\pi b)\) increases indefinitely, and \(\lambda_J\) of the massive branch tends to zero. This limit cannot be directly identified with the photon: the massless state has no rest frame and no natural frequency \(\omega_0\), from which formula (28) was derived.
8. What does a slit mean geometrically?
The physical plane of the slit should not be identified with the idempotent plane \((\em,i\em)\). The slit is the outer boundary of the setup and limits the set of admissible motion extensions in ordinary space.
Let \(\xi\) denote the coordinate of a point inside the hole. Each such point defines an admissible one-dimensional extension.
\[ \tag{34} \ell_\xi: \qquad \text{source} \longrightarrow \xi \longrightarrow \text{screen point}. \] In each individual event, the particle's motion remains one-dimensional: there is one source, one transit point, and one detection point. However, to calculate the distribution, the family of all geometrically admissible extensions \(\ell_\xi\) is considered.
The state associated with one such extension can be written in the form
\[ \tag{35} J_\xi = \ep e^{i\pi b_\xi} + \em e^{i\phi_\xi}, \qquad |J_\xi|=1. \] The external component determines the admissibility of the path, its direction, and speed. The difference in the lengths of the admissible paths is transformed into an internal phase difference \(\phi_\xi\). Therefore, the effect of the gap can be briefly expressed by the following rule:
\[ \tag{36} \boxed{ \text{the gap restricts extensions in }\ep, \qquad \text{the difference in their lengths manifests itself as phases in }\em }. \] The slit does not divide the real particle into parts. It creates a geometric set of alternatives, which is used to calculate the distribution of possible registrations of the entire operator \(J\).
9. Far-Field Path Difference
Consider a uniformly open slit of width \(d\):
\[ \tag{37} -\frac d2 \leq \xi \leq \frac d2. \] The screen is far enough away that the rays from different points on the slit to the chosen observation point can be considered nearly parallel. This is the far-field approximation, or Fraunhofer approximation.
If the observation direction forms an angle \(\theta\) with the normal to the slit, then the path from point \(\xi\) differs from the central path by approximately
\[ \tag{38} \Delta\ell_\xi \simeq \xi\sin\theta. \] According to formula (26), the corresponding internal phase difference is
\[ \tag{39} \Delta\phi_\xi = -\frac{2\pi}{\lambda_J} \xi\sin\theta. \] The sign determines the direction of phase counting, but does not affect the final probability, since the square of the resulting amplitude modulus is then taken.
10. Phase Summation by Slit Width
For a uniformly illuminated slit, all small sections of the aperture are considered equal. The phase-sensitive sum of the admissible extensions is written as an integral.
\[ \tag{40} \mathcal A_J(\theta) = \int_{-d/2}^{d/2} \em \exp \left( -i\frac{2\pi}{\lambda_J} \xi\sin\theta \right) d\xi. \] The idempotent \(\em\) shows that the internal phases of the alternatives are summed. It is constant with respect to \(\xi\), so it is taken outside the integral sign:
\[ \tag{41} \mathcal A_J(\theta) = \em \int_{-d/2}^{d/2} e^{-iq\xi}\,d\xi, \qquad q=\frac{2\pi}{\lambda_J}\sin\theta. \] Integration gives
\[ \tag{42} \begin{aligned} \mathcal A_J(\theta) &= \em \left[ \frac{e^{-iq\xi}}{-iq} \right]_{-d/2}^{d/2} \\[1mm] &= \em \frac{2\sin(qd/2)}{q}. \end{aligned} \] After normalizing to the amplitude at the center, where \(\theta=0\), we get
\[ \tag{43} \boxed{ \frac{\mathcal A_J(\theta)}{\mathcal A_J(0)} = \frac{ \sin \left( \dfrac{\pi d\sin\theta}{\lambda_J} \right) }{ \dfrac{\pi d\sin\theta}{\lambda_J} } }. \] Thus, the well-known function \(\operatorname{sinc}\) arises not from a predetermined de Broglie wave, but from the continuous addition of geometrically obtained phases across the aperture width.
