In a previous paper, an operator was introduced that combines the internal periodic state of an object and its external motion: \[ \tag{1} J(t) = e^{i\theta} \j^{\varpi t}. \] This operator can be viewed as the product of two independent components. The first determines the external state of the object, and the second, its internal periodic dynamics.

This separation allows us to consider the motion, energy, and wave properties of an object not as independent and opposed phenomena, but as different manifestations of a single geometric structure. Within the framework of the proposed model, a wave and a particle are not two different objects. They correspond to different limiting states of a single operator.

This material is an appendix to the work

• New Cartesian Basis with Added Energy and Momentum;
• and develops the ideas presented in the article: Unified Geometric Model for Waves and Particles.

Consider the complete operator \[ \tag{2} J(t) = e^{i\theta} \j^{\varpi t}. \] It consists of two factors: \[ \tag{3} J_{\mathrm{ext}} = e^{i\theta}, \] \[ \tag{4} J_{\mathrm{int}}(t) = \j^{\varpi t}. \] Therefore, \[ \tag{5} J(t) = J_{\mathrm{ext}} J_{\mathrm{int}}(t). \]

The operator \[ J_{\mathrm{ext}} = e^{i\theta} \] will be called the external operator. It determines the orientation of the object relative to external motion, as well as the relationship between rest energy, momentum, and total energy.

The operator \[ J_{\mathrm{int}}(t) = \j^{\varpi t} \] will be called the internal operator. It determines the object's eigenstate, its internal frequency, and the energy associated with it.

Thus, the complete operator allows for a natural division: \[ \tag{6} \boxed{ J(t) = J_{\mathrm{ext}} J_{\mathrm{int}}(t) }. \] Each of these operators can be worked with separately without losing the overall structure of the object.

The internal operator is written as \[ \tag{7} J_{\mathrm{int}}(t) = \j^{\varpi t}. \] Using the fractional power representation of the hyperbolic unit, we obtain \[ \tag{8} \j^{\varpi t} = \ep + \em e^{i\omega t}, \qquad \omega = \pi\varpi. \]

Therefore, \[ \tag{9} J_{\mathrm{int}}(t) = \ep + \em\cos\omega t + i\em\sin\omega t. \] The first component \(\ep\) remains constant, and the second component rotates in the complex plane. \[ \left\{ \em, i\em \right\}. \]

The internal state is characterized by its natural frequency \(\omega\), but does not directly depend on the angle of external motion \(\theta\). Therefore, the internal dynamics of an object can be studied separately by setting \[ \tag{10} \theta = 0. \] In this case, the complete operator takes the form \[ \tag{11} J(t) = \j^{\varpi t}. \]

This representation corresponds to considering the object in its own frame of reference, when external motion is ignored and only the internal periodic process is analyzed.

If the internal frequency corresponds to the object's natural energy, then we can write \[ \tag{12} E_0 = \hbar\omega. \] For a massive particle, the self-energy is identified with the rest energy: \[ \tag{13} E_0 = mc^2. \] Therefore, \[ \tag{14} \hbar\omega = mc^2. \]

Thus, the internal operator is responsible for the eigenstate of the object: \[ \tag{15} J_{\mathrm{int}}(t) \quad \longleftrightarrow \quad \omega, \; E_0, \; m. \]

Now let's temporarily exclude the internal rotation and consider only the external operator: \[ \tag{16} J_{\mathrm{ext}} = e^{i\theta}. \] Expanding the complex exponential gives \[ \tag{17} e^{i\theta} = \cos\theta + i\sin\theta. \]

The angle \(\theta\) characterizes the external state of the object. In the previously proposed geometric model, it is related to the velocity by the relation \[ \tag{18} \sin\theta = \beta = \frac{v}{c}. \] Hence \[ \tag{19} \cos\theta = \sqrt{1-\beta^2} = \frac{1}{\gamma}, \] where \[ \tag{20} \gamma = \frac{1} {\sqrt{1-\beta^2}} \] is the Lorentz factor.

