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# Phase Mediation of Forces in Quantum Systems of Matter

Kenneth Hennacy

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# Phase Mediation of Forces in Quantum Systems of Matter

Kenneth Hennacy

► To cite this version:

Kenneth Hennacy. Phase Mediation of Forces in Quantum Systems of Matter. 2024. hal-04593487

HAL Id: hal-04593487

https://hal.science/hal-04593487v1

Preprint submitted on 29 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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# Phase Mediation of Forces in Quantum Systems of Matter: An Alternative to Spontaneous Collapse Theories

Ken A. Hennacy

*Cybioms, Rockville, 20850, Maryland, U.S.A.*

Given the uncertainty surrounding hypothetical mechanisms of wavefunction collapse, a different approach is proposed for unifying classical and quantum descriptions of matter. Here, a refinement to Newtonian physics and the standard quantum wave formulation is introduced that retains wavelike behavior through the mediation of phase. Quantum phenomenon such as tunneling and superposition are cast into a classical framework in which particle trajectories are continuous and virtual exchanges of energy occur with the background. Five experiments in published papers are re-examined with this approach: Young's double slit interference, collisions between fundamental particles, tunneling conductance for weakly applied voltages, tunneling barrier thickness determinations from typical I-V measurements, and the measurements of radial nodes in hydrogen. Theoretical results match features in the experimental data both accounted for and not addressed by Schrödinger's theory. Further analysis suggests that the success of semi-classical analysis does not require the concept of wavefunction collapse but instead characterizes resonant phenomena and the non-local averaging of quantum state variables as the system's phases evolve. For a certain class of problems, the unbounded growth of uncertainty is controllable in a stricter sense than standard quantum theory allows, and a more detailed picture of what is happening can be obtained.

## I. INTRODUCTION

In 1926, Schrödinger presented his formulation of de Broglie's postulate depicting matter as a bundle of waves traveling with a group velocity [1]. Though unmatched in its predictions over the past century, paradoxes exist with the theory, such as an inconsistency between the relativistic calculation of the phase velocity ($\frac{c^2}{\nu}$) and non-relativistic calculation ($\frac{\bar{c}^2}{\nu}$), which describes the fundamental motion of waves. The result is a highly dispersive representation of particle wavepackets that has led to controversial predictions such as a superluminal speed in barrier tunneling [2], which has since been experimentally disproven [3].

Theories such as Ghirardi-Rimini-Weber (GRW) and Continuous Spontaneous Localization (CSL) [4],[5] are other examples of the uncertainty surrounding key concepts of quantum wave mechanics due to their prediction of radiation from a particle's sudden, jarring localization otherwise known as wavefunction collapse. This also has not been observed [6] to date. While experimentalists continue to explore the limits of quantum behavior by studying transitions between phase coherent and decoherent regimes, such as with Carbon 60 [7], the representation of particle-wave duality by a superposition of waves of arbitrary spatial extent along with mechanisms of wavefunction collapse so far lack the evidence for their existence needed to avoid contradictions with the ontological interpretation of quantum mechanics, according to Pusey et al. [8]. One may even go so far as to view wave mechanics similarly to that of Laplace or Fourier transforms: the purpose is to simplify analysis, not depict an underlying reality.

Then there are open questions as to whether or not fundamental particles are singularities; what it means for matter to go "dark". Shouldn't wave mechanics have

something to say about this?

As to whether particles are guided by waves instead, or something in-between, this debate has been persistent for nearly a century [9]. What has not been discussed is that wave-guidance or pilot-theory was inspired by the electromagnetic theory of light, whose quantum properties are the inspiration behind de Broglie's postulate, yet the waves for massive particles aren't physically manifested with energy content and are depicted as scalar functions with no transverse component. Why?

The suggestion here is that the electromagnetic field associated with a photon's wave takes up a cross section perpendicular to the propagation direction of light. Hence, it is natural to think of this field impinging upon a diffraction grating with continuity assured by the field equations for a significant distance, still propagating as it is split up due to its transverse extension, and recombining with non-local effects of interference dictating where the resultant bundle of energy winds up. This replaces the notion that a photon travels different paths at the same time, yet it is not a strictly local depiction of the process, nor does it encompass all of space as Huygens' principle and quantum mechanics suggest it should.

The question then arises as to whether there is physics behind the dynamics of matter that gives rise to wave-like behavior in the same manner that electromagnetic theory gives rise to wave solutions. The dispersion of matter waves and non-commutation relations casts the uncertainty principle as a fundamental limitation that influences dynamics. Yet, what if this were necessary due to our inaccurate characterization of the dynamics of a system on a microscopic scale? If such physics did not violate Bell's theorem and it preserved the essence of wave mechanics, then the uncertainty principle could be removed as the basis of probability calculations and classical concepts could be restored such that localized properties evolve in a continuous, definite fashion.
# Phase Mediation of Forces in Quantum Systems of Matter: An Alternative to Spontaneous Collapse Theories

Ken A. Hennacy

*Cybioms, Rockville, 20850, Maryland, U.S.A.*

Given the uncertainty surrounding hypothetical mechanisms of wavefunction collapse, a different approach is proposed for unifying classical and quantum descriptions of matter. Here, a refinement to Newtonian physics and the standard quantum wave formulation is introduced that retains wavelike behavior through the mediation of phase. Quantum phenomenon such as tunneling and superposition are cast into a classical framework in which particle trajectories are continuous and virtual exchanges of energy occur with the background. Five experiments in published papers are re-examined with this approach: Young's double slit interference, collisions between fundamental particles, tunneling conductance for weakly applied voltages, tunneling barrier thickness determinations from typical I-V measurements, and the measurements of radial nodes in hydrogen. Theoretical results match features in the experimental data both accounted for and not addressed by Schrödinger's theory. Further analysis suggests that the success of semi-classical analysis does not require the concept of wavefunction collapse but instead characterizes resonant phenomena and the non-local averaging of quantum state variables as the system's phases evolve. For a certain class of problems, the unbounded growth of uncertainty is controllable in a stricter sense than standard quantum theory allows, and a more detailed picture of what is happening can be obtained.

## I. INTRODUCTION

In 1926, Schrödinger presented his formulation of de Broglie's postulate depicting matter as a bundle of waves traveling with a group velocity [1]. Though unmatched in its predictions over the past century, paradoxes exist with the theory, such as an inconsistency between the relativistic calculation of the phase velocity ($\frac{c^2}{\nu}$) and non-relativistic calculation ($\frac{\bar{\nu}}{2}$), which describes the fundamental motion of waves. The result is a highly dispersive representation of particle wavepackets that has led to controversial predictions such as a superluminal speed in barrier tunneling [2], which has since been experimentally disproven [3].

Theories such as Ghirardi-Rimini-Weber (GRW) and Continuous Spontaneous Localization (CSL) [4],[5] are other examples of the uncertainty surrounding key concepts of quantum wave mechanics due to their prediction of radiation from a particle's sudden, jarring localization otherwise known as wavefunction collapse. This also has not been observed [6] to date. While experimentalists continue to explore the limits of quantum behavior by studying transitions between phase coherent and decoherent regimes, such as with Carbon 60 [7], the representation of particle-wave duality by a superposition of waves of arbitrary spatial extent along with mechanisms of wavefunction collapse so far lack the evidence for their existence needed to avoid contradictions with the ontological interpretation of quantum mechanics, according to Pusey et al. [8]. One may even go so far as to view wave mechanics similarly to that of Laplace or Fourier transforms: the purpose is to simplify analysis, not depict an underlying reality.

Then there are open questions as to whether or not fundamental particles are singularities; what it means for matter to go "dark". Shouldn't wave mechanics have

something to say about this?

As to whether particles are guided by waves instead, or something in-between, this debate has been persistent for nearly a century [9]. What has not been discussed is that wave-guidance or pilot-theory was inspired by the electromagnetic theory of light, whose quantum properties are the inspiration behind de Broglie's postulate, yet the waves for massive particles aren't physically manifested with energy content and are depicted as scalar functions with no transverse component. Why?

The suggestion here is that the electromagnetic field associated with a photon's wave takes up a cross section perpendicular to the propagation direction of light. Hence, it is natural to think of this field impinging upon a diffraction grating with continuity assured by the field equations for a significant distance, still propagating as it is split up due to its transverse extension, and recombining with non-local effects of interference dictating where the resultant bundle of energy winds up. This replaces the notion that a photon travels different paths at the same time, yet it is not a strictly local depiction of the process, nor does it encompass all of space as Huygens' principle and quantum mechanics suggest it should.

The question then arises as to whether there is physics behind the dynamics of matter that gives rise to wave-like behavior in the same manner that electromagnetic theory gives rise to wave solutions. The dispersion of matter waves and non-commutation relations casts the uncertainty principle as a fundamental limitation that influences dynamics. Yet, what if this were necessary due to our inaccurate characterization of the dynamics of a system on a microscopic scale? If such physics did not violate Bell's theorem and it preserved the essence of wave mechanics, then the uncertainty principle could be removed as the basis of probability calculations and classical concepts could be restored such that localized properties evolve in a continuous, definite fashion.
This has been the motivation behind hidden variable theories, however in these studies[10] either the classical computation of probabilities that preserve locality do not match up to the predictions of quantum superposition, hence violating Bell's inequality, or the statistics of wave mechanics are duplicated, such as with Bohm's theory [11]. The concepts of particle-wave duality, uncertainty, and superposition have been used to argue that continuous trajectories are not realized when wave behavior is exhibited, however this conclusion is based on expectations such as 1) complete destructive interference rules out the presence of particles, 2) wave behavior is associated with particles not the forces that guide them and 3) wave mechanics does not make assumptions that are only approximately correct for the characterization of matter.