11. Registration Distribution
Now the quadratic registration rule adopted in formula (3) is applied. After normalization, the distribution has the form
\[ \tag{44} \boxed{ \frac{P(\theta)}{P(0)} = \left[ \frac{ \sin \left( \dfrac{\pi d\sin\theta}{\lambda_J} \right) }{ \dfrac{\pi d\sin\theta}{\lambda_J} } \right]^2 }. \] Minima occur when the numerator vanishes while the denominator remains finite:
\[ \tag{45} \frac{\pi d\sin\theta_n}{\lambda_J} = \pi n, \qquad n=\pm1,\pm2,\ldots \] Therefore, the positions of the minima are equal
\[ \tag{46} \boxed{ d\sin\theta_n = n\lambda_J = n\frac{\lambda_0}{\gamma\beta} = n\lambda_0\cot(\pi b) }. \] Formula (46) directly relates the observed angular pattern to the external motion index \(b\). The central maximum is located at \(\theta=0\), where the phases of all admissible extensions coincide.
12. How velocity changes the pattern width
Let the screen be located at a distance \(L\) from the slit. For small angles
\[ \tag{47} \sin\theta\simeq\theta\simeq\frac{y}{L}. \] The distance from the center to the first minimum is
\[ \tag{48} y_1 \simeq \frac{L\lambda_J}{d} = \frac{L\lambda_0}{d\gamma\beta}. \] The full width of the central maximum between minima \(n=-1\) and \(n=+1\) is
\[ \tag{49} \boxed{ \Delta y \simeq \frac{2L\lambda_J}{d} = \frac{2L\lambda_0}{d\gamma\beta} }. \] Therefore, as the speed of a massive particle increases:
1. the magnitude of \(\gamma\beta\) increases;
2. the spatial period of \(\lambda_J\) decreases;
3.The angles of the diffraction minima decrease;
4. The central maximum and the entire diffraction pattern narrow.
2. the spatial period of \(\lambda_J\) decreases;
3.The angles of the diffraction minima decrease;
4. The central maximum and the entire diffraction pattern narrow.
In the nonrelativistic limit
\[ \tag{50} \beta\ll1, \qquad \gamma\simeq1, \] therefore
\[ \tag{51} \lambda_J \simeq \frac{\lambda_0}{\beta}, \qquad \Delta y \simeq \frac{2L\lambda_0}{d\beta}. \] If a small velocity is doubled, the spatial period and the pattern width are approximately halved. In the relativistic region, the compression occurs faster due to the increase in \(\gamma\).
13. Why the phase projection is summed, but the whole J is recorded
In integral (40), the expression \(\em e^{i\phi_\xi}\) is summed. This does not mean that the particle loses its external component or transforms only into an internal phase. The complete state of each alternative remains normalized:
\[ \tag{52} J_\xi\overline J_\xi=1. \] The roles of the two components are different. The \((\ep,i\ep)\) plane defines the external conditions of the problem: the admissibility of the path, the direction after the slit, the velocity, the \(\xi\) coordinate, and the integration domain. The \((\em,i\em)\) plane carries internal periodicity, so it is precisely its phase differences that can mutually amplify or cancel each other out.
During registration, these two descriptions are reunited. The detector does not observe a single idempotent \(\em\); it registers the entire particle. The phase sum determines only the statistical weight of the location where such a single event can occur.
Therefore, the proposed interpretation consists of two statements:
\[ \tag{53} \boxed{ \begin{aligned} \text{interfere}\quad& \text{phase continuations of the internal component}, \\ \text{is registered}\quad& \text{the entire normalized operator }J. \end{aligned} } \] The first statement relates diffraction to the geometry of split space. The second reconciles this geometry with the local nature of a single event on the screen.
14. Comparison with the standard de Broglie wavelength
Only now that the spatial period has been obtained geometrically can an independent verification be performed using generally accepted physical quantities. In the article on mass geometry, the resting frequency is related to mass by the expression
\[ \tag{54} \omega_0=\frac{m_0c^2}{\hbar}. \] Substituting it into the proper spatial scale (29), we obtain
\[ \tag{55} \lambda_0 = \frac{2\pi c}{\omega_0} = \frac{2\pi\hbar}{m_0c} = \frac{h}{m_0c}. \] Now the geometric period (30) takes the form
\[ \tag{56} \lambda_J = \frac{h}{\gamma m_0c\beta}. \] The relativistic momentum of a massive particle is
\[ \tag{57} p=\gamma m_0v = \gamma m_0c\beta. \] Therefore,
\[ \tag{58} \boxed{ \lambda_J = \frac{h}{\gamma m_0c\beta} = \frac{h}{p} = \lambda_{\mathrm{dB}} }. \] This coincidence is the main result of the paper. The de Broglie formula was not the initial condition of the derivation. It emerged after the inherent internal periodicity was transferred to the laboratory space by relativistic geometry.