The total energy of a moving object is \[ \tag{21} E = \gamma mc^2. \] Taking into account expression (19), we obtain \[ \tag{22} E = \frac{mc^2} {\cos\theta}. \]

The momentum of an object is determined by the expression \[ \tag{23} pc = \gamma mvc. \] Since \[ \gamma\beta = \frac{\sin\theta} {\cos\theta} = \tan\theta, \] we obtain \[ \tag{24} pc = mc^2\tan\theta. \]

Thus, the exterior angle \(\theta\) determines simultaneously the velocity, momentum, and total energy: \[ \tag{25} \sin\theta = \frac{v}{c}, \qquad \cos\theta = \frac{mc^2}{E}, \qquad \tan\theta = \frac{pc}{mc^2}. \]

These relations directly imply the relativistic relationship between energy and momentum: \[ \tag{26} E^2 = m^2c^4 + p^2c^2. \]

Therefore, the external operator is responsible for the kinematic state of the object: \[ \tag{27} J_{\mathrm{ext}} \quad \longleftrightarrow \quad v, \; p, \; E. \]

The complete operator \[ J(t) = e^{i\theta} \j^{\varpi t} \] contains two different phases:

1. The external phase \(\theta\), which characterizes the motion of the object as a whole;

2. The internal phase \(\omega t\), which characterizes the object's proper periodic state.

If the angle \(\theta\) remains constant and time varies, then only the internal state changes: \[ \tag{28} J(t) = e^{i\theta_0} \j^{\varpi t}. \]

If the internal phase is fixed at a certain point in time \(t_0\) and the angle \(\theta\) varies, then only the external state changes: \[ \tag{29} J(\theta) = e^{i\theta} \j^{\varpi t_0}. \]

In the most general case, both phases can change: \[ \tag{30} J(t) = e^{i\theta(t)} \j^{\varpi t}. \] Then the object simultaneously possesses internal periodic dynamics and external motion with variable velocity.

This means that accelerating an object does not necessarily change its internal structure. The external energy can vary due to the angle \(\theta\), while the internal frequency \(\omega\) is constant.

Conversely, a change in the natural frequency can occur while the external state remains unchanged. In this case, the internal operator changes, but the angle \(\theta\) remains constant.

An important property of the model is the invariance of the absolute values ​​of both factors. For the external operator \[ \tag{31} \left| e^{i\theta} \right| = 1. \] For the internal operator, we also assume \[ \tag{32} \left| \j^{\varpi t} \right| = 1. \]

Therefore, the absolute value of the complete operator is \[ \tag{33} \left| J(t) \right| = \left| e^{i\theta} \right| \left| \j^{\varpi t} \right| = 1. \]

Therefore, the transition from one state to another does not change the object's full modulus: \[ \tag{34} \boxed{ \left| J(t) \right| = 1 }. \] Only its geometric projections and orientation relative to the chosen basis directions change.

It is precisely the conservation of modulus that allows us to consider a wave and a particle as different states of a single object, rather than as two independent physical entities.

Consider the first characteristic value: \[ \tag{35} \theta = 0. \] Then \[ \tag{36} e^{i\theta} = 1, \] and the complete operator takes the form \[ \tag{37} J(t) = \j^{\varpi t}. \]

From the expression \[ \sin\theta = \frac{v}{c} \] it follows that \[ \tag{38} v = 0. \] Also \[ \tag{39} p = 0, \qquad E = mc^2. \]

This state corresponds to a massive object at rest. There is no external motion, but the internal periodic process is preserved.

Therefore, the absence of external motion does not mean the absence of motion at all. An object can retain its internal rotation, natural frequency, and rest energy.

For intermediate values ​​of the angle \[ \tag{40} 0 < \theta < \frac{\pi}{2} \] we have \[ \tag{41} 0 < v < c. \]

The complete operator is of the form \[ \tag{42} J(t) = e^{i\theta} \j^{\varpi t}. \] The internal state of the object is preserved, but external motion appears.

The rest energy and momentum are orthogonal projections of the total energy: \[ \tag{43} mc^2 = E\cos\theta, \] \[ \tag{44} pc = E\sin\theta. \]

Therefore, \[ \tag{45} E^2 = \left( mc^2 \right)^2 + \left( pc \right)^2. \]

For small values ​​of \(\theta\), the projection of the rest energy predominates: \[ \tag{46} mc^2 \gg pc. \] Such an object exhibits predominantly corpuscular properties and can be localized in a limited region of space.

As \(\theta\) increases, the momentum component increases, and the relative share of the projection of the rest energy decreases: \[ \tag{47} \frac{mc^2}{E} = \cos\theta. \]

Thus, a moving particle represents an intermediate state between the limiting state of rest and the limiting wave state.