These assertions are challenged here with a theory that retains important aspects of Schrödinger's work, yet unlike alternative constructs, existing experiments can be used for verification since its predictions don't always agree with wave mechanics. It would be a tall task to address the entirety of a hundred years of development of quantum theory in one body of work. Instead, tackling smaller problems first will help guide the development of a more comprehensive framework.

Focus areas in this paper include the superposition of free particle states, two-slit interference, barrier tunneling, a phenomenon still raising questions [3], and the electron probability distribution of hydrogen that exposed mismatches between theory and experiment [12]. This study may also lead to other applications, such as aiding the investigation into how quantum dynamics transitions into classical deterministic or chaotic behavior [13], [14].

## II. THEORY FOR ONE-DIMENSIONAL SYSTEMS WITH CYLINDRICAL SYMMETRY

The main concept behind quantum mechanics is that quantum state $|\psi\rangle$ and the phases assigned to it do not directly determine the wave dynamics of a system's components. Instead, the exact dynamics of the system is treated as an unknown, however the evolution of $|\psi\rangle$ is found from the solution to an equation generated from the Hamiltonian of a system considered to be isolated from its environment. Probabilities for observables are computed from $|\psi\rangle$, weighted by component amplitudes that either depend on the system's history under observation or boundary conditions.

In this work, state $|\phi\rangle$ is determined both from equations describing the dynamics of a system's components and a coupled equation determining the probability distribution. Steady-state analysis is performed in place of seeking energy eigenstates. The thinking here is that probabilities are directly linked to phase space considerations and detectability, which is separate from computing the system's dynamics. Also, it is assumed that any system under study is in constant interaction with its environment. Hence, seemingly static systems are actually in quasi-equilibrium, with fluctuations in energy and the uncertainties associated with them taking place all the time. Rather than incorporate uncertainty into the analysis, a time-averaged calculation of the system's behavior is obtained.

Within this approach, wave behavior is directly attributed to observables of the system and phase is treated on an equal footing with amplitudes, consistent with the quantum concepts introduced by Einstein [15], de Broglie [16], Schrödinger, and Dirac [17], yet also the classical analysis of Newton [18]. The bra-ket projection of $|\phi\rangle$ onto the measurement apparatus state $\langle x|$ in wave mechanics and this formulation are equivalent. However, as measurements involve the mediation of phase between a classical system and the system under study, a modification to the computation of expectation values is made to establish common ground with classical theory, but this approach is not equivalent to a real vector-space quantum theory [19].

The initial focus in this section is on 1-D systems with cylindrical symmetry because this will highlight key concepts that can be difficult to isolate from more complex calculations of a system's 3-D dynamics. Also, while this body of work suggests that the Lagrangian and its ability to handle variable mass can be used to generate equations of motion with further modification by the use of the relations mentioned here, the emphasis in this work is on fundamental derivations using explicit forces.

To begin, it is postulated that the motion of a particle is characterized by a modulated transverse wave. Here, de Broglie's relation, $p_e = \frac{\hbar}{\lambda_e}$, pertains to the carrier envelope's horizontal periodicity with wavelength $\lambda_e$, while the associated momentum amplitude $p$ of the particle in one dimension obeys the following relationships in terms of modulated wavevector $\tilde{k}$:

$$ \pm p = \hbar\chi_{\pm}\tilde{k} = \hbar\alpha_{\parallel}\hat{\chi}_{\pm}\tilde{k} \equiv \pm\alpha_{\parallel}\hbar k_e \equiv \pm\alpha_{\parallel}p_e \\ = \frac{h}{\lambda} \qquad (1) $$

where $\lambda$ is defined further on.

The chiral operator $\hat{\chi}_{\pm}$ is associated with a spin's projection onto the direction of motion, known as helicity, and acts upon the wave characterization's envelope only, not the carrier component, analyzed separately. As will be shown, it provides a phase correction that is necessary to keep a particle moving in the same direction. The properties of $\hat{\chi}_{\pm}$ are as follows:

$$ \hat{\chi}_{\pm}f(\phi_e(x))g(\phi_e(x)) = f(\hat{\chi}_{\pm}\phi_e(x))g(\phi_e(x)) \\ = f\left(\pm\left(\frac{\pi}{2}\hat{x}\cdot\hat{x}_o - \phi_e(x)\right)\right)g(\phi_e(x)) \qquad (2) $$

$$ \hat{\chi}_{\pm}^{-1} = \hat{\chi}_{\pm} \qquad (3) $$

$$ \hat{\chi}_{\pm}\frac{d}{dt} = \frac{d}{dt}\hat{\chi}_{\pm} \qquad (3) $$

where $\hat{x}$ and $\hat{x}_o$ are positional unit vectors assigned to the quanta of the particle and carrier propagation, respectively, while $f$ and $g$ are phase-periodic functions
This has been the motivation behind hidden variable theories, however in these studies[10] either the classical computation of probabilities that preserve locality do not match up to the predictions of quantum superposition, hence violating Bell's inequality, or the statistics of wave mechanics are duplicated, such as with Bohm's theory [11]. The concepts of particle-wave duality, uncertainty, and superposition have been used to argue that continuous trajectories are not realized when wave behavior is exhibited, however this conclusion is based on expectations such as 1) complete destructive interference rules out the presence of particles, 2) wave behavior is associated with particles not the forces that guide them and 3) wave mechanics does not make assumptions that are only approximately correct for the characterization of matter.

These assertions are challenged here with a theory that retains important aspects of Schrödinger's work, yet unlike alternative constructs, existing experiments can be used for verification since its predictions don't always agree with wave mechanics. It would be a tall task to address the entirety of a hundred years of development of quantum theory in one body of work. Instead, tackling smaller problems first will help guide the development of a more comprehensive framework.

Focus areas in this paper include the superposition of free particle states, two-slit interference, barrier tunneling, a phenomenon still raising questions [3], and the electron probability distribution of hydrogen that exposed mismatches between theory and experiment [12]. This study may also lead to other applications, such as aiding the investigation into how quantum dynamics transitions into classical deterministic or chaotic behavior [13], [14].

## II. THEORY FOR ONE-DIMENSIONAL SYSTEMS WITH CYLINDRICAL SYMMETRY

The main concept behind quantum mechanics is that quantum state $|\psi\rangle$ and the phases assigned to it do not directly determine the wave dynamics of a system's components. Instead, the exact dynamics of the system is treated as an unknown, however the evolution of $|\psi\rangle$ is found from the solution to an equation generated from the Hamiltonian of a system considered to be isolated from its environment. Probabilities for observables are computed from $|\psi\rangle$, weighted by component amplitudes that either depend on the system's history under observation or boundary conditions.

In this work, state $|\phi\rangle$ is determined both from equations describing the dynamics of a system's components and a coupled equation determining the probability distribution. Steady-state analysis is performed in place of seeking energy eigenstates. The thinking here is that probabilities are directly linked to phase space considerations and detectability, which is separate from computing the system's dynamics. Also, it is assumed that any system under study is in constant interaction with its environment. Hence, seemingly static systems are ac-

tually in quasi-equilibrium, with fluctuations in energy and the uncertainties associated with them taking place all the time. Rather than incorporate uncertainty into the analysis, a time-averaged calculation of the system's behavior is obtained.

Within this approach, wave behavior is directly attributed to observables of the system and phase is treated on an equal footing with amplitudes, consistent with the quantum concepts introduced by Einstein [15], de Broglie [16], Schrödinger, and Dirac [17], yet also the classical analysis of Newton [18]. The bra-ket projection of $|\phi\rangle$ onto the measurement apparatus state $\langle x|$ in wave mechanics and this formulation are equivalent. However, as measurements involve the mediation of phase between a classical system and the system under study, a modification to the computation of expectation values is made to establish common ground with classical theory, but this approach is not equivalent to a real vector-space quantum theory [19].

The initial focus in this section is on 1-D systems with cylindrical symmetry because this will highlight key concepts that can be difficult to isolate from more complex calculations of a system's 3-D dynamics. Also, while this body of work suggests that the Lagrangian and its ability to handle variable mass can be used to generate equations of motion with further modification by the use of the relations mentioned here, the emphasis in this work is on fundamental derivations using explicit forces.

To begin, it is postulated that the motion of a particle is characterized by a modulated transverse wave. Here, de Broglie's relation, $p_e = \frac{\hbar}{\lambda_e}$, pertains to the carrier envelope's horizontal periodicity with wavelength $\lambda_e$, while the associated momentum amplitude $p$ of the particle in one dimension obeys the following relationships in terms of modulated wavevector $\tilde{k}$:

$$ \pm p = \hbar\chi_{\pm}\tilde{k} = \hbar\alpha_{\parallel}\hat{\chi}_{\pm}\tilde{k} \equiv \pm\alpha_{\parallel}\hbar k_e \equiv \pm\alpha_{\parallel}p_e \\ = \frac{h}{\lambda} \qquad (1) $$

where $\lambda$ is defined further on.