The model thus proposes the following meaning for the de Broglie wavelength: it is the spatial period of the internal phase of the entire operator \(J\), observed during the simultaneous comparison of different admissible paths of a moving massive particle.
15. Comparison with Real Experiments
The obtained geometric result can be compared with several types of real experiments. They test various aspects of the proposed description: the dependence of the spatial period on velocity, the locality of individual registration, the applicability of the wave law to complex objects, and the special position of the photon.
| Experiment | What was changed | What was observed | Relation to the geometry of \(J\) |
|---|---|---|---|
| Davisson-Germer experiment [2] | Electron velocity and momentum | Change in the directions of diffraction maxima | As \(\beta\) increases, \(\gamma\beta\) increases, and \(\lambda_J\) decreases |
| Tonomura experiment with an electron biprism [3] | Number of registered electrons | Individual local hits gradually form interference fringes | Each event registers an entire \(J\), and the distribution of events is determined by adding the phases of the admissible continuations |
| Diffraction of molecules \(C_{60}\) [4] | Velocity of massive molecules | InterferenceComplex objects and blurring of the image with a spread of velocities | Each velocity group has its own value of \(b\) and its own period \(\lambda_J=\lambda_0\cot(\pi b)\) |
| Diffraction of light of different frequencies | Frequency and wavelength of light | Long-wavelength light creates a wider image, while short-wavelength light creates a narrower one | For the massless branch, the scale is determined by the frequency: \(\lambda_\gamma=c/\nu\), and not by the change in velocity |
The table shows that different experiments do not confirm the same detail models, but successive parts of a larger picture. The Davisson-Germer experiment tests the dependence of the spatial period on motion; the Tonomura experiment combines a local event with an extended distribution; diffraction \(C_{60}\) demonstrates the universality of the phase law for massive objects; experiments with light separate the massless wave branch from the motion of a massive particle.
In the Davisson-Germer experiment, electrons were scattered by a nickel crystal. Changing the accelerating voltage changed the velocity and momentum of the electrons, and with them, the directions of the diffraction maxima. The resulting wavelength was consistent with the dependence \(\lambda=h/p\). This means that with increasing velocity, the observed diffraction scale actually decreases, as predicted by formulas (30), (46), and (49).
It is important to note that the Davisson-Germer experiment was not a single-slit experiment: the crystal served as the spatial periodic structure. However, the quantity being tested was the same—the spatial phase of the electron matter and its dependence on momentum. Therefore, this experiment confirms the main kinematic scale of the paper, although the specific distribution function for the crystal differs from the single-slit function \(\operatorname{sinc}^2\).
Another fundamental result was obtained in experiments with sequential accumulation of individual electrons. The experiment by Tonomura et al. used an electron biprism rather than two mechanical slits. Each electron appeared on the detector as a single point, but as the number of events increased, persistent interference fringes became visible.
This observation corresponds to the separation expressed by formula (53): an individual event remains local and integral, while the distribution of multiple events is determined by the phase relationships of the admissible continuations.
16. Atoms and Large Molecules
Diffraction is not unique to electrons. Interference experiments have also been performed with neutrons, atoms, and molecules. A particularly illustrative experiment is one involving molecules (C_{60}) passing through a material diffraction grating. Observing the interference of such complex objects shows that the spatial phase law should not depend on the particle's elementary nature.
For slow, massive objects, the relativistic factor is practically unity, therefore
\[ \tag{59} \lambda_J \simeq \frac{h}{m_0v}. \] Two experimentally important regularities follow from this. At the same speed, a heavier object has a shorter wavelength and a narrower pattern. At the same mass, an increase in speed also decreases the wavelength. The spread of velocities in the beam means a spread of \(\lambda_J\), so the maxima of different particle subgroups partially overlap, and the contrast of the picture is reduced.
In geometric terms, this means that each velocity group has its own value of the exponent \(b\) and, consequently, its own spatial scale \(\lambda_0\cot(\pi b)\).
17. Why the photon is considered separately
For a free photon in a vacuum, the velocity is not a variable parameter:
\[ \tag{60} v=c, \qquad \beta=1. \] You cannot simply substitute \(\beta=1\) into formula (30). It was derived from the natural frequency \(\omega_0\) and the natural time \(\tau\) of a massive particle. The photon has no rest frame, and therefore does not have a proper Compton periodicity of the same type.