Consider the limit \[ \tag{48} \theta \rightarrow \frac{\pi}{2}. \] Then \[ \tag{49} \sin\theta \rightarrow 1, \qquad \cos\theta \rightarrow 0. \] Hence, \[ \tag{50} v \rightarrow c. \]

In this limit, the projection of the total energy onto the direction of the rest energy tends to zero: \[ \tag{51} mc^2 = E\cos\theta \rightarrow 0. \] At the same time, the momentum component becomes equal to the total energy: \[ \tag{52} pc = E\sin\theta \rightarrow E. \]

We obtain the relationship characteristic of a massless wave \[ \tag{53} E = pc. \]

The external operator tends to the value \[ \tag{54} e^{i\theta} \rightarrow i. \] Therefore, the full operator in the wave limit takes the form \[ \tag{55} J_{\mathrm{wave}}(t) = i\j^{\varpi t}. \]

It should be emphasized that the internal operator \[ \j^{\varpi t} \] does not disappear. Only its external orientation changes.

Thus, the wave state arises not as a result of the destruction of the internal structure of the object, but as a result of the extreme rotation of the external operator by an angle \[ \frac{\pi}{2}. \]

Let the object be at rest in its initial state: \[ \tag{56} \theta = 0. \] Then \[ \tag{57} J_{\mathrm{particle}}(t) = \j^{\varpi t}. \]

When external energy is imparted to the object, the angle \(\theta\) increases: \[ \tag{58} 0 \longrightarrow \theta \longrightarrow \frac{\pi}{2}. \] The outer operator changes continuously: \[ \tag{59} 1 \longrightarrow e^{i\theta} \longrightarrow i. \]

Accordingly, the complete operator goes through a sequence of states \[ \tag{60} \j^{\varpi t} \longrightarrow e^{i\theta}\j^{\varpi t} \longrightarrow i\j^{\varpi t}. \]

The operator's absolute value remains unchanged: \[ \tag{61} \left| \j^{\varpi t} \right| = \left| e^{i\theta}\j^{\varpi t} \right| = \left| i\j^{\varpi t} \right| = 1. \]

Therefore, the particle-wave transition can be interpreted as a continuous change in the external orientation of the same mathematical object.

The internal state can be preserved during such a transition: \[ \tag{62} J_{\mathrm{int}}(t) = \j^{\varpi t} = \mathrm{const} \quad \text{in form}. \] Only the external factor changes \[ e^{i\theta}. \]

The inverse transition corresponds to a decrease in the angle: \[ \tag{63} \frac{\pi}{2} \longrightarrow \theta \longrightarrow 0. \] Then the external operator changes according to the law \[ \tag{64} i \longrightarrow e^{i\theta} \longrightarrow 1. \]

The complete operator goes through the inverse sequence: \[ \tag{65} i\j^{\varpi t} \longrightarrow e^{i\theta}\j^{\varpi t} \longrightarrow \j^{\varpi t}. \]

As \(\theta\) decreases, a non-zero projection of the rest energy reappears: \[ \tag{66} mc^2 = E\cos\theta. \] At the same time, the momentum component decreases: \[ \tag{67} pc = E\sin\theta. \]

Thus, the object gradually transitions from a predominantly wave regime to a predominantly corpuscular regime.

Within the proposed model, such a transition does not require the emergence of a new mathematical object. The wave and corpuscular states are described by a single operator, differing only in the value of the external angle \(\theta\).

The above regimes can be represented as a single diagram: \[ \tag{68} \begin{array}{c|c|c|c} \theta & v & J_{\mathrm{ext}} & \text{state} \\ \hline 0 & 0 & 1 & \text{particle at rest} \\ 0<\theta<\pi/2 & 0<v<c & e^{i\theta} & \text{moving particle} \\ \theta\rightarrow\pi/2 & v\rightarrow c & i & \text{wave limit} \end{array} \]

The table shows that there is no sharp mathematical boundary between the particle and the wave. There is a continuous sequence of intermediate states defined by the angle \(\theta\).

For small values ​​of \(\theta\), the projection of the rest energy predominates, and the object exhibits predominantly particle properties.

For values ​​of \(\theta\), close to \(\pi/2\), the momentum component predominates, and the object exhibits predominantly wave properties.

Therefore, the question "wave or particle?" within this model is replaced by another question:

What is the current value of the exterior angle \(\theta\) of the total operator?

The observed properties of an object are determined not only by its complete internal structure, but also by which projection of the total operator interacts with the measuring device.

If the experiment predominantly identifies the localized component, the object is registered as a particle.

If the experiment is sensitive to periodicity, phase, and amplitude distribution, the object exhibits wave properties.

However, in both cases, the complete mathematical object remains the same: \[ \tag{69} J(t) = e^{i\theta} \j^{\varpi t}. \]

The difference arises in the way it is projected and interacts with the external system, and not necessarily in a change in the fundamental nature of the object itself.