The chiral operator $\hat{\chi}_{\pm}$ is associated with a spin's projection onto the direction of motion, known as helicity, and acts upon the wave characterization's envelope only, not the carrier component, analyzed separately. As will be shown, it provides a phase correction that is necessary to keep a particle moving in the same direction. The properties of $\hat{\chi}_{\pm}$ are as follows:

$$ \hat{\chi}_{\pm}f(\phi_e(x))g(\phi_e(x)) = f(\hat{\chi}_{\pm}\phi_e(x))g(\phi_e(x)) \qquad (2) \\ = f\left(\pm\left(\frac{\pi}{2}\hat{x}\cdot\hat{x}_o - \phi_e(x)\right)\right)g(\phi_e(x)) $$

$$ \hat{\chi}_{\pm}^{-1} = \hat{\chi}_{\pm} \qquad (3) $$

$$ \hat{\chi}_{\pm}\frac{d}{dt} = \frac{d}{dt}\hat{\chi}_{\pm} \qquad (3) $$

where $\hat{x}$ and $\hat{x}_o$ are positional unit vectors assigned to the quanta of the particle and carrier propagation, respectively, while $f$ and $g$ are phase-periodic functions
dependent on the quanta's position in their simplest, irreducible form. For example, if functions f and g can be combined into a harmonic dependence on phase, this should be done first before applying the $\hat{\chi}_{\pm}$ operator. Hence, it is acceptable if g were to become unity. If f and g do not depend on the quanta's location or the phase dependence is not cyclical, then the $\hat{\chi}_{\pm}$ operator has no effect.

The velocity and its classical, phase decoherent average $\bar{v}$ is expressed by the following relationships in terms of modulated wavelength $\lambda$:

$$ \pm v = \hat{\chi}_{\pm} \tilde{\lambda} \nu \kappa \equiv \lambda \nu \kappa \equiv v_e \kappa \\ \bar{v} = \lambda_e \nu_e = (\lambda_a \alpha_{\parallel}) (\alpha_{\parallel}^{-1} \nu_a) = \lambda_a \nu_a \quad (4) $$

where $v_e$ represents a spatially varying phase velocity and $\kappa$ is an oscillating component due to the carrier. As shown, $\alpha_{\parallel}$ converts the traditional momentum de Broglie wavelength $\lambda_e$ and frequency $\nu_e$, characterizing the periodicity of the envelope, to the wavelength amplitude $\lambda_a$ and frequency amplitude $\nu_a$ characterizing the periodicity of the carrier. The quantities $\lambda$ and $\nu$ are functions in general, while the de Broglie and carrier parameters are constants only for a free particle.

The next relation is a classical correspondence limit of the well-known relationship between frequency and momentum for light to include free particles with rest mass $m_o$:

$$ h \nu_e \leftrightarrow \bar{p} \bar{v} : m_o \gamma \bar{v}^2 \quad (5) $$

Substituting $\bar{v} = \lambda_e \nu_e$ into (5) and introducing the conversion factor $\alpha_{\parallel}$ from (4) between $\lambda_e$, $\nu_e$, and variables $\lambda$, $\nu$ normalized by amplitudes $\lambda_a$ and $\nu_a$, with the introduction of quantum mass $m$ in place of $m_o$, the following variable-equivalent to (5) for particles that aren't necessarily free is generated:

$$ m = \frac{h}{\gamma \lambda_e^2(\vec{x}) \nu_e(\vec{x})} = \frac{h}{\alpha_{\parallel} \gamma \lambda^2 \nu} \quad (6) $$

where all quantities represented are real-valued.

Finally, with possible application to Monte Carlo calculations for many-body problems [20], the following modification of Newton's second law is proposed:

$$ \mathbf{F} = \frac{dp}{dt} = p_F \nu' \langle \phi'_F | \hat{F} | \phi \rangle = \frac{h \nu'}{\lambda_F} \langle \phi'_F | \hat{F} | \phi \rangle \quad (7) $$

where $\hat{F}$ is an operator and $\phi$ is the quantum phase of an interacting component, measured in radians, with $|\phi\rangle$ referring to the real value of its quantum state, unless otherwise noted, ranging between $\pm 1$ in value.

This introduces the impact of phase mediation with an interacting partner's field having a phase $\phi'_F$. As an additional convenience, the amplitude of $|\mathbf{F}| = \frac{dp}{dt}$ is equated to the exchange momentum $p_F$ multiplied by rate $\nu'$, such as contributed by virtual particle exchange,

expressed alternatively with de Broglie's relation, $\frac{h\nu'}{\lambda_F}$. The projection $\langle \phi'_F | \hat{F} | \phi \rangle$ is normalized in $\mathbb{R}$ such that $|\mathbf{F}|$ is equivalent to its amplitude in the classical correspondence limit when the phase dependency has become decoherent.

A modification to both classical and quantum theory, phase mediation is the process whereby a particle's quantum phase $\phi$ reacts to $\phi'_F$, and in turn, the interacting partner's field with phase $\phi'_F$ reacts to $\phi$. Over time, a steady-state equilibrium is achieved if the phase projection term is independent of time, which does not rule out the possibility that it depends on position. The interaction of phases is discussed for the free particle case further down, a constant applied force in Section IV, and a study of particle collisions in Section VI. Also, phase mediation is the suggested mechanism of entanglement that gives rise to Bose and Fermi statistics, for example, modifying particle-particle interactions through different phase/spin alignments to ensure the lowest exchange-force energy, as discussed in Appendix C.

Assuming a well-behaved or ergodic system [21] with cylindrical symmetry, the following is a 1-D steady-state equation for the particle's dynamics in terms of amplitudes using (7) combined with (1), while also specifying $\lambda(\vec{x}, t) = \lambda(\vec{x}(t))$:

$$ \frac{h \nu'}{\lambda_F} \langle \phi'_F | \hat{F} | \phi \rangle = \frac{d}{dt} \left( \frac{h}{\lambda} \right) = - \frac{h}{\lambda^2} \frac{d\lambda}{dt} = - \frac{h}{\lambda^2} \frac{d\lambda}{d\vec{x}} \cdot \frac{d\vec{x}}{dt} \\ = - \frac{h}{\lambda^2} \frac{d\lambda}{d\vec{x}} \cdot \vec{v}_e \quad (8) $$

The envelope velocity, $\vec{v}_e$, represents an average of the carrier's oscillation, which is a component solved for separately. Using $\vec{v}_e = \lambda \nu \hat{v}$ from (4) and rearranging terms, the simplified version of (8) is obtained:

$$ \frac{d\lambda}{d\vec{x}} \cdot \hat{v} = - \frac{\lambda}{\lambda_F} \frac{\nu'}{\nu} \langle \phi'_F | \hat{F} | \phi \rangle \quad (9) $$

where any time dependence involved with $\langle \phi'_F | \hat{F} | \phi \rangle$ contributes to noise.

To determine the equation for the probability of a single particle, the propagator $\langle \psi(x_o) | \psi(x) \rangle$ in Feynman's path integral formulation is used as a guide. Assuming $x_o$ is determined by a boundary condition, in one dimension the square of the norm of the propagator represents the probability that is being sought. The propagator is determined by a summation over all paths between $x_o$ and $x$ with weights assigned to the action $S$ of each path given by:

$$ w = e^{i \frac{S}{\hbar}} \quad (10) $$

It will now be shown why this propagator can be expressed with a superposition of solutions to (9). To start, a well-known relationship between the differential action and an action-angle variable will be derived. The action $S$ is equated to the work done by the force generated
dependent on the quanta's position in their simplest, irreducible form. For example, if functions $f$ and $g$ can be combined into a harmonic dependence on phase, this should be done first before applying the $\hat{\chi}_{\pm}$ operator. Hence, it is acceptable if $g$ were to become unity. If $f$ and $g$ do not depend on the quanta's location or the phase dependence is not cyclical, then the $\hat{\chi}_{\pm}$ operator has no effect.

The velocity and its classical, phase decoherent average $\bar{v}$ is expressed by the following relationships in terms of modulated wavelength $\lambda$:

$$ \pm v = \hat{\chi}_{\pm} \tilde{\lambda} \nu \kappa \equiv \lambda \nu \kappa \equiv v_e \kappa \\ \bar{v} = \lambda_e \nu_e = (\lambda_a \alpha_{\parallel}) (\alpha_{\parallel}^{-1} \nu_a) = \lambda_a \nu_a \quad (4) $$

where $v_e$ represents a spatially varying phase velocity and $\kappa$ is an oscillating component due to the carrier. As shown, $\alpha_{\parallel}$ converts the traditional momentum de Broglie wavelength $\lambda_e$ and frequency $\nu_e$, characterizing the periodicity of the envelope, to the wavelength amplitude $\lambda_a$ and frequency amplitude $\nu_a$ characterizing the periodicity of the carrier. The quantities $\lambda$ and $\nu$ are functions in general, while the de Broglie and carrier parameters are constants only for a free particle.

The next relation is a classical correspondence limit of the well-known relationship between frequency and momentum for light to include free particles with rest mass $m_o$:

$$ h \nu_e \leftrightarrow \bar{p} \bar{v} : m_o \gamma \bar{v}^2 \quad (5) $$

Substituting $\bar{v} = \lambda_e \nu_e$ into (5) and introducing the conversion factor $\alpha_{\parallel}$ from (4) between $\lambda_e$, $\nu_e$, and variables $\lambda$, $\nu$ normalized by amplitudes $\lambda_a$ and $\nu_a$, with the introduction of quantum mass $m$ in place of $m_o$, the following variable-equivalent to (5) for particles that aren't necessarily free is generated:

$$ m = \frac{h}{\gamma \lambda_e^2(\vec{x}) \nu_e(\vec{x})} = \frac{h}{\alpha_{\parallel} \gamma \lambda^2 \nu} \quad (6) $$

where all quantities represented are real-valued.