For the photonic, massless branch, the spatial period is determined directly by the wave frequency:
\[ \tag{61} \boxed{ \lambda_\gamma = \frac{c}{\nu} = \frac{2\pi c}{\omega} }. \] Therefore, the diffraction pattern of light in a vacuum can be changed not by changing the photon velocity, but by changing the frequency. For a single slit
\[ \tag{62} d\sin\theta_n = n\lambda_\gamma = n\frac{c}{\nu}. \] Longer wavelength light produces a broader pattern, while increasing frequency narrows it. Thus, the massive and massless branches lead to the same phase summation geometry at the slit, but the source of their spatial period is different:
\[ \tag{63} \boxed{ \begin{aligned} |\beta| < 1: \qquad&\lambda_J=\frac{\lambda_0}{\gamma\beta}, \\[1mm] |\beta|=1: \qquad& \lambda_\gamma=\frac{c}{\nu}. \end{aligned} } \] 18. What is derived from geometry, and what remains a hypothesis?
To evaluate the result, it is necessary to strictly distinguish three levels of reasoning.
The first level is the geometric derivation within the model. The spatial phase is obtained from the proper rotation \(\em e^{i\omega_0\tau}\), time transformation, and coupling \(\beta=\sin(\pi b)\)
\[ \tag{64} \Delta\phi = -\frac{\omega_0}{c} \tan(\pi b)\,\Delta\ell \] and spatial Period
\[ \tag{65} \lambda_J = \lambda_0\cot(\pi b). \] The second level is a known experimental law. The diffraction of massive particles is determined by the wavelength \(h/p\), which decreases with increasing momentum. After coupling the natural frequency with the mass, the geometric period exactly reproduces this law.
The third level is a new interpretation. The model associates the external path limitation with the \(\ep\)-plane, the internal phase difference with the \(\em\)-plane, and individual registration with the integer operator \(J\). This interpretation has not yet been clearly demonstrated by experiment: the same observed patterns are described by standard quantum mechanics.
Furthermore, the \(P\propto|\mathcal A_J|^2\) rule in this paper is adopted, rather than derived from the unit norm \(J\). A complete theory requires deriving a quadratic rule from the interaction of the particle operator with the detector operator.
Therefore, the agreement of formula (58) should be considered an important test of the internal consistency of the geometry, but not definitive proof of the uniqueness of the proposed model.
Conclusion
The paper set out to obtain the diffraction scale of a massive particle without initially introducing momentum and the de Broglie wavelength. The initial process was the internal rotation of the operator \(J=\j^a(-\j)^b\), determined by the natural frequency \(\omega_0\).
The relativistic transformation of the proper time showed that the moving internal phase acquires a spatial component with a coefficient
\[ \tag{66} k_J = \frac{\omega_0}{c}\gamma\beta = \frac{\omega_0}{c}\tan(\pi b). \] This is where the spatial period directly originated.
\[ \tag{67} \boxed{ \lambda_J = \frac{2\pi}{k_J} = \frac{\lambda_0}{\gamma\beta} = \lambda_0\cot(\pi b) }. \] Integrating the phases over the width of one slit resulted in a distribution of \(\operatorname{sinc}^2\), and the position of the minima turned out to be inversely proportional to \(\gamma\beta\). Therefore, as the velocity of the massive particle increases, the diffraction pattern narrows. After independently introducing the constraint \(\omega_0=m_0c^2/\hbar\), the geometric period precisely coincided with the de Broglie wavelength:
\[ \tag{68} \boxed{ \lambda_J=\frac{h}{p} }. \] In the proposed geometric picture, the slit does not bisect the particle and does not literally force it to travel along several spatial trajectories simultaneously. In each individual event, the particle maintains its integrity, and its motion remains one-dimensional. The slit merely creates a multitude of permissible continuations of this motion, each of which corresponds to its own accumulation of the internal phase.
Materials used
- Wikipedia. Diffraction.
- C. Davisson, L. H. Germer. Diffraction of Electrons by a Crystal of Nickel, Physical Review 30, 705 (1927).
- A. Tonomura et al. Demonstration of single-electron buildup of an interference pattern, American Journal of Physics 57, 117 (1989).
- M. Arndt et al. Wave-particle duality of C60 molecules, Nature 401, 680–682 (1999).