This allows for a geometric interpretation of wave-particle duality without introducing two opposing entities.

Since the full operator is a product of independent factors, operations on its interior and exterior parts can be performed separately.

When studying the interior dynamics, the exterior angle is fixed: \[ \tag{70} \theta = \theta_0, \] and the derivative is considered \[ \tag{71} \frac{dJ}{dt} = e^{i\theta_0} \frac{d}{dt} \j^{\varpi t}. \]

When studying external acceleration, the internal frequency can be considered constant: \[ \tag{72} \omega = \omega_0, \] and the change in the total operator is determined by the derivative of the angle: \[ \tag{73} \frac{dJ}{dt} = i\frac{d\theta}{dt} e^{i\theta} \j^{\varpi t} + e^{i\theta} \frac{d}{dt} \j^{\varpi t}. \]

The first term describes the change in the external state: \[ \tag{74} \left( \frac{dJ}{dt} \right)_{\mathrm{ext}} = i\dot\theta e^{i\theta} \j^{\varpi t}. \]

The second term describes the internal motion: \[ \tag{75} \left( \frac{dJ}{dt} \right)_{\mathrm{int}} = e^{i\theta} \frac{d}{dt} \j^{\varpi t}. \]

Thus, \[ \tag{76} \frac{dJ}{dt} = \left( \frac{dJ}{dt} \right)_{\mathrm{ext}} + \left( \frac{dJ}{dt} \right)_{\mathrm{int}}. \]

This separation allows us to independently study the acceleration of the object and its own internal dynamics, while maintaining a unified description of the complete state.

Three main modes are possible within the complete operator.

The first mode — only the internal phase changes: \[ \tag{77} J(t) = e^{i\theta_0} \j^{\varpi t}. \] This mode corresponds to an internal periodic process with a constant external velocity.

The second mode — only the external angle changes: \[ \tag{78} J(t) = e^{i\theta(t)} \j^{\varpi t_0}. \] This corresponds to the acceleration or deceleration of an object with a fixed internal state.

The third mode—the external and internal phases change simultaneously: \[ \tag{79} J(t) = e^{i\theta(t)} \j^{\varpi(t)t}. \] This mode corresponds to an interaction in which not only the object's motion changes, but also its own internal energy.

Therefore, a change in the total energy does not always mean a change in the internal energy. It is necessary to distinguish between:

— a change in the external energy due to the angle \(\theta\);

— a change in the internal energy due to the frequency \(\omega\);

— a joint change in both components.

The main result of the proposed approach is that a wave and a particle are described by the same operator: \[ \tag{80} \boxed{ J(t) = e^{i\theta} \j^{\varpi t} }. \]

In it, the external factor \[ e^{i\theta} \] determines the velocity, momentum, and the ratio of the rest energy to the total energy.

The internal factor \[ \j^{\varpi t} \] determines the natural frequency, internal energy, and periodic structure of the object.

The particle corresponds to a state in which a significant projection of the total energy in the direction of the rest energy is conserved: \[ \tag{81} mc^2 = E\cos\theta \neq 0. \]

The wave corresponds to the limiting state: \[ \tag{82} \cos\theta \rightarrow 0, \qquad E \rightarrow pc. \]

In this case, the absolute value of the complete operator remains unchanged, and the transition is accomplished by rotating the external component.

The complete operator \[ J(t) = e^{i\theta} \j^{\varpi t} \] is naturally divided into external and internal operators: \[ J_{\mathrm{ext}} = e^{i\theta}, \qquad J_{\mathrm{int}}(t) = \j^{\varpi t}. \]

The internal operator describes the intrinsic periodic dynamics of an object, its frequency, rest energy, and mass. The external operator describes the velocity, momentum, and total energy relative to an external reference frame.

Separating the operators allows us to separately study the internal state of an object and its external motion. The complete operator preserves the unity of both components.

The angle \(\theta\) determines the nature of the external state. At \[ \theta = 0 \] the object is at rest and exhibits predominantly particle properties.

At \[ 0 < \theta < \frac{\pi}{2} \] the object is a moving massive particle, containing both rest energy and momentum.

In the limit \[ \theta \rightarrow \frac{\pi}{2} \] the projection of the rest energy tends to zero, the velocity tends to the speed of light, and the wave relation \[ E = pc. \]

Thus, the transition from particle to wave and back can be represented as a continuous change in the external angle \(\theta\), while the internal operator retains its general form.

Within the proposed model, a wave and a particle are not opposing physical entities, but different geometric states of a single mathematical object.