Finally, with possible application to Monte Carlo calculations for many-body problems [20], the following modification of Newton's second law is proposed:

$$ \mathbf{F} = \frac{dp}{dt} = p_F \nu' \langle \phi'_F | \hat{F} | \phi \rangle = \frac{h \nu'}{\lambda_F} \langle \phi'_F | \hat{F} | \phi \rangle \quad (7) $$

where $\hat{F}$ is an operator and $\phi$ is the quantum phase of an interacting component, measured in radians, with $|\phi\rangle$ referring to the real value of its quantum state, unless otherwise noted, ranging between $\pm 1$ in value.

This introduces the impact of phase mediation with an interacting partner's field having a phase $\phi'_F$. As an additional convenience, the amplitude of $|\mathbf{F}| = \frac{dp}{dt}$ is equated to the exchange momentum $p_F$ multiplied by rate $\nu'$, such as contributed by virtual particle exchange,

expressed alternatively with de Broglie's relation, $\frac{h\nu'}{\lambda_F}$. The projection $\langle \phi'_F | \hat{F} | \phi \rangle$ is normalized in $\mathbb{R}$ such that $|\mathbf{F}|$ is equivalent to its amplitude in the classical correspondence limit when the phase dependency has become decoherent.

A modification to both classical and quantum theory, phase mediation is the process whereby a particle's quantum phase $\phi$ reacts to $\phi'_F$, and in turn, the interacting partner's field with phase $\phi'_F$ reacts to $\phi$. Over time, a steady-state equilibrium is achieved if the phase projection term is independent of time, which does not rule out the possibility that it depends on position. The interaction of phases is discussed for the free particle case further down, a constant applied force in Section IV, and a study of particle collisions in Section VI. Also, phase mediation is the suggested mechanism of entanglement that gives rise to Bose and Fermi statistics, for example, modifying particle-particle interactions through different phase/spin alignments to ensure the lowest exchange-force energy, as discussed in Appendix C.

Assuming a well-behaved or ergodic system [21] with cylindrical symmetry, the following is a 1-D steady-state equation for the particle's dynamics in terms of amplitudes using (7) combined with (1), while also specifying $\lambda(\vec{x}, t) = \lambda(\vec{x}(t))$:

$$ \frac{h \nu'}{\lambda_F} \langle \phi'_F | \hat{F} | \phi \rangle = \frac{d}{dt} \left( \frac{h}{\lambda} \right) = - \frac{h}{\lambda^2} \frac{d\lambda}{dt} = - \frac{h}{\lambda^2} \frac{d\lambda}{d\vec{x}} \cdot \frac{d\vec{x}}{dt} \\ = - \frac{h}{\lambda^2} \frac{d\lambda}{d\vec{x}} \cdot \vec{v}_e \quad (8) $$

The envelope velocity, $\vec{v}_e$, represents an average of the carrier's oscillation, which is a component solved for separately. Using $\vec{v}_e = \lambda \nu \hat{v}$ from (4) and rearranging terms, the simplified version of (8) is obtained:

$$ \frac{d\lambda}{d\vec{x}} \cdot \hat{v} = - \frac{\lambda}{\lambda_F} \frac{\nu'}{\nu} \langle \phi'_F | \hat{F} | \phi \rangle \quad (9) $$

where any time dependence involved with $\langle \phi'_F | \hat{F} | \phi \rangle$ contributes to noise.

To determine the equation for the probability of a single particle, the propagator $\langle \psi(x_o) | \psi(x) \rangle$ in Feynman's path integral formulation is used as a guide. Assuming $x_o$ is determined by a boundary condition, in one dimension the square of the norm of the propagator represents the probability that is being sought. The propagator is determined by a summation over all paths between $x_o$ and $x$ with weights assigned to the action $S$ of each path given by:

$$ w = e^{i \frac{S}{\hbar}} \quad (10) $$

It will now be shown why this propagator can be expressed with a superposition of solutions to (9). To start, a well-known relationship between the differential action and an action-angle variable will be derived. The action $S$ is equated to the work done by the force generated
from a conservative potential, $V(x)$:

$$ S = \int L dt' = \int_{t_o}^{t_o+dt} (T-V)dt' = \int_{x_o}^{x_o+dx} \frac{dp_e}{dt} dx' \\ = \int_{p_o}^{p_o+dp} v dp_e = p_e \frac{dx}{dt} \Big|_{t_o}^{t_o+dt} - \int_{t_o}^{t_o+dt} p_e dv \quad (11) $$

where $T$ is the kinetic energy and an integration by parts is performed on the second line. Forcing the remaining integration over an infinitesimal change in time, the differential action is obtained by removing the integral on the LHS and multiplying by $dt$ on the RHS:

$$ dS = Ldt = p_e dx = \hbar \hat{\chi}_{\pm} \vec{k} \cdot d\vec{x} \quad (12) $$

where $p_e dx$ is dominant due to the second term on the RHS of (11) having a square of differentials $dvdt$. Here, $k_e \equiv \hat{\chi}_{\pm} \vec{k} \cdot \hat{x}$.

Since the differential action is linearly dependent on a wavevector, the norm of the path integration of the superposition of solutions to (9) expressed as a modulated wavevector $\tilde{k}$ evaluated at the position of the quanta is equivalent to the norm of the superposition of path integrations of $dS$ that give weighting factors (10) for the quanta. The proof of this is a multi-path extension of the modified Huygen's principle derivation for transverse waves of Section VII along with a generalization of the section's interference calculation, along with the squaring the norm of the superposition of terms in the form of (10), assuming constant wavevectors (solutions for the free particle case) in the exponential, which can be generalized based on the work in Section IV.

With the probability now a function of the norm of a superimposed weighting factor, with a dependence on complex function $\tilde{S}$:

$$ P = P\left(\Sigma e^{i\frac{\tilde{S}}{\hbar}}\right) \equiv P(|\tilde{S}|) \equiv P\left(\int \hat{\chi}_{\pm} \tilde{k} \cdot d\vec{x}\right) \quad (13) $$

the steady-state probability for detection of a particle inside a bin, $x$ to $x+dx$, is determined by the following:

$$ dP(\vec{x}) = 2P(\vec{x}) \frac{d|\tilde{S}|}{\hbar} = 2P(\vec{x})k_e(x)dx = 2P(\vec{x})\hat{\chi}_{\pm} \tilde{k} \cdot d\vec{x} \quad (14) $$

where the wavevectors $k_e$ and $\tilde{k}$ are real-valued. The factor of two is due to the square of the norm of the superimposed weighting factors, and it also can be attributed to the recognition that both positive and negative phase solutions yield the same action.

Another way to view this is to express $k_e = \frac{2\pi\nu}{v}$, which is derivable from (1) and (5), as the time $dt$ the particle is detectable inside a bin multiplying the envelope frequency, as shown:

$$ \hat{\chi}_{\pm} \frac{dP}{P} \propto k_e dx = k_e v dt \propto \nu_e dt = \nu_e \frac{dx}{v} \quad (15) $$

where $\hat{\chi}_{\pm}$ was applied on both sides of (14) and the property $\hat{\chi}_{\pm}^2 = 1$ was used.

Due to the 90° shift on phase that the $\hat{\chi}_{\pm}$ operator introduces when the quanta moves parallel to the carrier's direction of propagation in steady-state equilibrium, the LHS is proportional rather than inversely proportional to the probability of detection, $P$, by an ideal detector. The result is a probability that increases with both $dt$ and $\nu_e$: a hybrid between a classical phase-space calculation and the basic hypothesis behind Feynman's path integral formulation involving action whereby non-classical contributions to the probability for the quanta interfere less with decreases in $\nu_e$.

As discussed further on, this does not rule out interference nodes which occur in the limit that $dt \to 0$. In Section V, it is shown that frequency $\nu(x)$ is connected to the transverse amplitude of oscillation, which determines the size and speed of the particle's wave. The higher $\nu(x)$ is, the higher the amplitude and the more likely the particle's quanta becomes detectable within an arbitrarily small bin at an arbitrary location due to a slower propagation speed. In the other limit, for small transverse amplitude, $\nu(x)$ is small, but the wavelength, $\lambda(x)$, is large.

The solution to (14) is given by the following for a particle traveling in the $\vec{x}$ direction:

$$ P(\vec{x}) = |A_{\phi} \cdot \langle \vec{x} | \phi \rangle|^2 \quad (16) \\ = [A_R \operatorname{Re}(\langle \vec{x} | \phi \rangle)]^2 + [A_I \operatorname{Im}(\langle \vec{x} | \phi \rangle)]^2 $$

$$ \langle \vec{x} | \phi \rangle = e^{\pm \hat{\chi}_{\pm} (\int^x \tilde{k}(x') \cdot d\vec{x}' - i\phi_o)} \quad (17) $$

where $A_{\phi} = A_R + iA_I$ is complex in general and represents a statistical quantity. More on the meaning of this constant vector is presented further down. Its operation with $|\phi\rangle$ is a dyadic dot product such that $(1, i)$ are treated as orthogonal unit vectors in $\mathbb{C}$, which remain after the dot product is completed.

The theory so far states that the probability and wavevector from (9) are calculated first before state $|\phi\rangle$ is determined. However, the real-valued function $\tilde{k} \cdot \hat{x}$ representing the horizontal periodicity of the carrier envelope or de Broglie wavevector can be determined with the requirement that it be consistent with (9) and (17) such that (14) holds. Just as with Schrödinger's theory, it is possible to form an equation for $|\phi\rangle$ directly, provided the correct operators are used. Further down in this section, the Hamiltonian operator is derived for a free particle. The energy eigenstates of this operator can be used to calculate $|\phi\rangle$ as an alternative approach to the one presented here.

If one were to use standard wave mechanics to describe a free particle in a region of constant potential $V_o \equiv \pm h\nu'$ with energy $E - m_o c^2 > V_o$, the solution would be a momentum eigenstate of form:

$$ \langle x | \psi \rangle = e^{\pm ik_e x} \quad (18) $$

with a probability density $P = 1$.
from a conservative potential, $V(x)$:

$$ S = \int L dt' = \int_{t_o}^{t_o+dt} (T-V)dt' = \int_{x_o}^{x_o+dx} \frac{dp_e}{dt} dx' \\ = \int_{p_o}^{p_o+dp} v dp_e = p_e \frac{dx}{dt} \Big|_{t_o}^{t_o+dt} - \int_{t_o}^{t_o+dt} p_e dv \quad (11) $$

where $T$ is the kinetic energy and an integration by parts is performed on the second line. Forcing the remaining integration over an infinitesimal change in time, the differential action is obtained by removing the integral on the LHS and multiplying by $dt$ on the RHS:

$$ dS = L dt = p_e dx = \hbar \hat{\chi}_{\pm} \vec{k} \cdot d\vec{x} \quad (12) $$

where $p_e dx$ is dominant due to the second term on the RHS of (11) having a square of differentials $dvdt$. Here, $k_e \equiv \hat{\chi}_{\pm} \vec{k} \cdot \hat{x}$.

Since the differential action is linearly dependent on a wavevector, the norm of the path integration of the superposition of solutions to (9) expressed as a modulated wavevector $\tilde{k}$ evaluated at the position of the quanta is equivalent to the norm of the superposition of path integrations of $dS$ that give weighting factors (10) for the quanta. The proof of this is a multi-path extension of the modified Huygen's principle derivation for transverse waves of Section VII along with a generalization of the section's interference calculation, along with the squaring the norm of the superposition of terms in the form of (10), assuming constant wavevectors (solutions for the free particle case) in the exponential, which can be generalized based on the work in Section IV.

With the probability now a function of the norm of a superimposed weighting factor, with a dependence on complex function $\tilde{S}$:

$$ P = P \left( \Sigma e^{i \frac{\tilde{S}}{\hbar}} \right) \equiv P(|\tilde{S}|) \equiv P \left( \int \hat{\chi}_{\pm} \tilde{k} \cdot d\vec{x} \right) \quad (13) $$

the steady-state probability for detection of a particle inside a bin, $x$ to $x+dx$, is determined by the following:

$$ dP(\vec{x}) = 2P(\vec{x}) \frac{d|\tilde{S}|}{\hbar} = 2P(\vec{x})k_e(x)dx = 2P(\vec{x})\hat{\chi}_{\pm} \tilde{k} \cdot d\vec{x} \quad (14) $$

where the wavevectors $k_e$ and $\tilde{k}$ are real-valued. The factor of two is due to the square of the norm of the superimposed weighting factors, and it also can be attributed to the recognition that both positive and negative phase solutions yield the same action.

Another way to view this is to express $k_e = \frac{2\pi\nu}{v}$, which is derivable from (1) and (5), as the time $dt$ the particle is detectable inside a bin multiplying the envelope frequency, as shown:

$$ \hat{\chi}_{\pm} \frac{dP}{P} \propto k_e dx = k_e v dt \propto \nu_e dt = \nu_e \frac{dx}{v} \quad (15) $$

where $\hat{\chi}_{\pm}$ was applied on both sides of (14) and the property $\hat{\chi}_{\pm}^2 = 1$ was used.

Due to the 90° shift on phase that the $\hat{\chi}_{\pm}$ operator introduces when the quanta moves parallel to the carrier's direction of propagation in steady-state equilibrium, the LHS is proportional rather than inversely proportional to the probability of detection, $P$, by an ideal detector. The result is a probability that increases with both $dt$ and $\nu_e$: a hybrid between a classical phase-space calculation and the basic hypothesis behind Feynman's path integral formulation involving action whereby non-classical contributions to the probability for the quanta interfere less with decreases in $\nu_e$.

As discussed further on, this does not rule out interference nodes which occur in the limit that $dt \to 0$. In Section V, it is shown that frequency $\nu(x)$ is connected to the transverse amplitude of oscillation, which determines the size and speed of the particle's wave. The higher $\nu(x)$ is, the higher the amplitude and the more likely the particle's quanta becomes detectable within an arbitrarily small bin at an arbitrary location due to a slower propagation speed. In the other limit, for small transverse amplitude, $\nu(x)$ is small, but the wavelength, $\lambda(x)$, is large.

The solution to (14) is given by the following for a particle traveling in the $\vec{x}$ direction:

$$ P(\vec{x}) = |A_{\phi} \cdot \langle \vec{x} | \phi \rangle|^2 \quad (16) \\ = [A_R \operatorname{Re}(\langle \vec{x} | \phi \rangle)]^2 + [A_I \operatorname{Im}(\langle \vec{x} | \phi \rangle)]^2 $$

$$ \langle \vec{x} | \phi \rangle = e^{\pm \hat{\chi}_{\pm} (\int^x \tilde{k}(x') \cdot d\vec{x}' - i\phi_o)} \quad (17) $$

where $A_{\phi} = A_R + iA_I$ is complex in general and represents a statistical quantity. More on the meaning of this constant vector is presented further down. Its operation with $|\phi\rangle$ is a dyadic dot product such that $(1, i)$ are treated as orthogonal unit vectors in $\mathbb{C}$, which remain after the dot product is completed.

The theory so far states that the probability and wavevector from (9) are calculated first before state $|\phi\rangle$ is determined. However, the real-valued function $\tilde{k} \cdot \hat{x}$ representing the horizontal periodicity of the carrier envelope or de Broglie wavevector can be determined with the requirement that it be consistent with (9) and (17) such that (14) holds. Just as with Schrödinger's theory, it is possible to form an equation for $|\phi\rangle$ directly, provided the correct operators are used. Further down in this section, the Hamiltonian operator is derived for a free particle. The energy eigenstates of this operator can be used to calculate $|\phi\rangle$ as an alternative approach to the one presented here.

If one were to use standard wave mechanics to describe a free particle in a region of constant potential $V_o \equiv \pm h\nu'$ with energy $E - m_o c^2 > V_o$, the solution would be a momentum eigenstate of form:

$$ \langle x | \psi \rangle = e^{\pm ik_e x} \quad (18) $$

with a probability density $P = 1$.
Solving this case with the theory presented here, the arbitrariness of the gauge allows one to set $\nu'$ and the RHS of (9) to zero. This is one of the reasons why the force associated with a constant potential is virtual. The choice of gauge takes the phase projection term $\langle\phi'_F|\hat{F}|\phi\rangle$ out of the solution to the differential equation, which will be important for further analysis discussed below. Given the particle's quanta position is not an explicit dynamical variable in this example, (9) corresponds to a momentum eigenvalue problem in Schrödinger's treatment.

Its solution for $\nu' = 0$ is a complex constant in $\mathbb{C}$, however since $\phi$ must be bounded everywhere, $\tilde{k} \equiv i\tilde{k}_c$. Using $\vec{x}$ to represent the location of the particle's quanta and $\hat{x}_o$ as the unit vector normal to the transverse oscillation of its carrier, the consideration of (2) and (17) defines the following phases:

$$ \phi(x) = \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - \phi_e(x) \quad (19) $$

$$ \phi_e(x) = \tilde{k}_c \cdot \vec{x} - \phi_o = k_e x - \phi_o \quad (20) $$

The wavevector amplitude is determined after first substituting $\tilde{k} \equiv i\tilde{k}_c$ into (17), yielding $\langle x|\phi \rangle = e^{\pm i\phi(x)}$, matching Schrödinger's momentum eigenstate (18) with allowable phase offsets. It is then necessary to find $A_\phi$ to calculate the probability.

To start, a free particle in a region of constant potential $V_o \equiv \pm h\nu'$ with energy $E - m_o c^2 > V_o$ is likely to have an arbitrary value for $\phi_o$. Representing a statistical ensemble of these particles, one could attempt to create a superposition of states $|\phi\rangle$ with different offsets, however for every $|\phi\rangle$ there would be a matching $|-\phi\rangle$ that would nullify it. Instead, if $A_\phi = A_R$ for each particle state, then a summation over particles of the square of their norms $\cos^2\phi(x)$ with arbitrary offsets would have matching terms $\cos^2(\phi(x) - \pi/2)$, giving the result $P = 1$.

This is the reason a statistical ensemble referred to by $A_\phi$ is aligned with probabilities, not states $|\phi\rangle$. Hence, to represent the arbitrary collection of phases with a single solution, both the real and imaginary component of $A_\phi$ is required. With this, $\cos^2\phi(x)$ matches up to $\sin^2\phi(x)$ if the real and imaginary coefficients are equal, and this gives $P = 1$. Though one phase is attributed to $\phi(x)$ (more will be said about this later), implicitly all phase offsets are represented if the $A_\phi$ coefficient is chosen such that there is an equal chance to detect a particle at any location. If $|A_R| \neq |A_I|$, this represents a bias of phase offsets due to some interaction or boundary condition and if $A_\phi = A_R$ or $A_I$, then a specific phase offset is being referenced.

In consideration of a single particle, then, with $\hat{x}$ and $\hat{x}_o$ as constants, it has a specific phase offset and its probability of location is of form:

$$ P(x) = [A_R \operatorname{Re}(\langle x|\phi \rangle)]^2 = N \cos^2 \phi(x) \quad (21) $$

where $N = A_R^2$ was an arbitrary choice over using the imaginary $A_\phi$ coefficient with a different phase offset, $\phi_o$.

Suppose the quanta of a particle were to experience an impulse and move perpendicular to its original motion at its peak probability location while the carrier was still traveling unaffected (there is a delayed response as discussed in Section VII). Originally, $\hat{x} \parallel \hat{x}_o$, so the probability was $\sin^2(k_e x - \phi_o)$, but now that $\hat{x} \cdot \hat{x}_o = 0$, it becomes $\cos^2(k_e x - \phi'_o)$ where:

$$ \phi'_o = \phi_o + \frac{\pi}{2} \quad (22) $$

This sudden shift of phase offset ensures that the probability remains continuous, however another way to view the transition is to suggest that $A_\phi$ became imaginary. Then, $\phi_o$ is unaffected and the probability distribution remains the same at $\sin^2(k_e x - \phi_o)$.

Since (21) is also a solution inside a square well potential with $\hat{x} \cdot \hat{x}_o = 1$ and $\phi_o$ chosen to satisfy boundary conditions, it applies to a free particle with or without boundaries present, representative of its unique, intrinsic dynamics (examined further on), in contrast to arbitrary, hypothetical wave packet constructs in Schrödinger's theory that characterize measurement uncertainty. In this sense, the particle's dynamics are "localized" with the inclusion of non-local time-averaged influences from the boundaries on its phase offset $\phi_o$ when steady-state analysis is performed. The solution thus satisfies the criteria that a box of infinite length and free space are equivalent, unlike Schrödinger's treatment which divides the two problems into a plane wave solution and particle-in-a-box solution.

As for the carrier term, the same arguments for the wavevector apply, only it will be assumed that propagation is along the x-axis with displacement $x_o$ and a wavevector given by $\tilde{k} \equiv ik_a \hat{x}_o$. The resultant solution for the carrier phase component in $\mathbb{R}$ can be expressed as:

$$ \kappa = 1 + \sin(k_a x_o - \phi_a) \quad (23) $$

where $\kappa$ is a multiplier of the wavelength amplitude consistent with standard modulated wave formulations [22] only here, the carrier behaves as the modulating component to the particle's motion.

Equating the derivative of $P(x)$ in (21) to $2\hat{\chi}_\pm \tilde{k} \cdot \hat{x} P(x)$ in (14) and attaching the $\alpha_\parallel$ coefficient from (1) identifies the following amplitude solution to (9) in $\mathbb{R}$ for steady-state equilibrium:

$$ \begin{aligned} k(x) &= \alpha_\parallel (\hat{k}_c \cdot \hat{x}) k_c \hat{\chi}_\pm \tan \phi(x) \\ &= \pm \alpha_\parallel k_e \tan \phi_e(x) \\ &= \pm k_a \tan(k_e x - \phi_o) \end{aligned} \quad (24) $$

$$ \lambda(x) = \frac{2\pi}{k(x)} = \pm \lambda_a \cot(k_e x - \phi_o) \quad (25) $$

where $\hat{\chi}_\pm$ was multiplied on both sides of (14) and its square set to 1 on the RHS, $\lambda_c = \frac{2\pi}{k_c}$, and for consistency
Solving this case with the theory presented here, the arbitrariness of the gauge allows one to set $\nu'$ and the RHS of (9) to zero. This is one of the reasons why the force associated with a constant potential is virtual. The choice of gauge takes the phase projection term $\langle\phi'_F|\hat{F}|\phi\rangle$ out of the solution to the differential equation, which will be important for further analysis discussed below. Given the particle's quanta position is not an explicit dynamical variable in this example, (9) corresponds to a momentum eigenvalue problem in Schrödinger's treatment.

Its solution for $\nu' = 0$ is a complex constant in $\mathbb{C}$, however since $\phi$ must be bounded everywhere, $\tilde{k} \equiv i\tilde{k}_c$. Using $\vec{x}$ to represent the location of the particle's quanta and $\hat{x}_o$ as the unit vector normal to the transverse oscillation of its carrier, the consideration of (2) and (17) defines the following phases:

$$ \phi(x) = \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - \phi_e(x) \quad (19) $$

$$ \phi_e(x) = \tilde{k}_c \cdot \vec{x} - \phi_o = k_e x - \phi_o \quad (20) $$

The wavevector amplitude is determined after first substituting $\tilde{k} \equiv i\tilde{k}_c$ into (17), yielding $\langle x|\phi \rangle = e^{\pm i\phi(x)}$, matching Schrödinger's momentum eigenstate (18) with allowable phase offsets. It is then necessary to find $A_\phi$ to calculate the probability.

To start, a free particle in a region of constant potential $V_o \equiv \pm h\nu'$ with energy $E - m_o c^2 > V_o$ is likely to have an arbitrary value for $\phi_o$. Representing a statistical ensemble of these particles, one could attempt to create a superposition of states $|\phi\rangle$ with different offsets, however for every $|\phi\rangle$ there would be a matching $|-\phi\rangle$ that would nullify it. Instead, if $A_\phi = A_R$ for each particle state, then a summation over particles of the square of their norms $\cos^2\phi(x)$ with arbitrary offsets would have matching terms $\cos^2(\phi(x) - \pi/2)$, giving the result $P = 1$.

This is the reason a statistical ensemble referred to by $A_\phi$ is aligned with probabilities, not states $|\phi\rangle$. Hence, to represent the arbitrary collection of phases with a single solution, both the real and imaginary component of $A_\phi$ is required. With this, $\cos^2\phi(x)$ matches up to $\sin^2\phi(x)$ if the real and imaginary coefficients are equal, and this gives $P = 1$. Though one phase is attributed to $\phi(x)$ (more will be said about this later), implicitly all phase offsets are represented if the $A_\phi$ coefficient is chosen such that there is an equal chance to detect a particle at any location. If $|A_R| \neq |A_I|$, this represents a bias of phase offsets due to some interaction or boundary condition and if $A_\phi = A_R$ or $A_I$, then a specific phase offset is being referenced.

In consideration of a single particle, then, with $\hat{x}$ and $\hat{x}_o$ as constants, it has a specific phase offset and its probability of location is of form:

$$ P(x) = [A_R \operatorname{Re}(\langle x|\phi \rangle)]^2 = N \cos^2 \phi(x) \quad (21) $$

where $N = A_R^2$ was an arbitrary choice over using the imaginary $A_\phi$ coefficient with a different phase offset, $\phi_o$.

Suppose the quanta of a particle were to experience an impulse and move perpendicular to its original motion at its peak probability location while the carrier was still traveling unaffected (there is a delayed response as discussed in Section VII). Originally, $\hat{x} \parallel \hat{x}_o$, so the probability was $\sin^2(k_e x - \phi_o)$, but now that $\hat{x} \cdot \hat{x}_o = 0$, it becomes $\cos^2(k_e x - \phi'_o)$ where:

$$ \phi'_o = \phi_o + \frac{\pi}{2} \quad (22) $$

This sudden shift of phase offset ensures that the probability remains continuous, however another way to view the transition is to suggest that $A_\phi$ became imaginary. Then, $\phi_o$ is unaffected and the probability distribution remains the same at $\sin^2(k_e x - \phi_o)$.

Since (21) is also a solution inside a square well potential with $\hat{x} \cdot \hat{x}_o = 1$ and $\phi_o$ chosen to satisfy boundary conditions, it applies to a free particle with or without boundaries present, representative of its unique, intrinsic dynamics (examined further on), in contrast to arbitrary, hypothetical wave packet constructs in Schrödinger's theory that characterize measurement uncertainty. In this sense, the particle's dynamics are "localized" with the inclusion of non-local time-averaged influences from the boundaries on its phase offset $\phi_o$ when steady-state analysis is performed. The solution thus satisfies the criteria that a box of infinite length and free space are equivalent, unlike Schrödinger's treatment which divides the two problems into a plane wave solution and particle-in-a-box solution.

As for the carrier term, the same arguments for the wavevector apply, only it will be assumed that propagation is along the x-axis with displacement $x_o$ and a wavevector given by $\tilde{k} \equiv ik_a \hat{x}_o$. The resultant solution for the carrier phase component in $\mathbb{R}$ can be expressed as:

$$ \kappa = 1 + \sin(k_a x_o - \phi_a) \quad (23) $$

where $\kappa$ is a multiplier of the wavelength amplitude consistent with standard modulated wave formulations [22] only here, the carrier behaves as the modulating component to the particle's motion.

Equating the derivative of $P(x)$ in (21) to $2\hat{\chi}_\pm \tilde{k} \cdot \hat{x} P(x)$ in (14) and attaching the $\alpha_\parallel$ coefficient from (1) identifies the following amplitude solution to (9) in $\mathbb{R}$ for steady-state equilibrium:

$$ \begin{aligned} k(x) &= \alpha_\parallel (\hat{k}_c \cdot \hat{x}) k_c \hat{\chi}_\pm \tan \phi(x) \\ &= \pm \alpha_\parallel k_e \tan \phi_e(x) \\ &= \pm k_a \tan(k_e x - \phi_o) \end{aligned} \quad (24) $$

$$ \lambda(x) = \frac{2\pi}{k(x)} = \pm \lambda_a \cot(k_e x - \phi_o) \quad (25) $$

where $\hat{\chi}_\pm$ was multiplied on both sides of (14) and its square set to 1 on the RHS, $\lambda_c = \frac{2\pi}{k_c}$, and for consistency
with the carrier the following definitions are made:

$$
\begin{align}
\lambda_a &= \frac{2\pi}{k_a} \equiv (\hat{k}_c \cdot \hat{x})\lambda_c \nonumber \\
\lambda_e &= \frac{2\pi}{k_e} \equiv \frac{\lambda_c}{\hat{k}_c \cdot \hat{x}} \tag{26} \\
\alpha_{\parallel} &= \frac{1}{(\hat{k}_c \cdot \hat{x})^2} = \frac{k_a}{k_e} = \frac{\lambda_e}{\lambda_a} = \frac{\lambda_c^2}{\lambda_a^2} \nonumber
\end{align}
$$

In standard wave mechanics, $\alpha_{\parallel} = 1$, which from (26) states that $\lambda_e = \lambda_a$, i.e. the envelope and transverse oscillation within it have the same frequency. However, one interesting problem that is rarely discussed is that Schrödinger's theory states that a particle's "frequency" is proportional to its energy, but this frequency isn't the same as the one paired-up with the de Broglie wavelength. One only needs to consider the case when $\lambda \to \infty$ for low momentum, and $\nu$ must go to zero to have zero velocity, however $h\nu$ must approach the mass-energy of the particle. This discrepancy is responsible for the speed-of-light violating calculation of the phase velocity in Schrödinger's theory. To prove the problem is addressed in this framework, some preliminary calculations need to be performed.

Having no field lines, the force **F** associated with a constant potential $V_o$ and work done is virtual, i.e. for $E - m_o c^2 > V_o$ the interaction strength is proportional to the particle's momentum, which determines the interaction wavelength, $\lambda_F$. As to whether such virtual interactions with the potential as the medium could make non-local exchanges possible, thus satisfying Bell's Theorem [23], that is an open question discussed further in Section VI.

In summary, virtual force **F** for $E - m_o c^2 > V_o$ should have these properties:

$$
\begin{align}
\lambda_F \propto \lambda \tag{27} \\
\mathbf{F}'\text{-s direction} = (\text{sign of spin's direction}) \times (\text{sign of } |\phi\rangle) \tag{28}
\end{align}
$$

$$
\lambda \to 0 \implies \mathbf{F} \to 0 \tag{29}
$$

thus insuring no net work is done. The virtual force's
dependency on helicity is argued in Appendix A, but for
now these properties will be used to compute $\lambda_F$.

From the analysis in Section VI, the background poten-
tial has a step function response to the quantum phase of
a particle. As such, the Fourier component that survives
the mediation process gives $\phi'_F$ the following state:

$$
\langle \phi'_{F} | x' \rangle = \frac{4}{\pi} \sin \phi(x') = \frac{4}{\pi} \cos \phi_{e}(x') \quad (30)
$$

assuming $\hat{x} \parallel \hat{x}_o$. The property referenced by (28) sug-
gests the following operator for the force assigned to a
constant potential:

$$
F \hat{F} = \frac{h \nu'}{\lambda_F} \hat{h}_s \cdot \hat{\chi}_{\pm} \quad (31)
$$

where $\hat{h}_s$ represents the helicity's direction. Since $\hat{\chi}_{\pm}$ introduces up to a 90° correction to the particle's phase rotation, the phase projection of the spin's interaction with the constant potential then becomes:

$$
\langle x' | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle = \operatorname{sgn} (\sin \phi_e(x')) \cos \phi_e(x') \quad (32)
$$

where the ± sign of $\hat{\chi}_{\pm}$ now has an explicit functional dependence.

The work done by |**F**| flips every quarter cycle. To compute its amplitude, the basis states $|x'\rangle \langle x'|$ are inserted into (7), then the work expression is equated to the RHS:

$$
\int_0^{\frac{\lambda}{4}} |\mathbf{F}| \, dx' = \int_0^{\frac{\lambda}{4}} \frac{h\nu'}{\lambda_F} \langle \phi'_F | x' \rangle \langle x' | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle \, dx' = h\nu_a \quad (33)
$$

where $h\nu_a$ is the available interaction energy per quarter cycle. Substituting into (33) the phase state terms using (20) with a change of variables to $x'' = k_e x' - \phi_o$, one obtains:

$$
\frac{h\nu'}{\lambda_F} \frac{4}{\pi k_e} \int_0^{\frac{\pi}{2}} \cos^2(x'') dx'' = \frac{h\nu'}{\lambda_F k_e} = \frac{h\nu'}{\lambda_F \alpha_{\parallel}^{-1} k} = h\nu_a \quad (34)
$$

Thus, for a region of constant potential $V_o \equiv \pm h\nu'$, its associated virtual force has up to energy $h\nu_a$ for exchange, but since $\vec{k}_e \neq \vec{k}_a \equiv \vec{k}$, and amplitudes are involved in the calculation, the actual exchange energy is:

$$
h\nu_x = h\alpha_{\parallel}^{-1}\nu_a \quad (35)
$$

with momentum exchange wavelength $\lambda_F$ given by:

$$
\lambda_F = \pm \frac{\nu'}{\nu_x} \frac{\lambda}{2\pi} \qquad (36)
$$

As an aside, this derivation did not explicitly represent
the ± signs attached to $\nu'$ in (33) as they cancel out in
(34).

Recalling the discussion leading up to (21), the statis-
tics attributed to the arbitrary phase offsets for free par-
ticles should also be possible for the solution to (9) in
order to have consistency between the probability calcu-
lation and the modified equation of Newton’s second law.
Yet, in steady-state problems, specific phase offsets are
necessarily assigned to boundary conditions.

Consider, however, that there is one component that isn't represented explicitly in the statistical analysis or the dynamic equations. That is time. For traveling waves, the phase $\phi_e$ is written more generally as the following:

$$
\phi_e(x,t) = k_e x - \phi_x - \omega_e t + \phi_t \quad (37)
$$

where $\omega_e = 2\pi\nu_e$ and $\phi_x$ is a constant determined by
boundary conditions while $\phi_t$ is associated with time-
dependent or initial conditions.

The statistical averaging that must be done for char-
acterizing multiple particles in steady-state problems has
with the carrier the following definitions are made:

$$
\begin{align}
\lambda_a &= \frac{2\pi}{k_a} \equiv (\hat{k}_c \cdot \hat{x})\lambda_c \nonumber \\
\lambda_e &= \frac{2\pi}{k_e} \equiv \frac{\lambda_c}{\hat{k}_c \cdot \hat{x}} \tag{26} \\
\alpha_{\parallel} &= \frac{1}{(\hat{k}_c \cdot \hat{x})^2} = \frac{k_a}{k_e} = \frac{\lambda_e}{\lambda_a} = \frac{\lambda_c^2}{\lambda_a^2} \nonumber
\end{align}
$$

In standard wave mechanics, $\alpha_{\parallel} = 1$, which from (26) states that $\lambda_e = \lambda_a$, i.e. the envelope and transverse oscillation within it have the same frequency. However, one interesting problem that is rarely discussed is that Schrödinger's theory states that a particle's "frequency" is proportional to its energy, but this frequency isn't the same as the one paired-up with the de Broglie wavelength. One only needs to consider the case when $\lambda \to \infty$ for low momentum, and $\nu$ must go to zero to have zero velocity, however $h\nu$ must approach the mass-energy of the particle. This discrepancy is responsible for the speed-of-light violating calculation of the phase velocity in Schrödinger's theory. To prove the problem is addressed in this framework, some preliminary calculations need to be performed.

Having no field lines, the force **F** associated with a constant potential $V_o$ and work done is virtual, i.e. for $E - m_o c^2 > V_o$ the interaction strength is proportional to the particle's momentum, which determines the interaction wavelength, $\lambda_F$. As to whether such virtual interactions with the potential as the medium could make non-local exchanges possible, thus satisfying Bell's Theorem [23], that is an open question discussed further in Section VI.

In summary, virtual force **F** for $E - m_o c^2 > V_o$ should have these properties:

$$
\begin{align}
\lambda_F \propto \lambda \tag{27} \\
\mathbf{F}'\text{-s direction} = (\text{sign of spin's direction}) \times (\text{sign of } |\phi\rangle) \tag{28}
\end{align}
$$

$$
\lambda \to 0 \implies \mathbf{F} \to 0 \tag{29}
$$

thus insuring no net work is done. The virtual force's
dependency on helicity is argued in Appendix A, but for
now these properties will be used to compute $\lambda_F$.

From the analysis in Section VI, the background poten-
tial has a step function response to the quantum phase of
a particle. As such, the Fourier component that survives
the mediation process gives $\phi'_F$ the following state:

$$
\langle \phi'_{F} | x' \rangle = \frac{4}{\pi} \sin \phi(x') = \frac{4}{\pi} \cos \phi_{e}(x') \quad (30)
$$

assuming $\hat{x} \parallel \hat{x}_o$. The property referenced by (28) sug-
gests the following operator for the force assigned to a
constant potential:

$$
F \hat{F} = \frac{h \nu'}{\lambda_F} \hat{h}_s \cdot \hat{\chi}_{\pm} \quad (31)
$$

where $\hat{h}_s$ represents the helicity's direction. Since $\hat{\chi}_{\pm}$ introduces up to a 90° correction to the particle's phase rotation, the phase projection of the spin's interaction with the constant potential then becomes:

$$
\langle x' | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle = \operatorname{sgn} (\sin \phi_e(x')) \cos \phi_e(x') \quad (32)
$$

where the ± sign of $\hat{\chi}_{\pm}$ now has an explicit functional dependence.

The work done by $|\mathbf{F}|$ flips every quarter cycle. To compute its amplitude, the basis states $|x'\rangle \langle x'|$ are inserted into (7), then the work expression is equated to the RHS:

$$
\int_0^{\frac{\lambda}{4}} |\mathbf{F}| \, dx' = \int_0^{\frac{\lambda}{4}} \frac{h\nu'}{\lambda_F} \langle \phi'_F | x' \rangle \langle x' | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle \, dx' = h\nu_a \quad (33)
$$

where $h\nu_a$ is the available interaction energy per quarter cycle. Substituting into (33) the phase state terms using (20) with a change of variables to $x'' = k_e x' - \phi_o$, one obtains:

$$
\frac{h\nu'}{\lambda_F} \frac{4}{\pi k_e} \int_0^{\frac{\pi}{2}} \cos^2(x'') dx'' = \frac{h\nu'}{\lambda_F k_e} = \frac{h\nu'}{\lambda_F \alpha_{\parallel}^{-1} k} = h\nu_a \quad (34)
$$

Thus, for a region of constant potential $V_o \equiv \pm h\nu'$, its associated virtual force has up to energy $h\nu_a$ for exchange, but since $\vec{k}_e \neq \vec{k}_a \equiv \vec{k}$, and amplitudes are involved in the calculation, the actual exchange energy is:

$$
h\nu_x = h\alpha_{\parallel}^{-1}\nu_a \quad (35)
$$

with momentum exchange wavelength $\lambda_F$ given by:

$$
\lambda_F = \pm \frac{\nu'}{\nu_x} \frac{\lambda}{2\pi} \quad (36)
$$

As an aside, this derivation did not explicitly represent
the ± signs attached to $\nu'$ in (33) as they cancel out in
(34).

Recalling the discussion leading up to (21), the statis-
tics attributed to the arbitrary phase offsets for free par-
ticles should also be possible for the solution to (9) in
order to have consistency between the probability calcu-
lation and the modified equation of Newton’s second law.
Yet, in steady-state problems, specific phase offsets are
necessarily assigned to boundary conditions.

Consider, however, that there is one component that isn't represented explicitly in the statistical analysis or the dynamic equations. That is time. For traveling waves, the phase $\phi_e$ is written more generally as the following:

$$
\phi_e(x, t) = k_e x - \phi_x - \omega_e t + \phi_t \quad (37)
$$

where $\omega_e = 2\pi\nu_e$ and $\phi_x$ is a constant determined by
boundary conditions while $\phi_t$ is associated with time-
dependent or initial conditions.

The statistical averaging that must be done for char-
acterizing multiple particles in steady-state problems has
to account for variations in $\phi_t$. Assuming the constant potential is dominating the phase mediation process, its phase in time will dictate the end result, while particles injected into the region where the constant potential becomes influential will interact with it at arbitrary times, so $\phi_t$ is random. Hence, the steady-state projection term in the modified second law equation actually takes the following form:

$$ \langle \phi'_{F} | x' \rangle \langle x' | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle = \operatorname{sgn} (\sin \phi_e(x')) \cos^2 \phi_e(x + \phi_t) \quad (38) $$

where the $\frac{4}{\pi}$ normalization is now absent because it was absorbed into the calculation of the force's amplitude.

The discussion leading to (21) regarding the interpretation of $A_\phi$ can now be applied to the statistical average of the solutions to (9). Since the operators and quantities involved in Newton's modified second law don't depend on $\phi_t$, a statistical summation can be done over equations to represent the statistical average for $\lambda$. Matching up solutions with a 180° phase shift relative to each other, the projection term of $\langle \phi'_F | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle$ is zero for the sum. However, adding $\phi_t = \frac{\pi}{2}$ phase shifts to one of the solutions in each pair used in the averaging, there are two separate classes of solutions, designated as $\phi_{\pm}$ corresponding to the $\hat{\chi}_{\pm}$ operator where:

$$ \left| \langle \phi'_{F} | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi_{\pm} + \phi_t \rangle \right| = \pm 1 \quad (39) $$

with departures from the average used to characterize noise or transient behavior.

The equation to solve can now be represented more explicitly. Referring to (9) with the $\hat{\chi}_{\pm}$ operator in $\lambda$ remaining after substitution of (36), as well as using (19) for $\phi(x)$ and (39) for the phase projection term, the steady-state equation for the amplitude of $\lambda$ becomes:

$$ \frac{d\lambda}{d\vec{x}} \cdot \hat{v} = \frac{d}{d\vec{x}} (\hat{\chi}_{\pm} \lambda(\phi_e)) \cdot \hat{v} = \mp 2\pi \hat{\chi}_{\pm} \frac{\nu_x}{\nu(x)} \quad (40) $$

$$ \left( \frac{d}{d\vec{x}} \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o(\vec{x}) \right) \cdot \hat{v} - k_e \right) \hat{\chi}_{\pm} \frac{d\lambda}{d\phi_e} = \mp 2\pi \frac{\nu_x}{\hat{\chi}_{\pm} \nu(x)} \quad (41) $$

$$ \left( \frac{d}{d\vec{x}} \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o(\vec{x}) \right) \cdot \hat{v} - k_e \right) \frac{d\lambda}{d\phi_e} = \mp 2\pi \frac{\nu_x}{\nu(x)} \quad (41) $$

where the $\hat{\chi}_{\pm}$ operators cancel out due to their identical action.

From (26) and (35), the following is a relationship between constants for $E - m_o c^2 > V_o$, with a particle becoming either weakly or strongly interacting in the Newtonian limit:

$$ \nu_x = \alpha_{\parallel}^{-1} \nu_a = (\hat{k}_c \cdot \hat{x})^2 \nu_a = \frac{\lambda_a}{\lambda_e} \nu_a \quad (42) $$

For the weakly interacting case, $\lambda_e \to \infty$ and $\alpha_{\parallel}^{-1}$ is small, with the particle's path following a spacetime geodesic since no oscillation occurs in a finite amount of time. The phase in (25) is effectively pinned to $\phi_o$.

Without phase evolution, phenomenon such as interference will not happen.

For the strongly interacting case when $\lambda_e \to 0$, $\alpha_{\parallel}^{-1}$ is large. The phase is decoherent in this regime since $\hbar\alpha_{\parallel}^{-1}\nu_a$, which as shown below is equivalent to $\hbar\nu_e$, exceeds the work-energy that can be provided by the background. More is discussed on this for the case of a constant force in the classical limit when $\phi_o$ effectively drifts to track $k_e x$. Without phase coherence, the particle exhibits only classical Newtonian phenomenon as well.

It will now be shown that the phase rotation caused by $\hat{\chi}_{\pm}$ ensures classical correspondence for the velocity. Substituting $\lambda(x)$ from (25) into (41), then solving for $\nu(x)$ using (20), (26), and (42) gives:

$$ \nu(x) = \nu_a \sin^2 \phi_e(x) \quad (43) $$

The velocity of the statistical ensemble of particles then takes the following form with $k_e \equiv \hat{k}_c \cdot \hat{x}$:

$$ v(x) = \hat{\chi}_{\pm} \lambda \nu_{\kappa} \quad (44) $$

$$ = \lambda_a \nu_a \kappa \hat{\chi}_{\pm} \sin \phi_e(x) \cos \phi_e(x) $$

$$ = \pm \lambda_a \nu_a \kappa \sin \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - k_e x + \phi_o \right) \cos (k_e x - \phi_o) \quad (45) $$

The $\hat{\chi}_{\pm}$ operator introduced support for even and odd solutions to (40), and without it the velocity in (45) would represent a pure vibration, such as the case when an impulse force causes the quanta to move perpendicular to the carrier's phase propagation. When the quanta is moving parallel to the carrier's propagation, $\hat{x} \cdot \hat{x}_o = 1$, and the following result is obtained from (23) and (45):

$$ \begin{aligned} v_e &= \bar{v} \cos^2(k_e x - \phi_o) \\ v_{x_o}(x) &= \pm v_e (1 + \sin(k_a x - \phi_a)) \end{aligned} \quad (46) $$

where $\bar{v} = \lambda_a \nu_a$. In Section IV, V, and further below, the behavior of this velocity is discussed for both the coherent and decoherent regimes. Though $\hat{\chi}_{\pm}$ introduced sign ambiguity into the velocity, boundary conditions ultimately choose which sign applies.

The important takeaway from this result is that the frequency and probability are in phase with one another when the quanta is moving in the direction of the carrier's propagation, while being out of phase with the velocity. The probability responds to an increase or decrease in the amplitude and forward velocity of the oscillation, consistent with the probability's dependence on $dt$ and $\nu_e$, as examined further in Section V.

To determine the averages of other quantities, it is first noted that the frequency amplitude $\nu_a$ is identical to $\nu$ in Schrödinger's theory:

$$ h\nu_a = m_o\gamma c^2 \quad (47) $$

Then the velocity amplitude of a particle is cast as a vector, having separate longitudinal and transverse components such that their combined velocity is the speed of
to account for variations in $\phi_t$. Assuming the constant potential is dominating the phase mediation process, its phase in time will dictate the end result, while particles injected into the region where the constant potential becomes influential will interact with it at arbitrary times, so $\phi_t$ is random. Hence, the steady-state projection term in the modified second law equation actually takes the following form:

$$ \langle \phi'_{F} | x' \rangle \langle x' | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle = \operatorname{sgn} (\sin \phi_e(x')) \cos^2 \phi_e(x + \phi_t) \quad (38) $$

where the $\frac{4}{\pi}$ normalization is now absent because it was absorbed into the calculation of the force's amplitude.

The discussion leading to (21) regarding the interpretation of $A_\phi$ can now be applied to the statistical average of the solutions to (9). Since the operators and quantities involved in Newton's modified second law don't depend on $\phi_t$, a statistical summation can be done over equations to represent the statistical average for $\lambda$. Matching up solutions with a 180° phase shift relative to each other, the projection term of $\langle \phi'_F | \hat{h}_s \cdot \hat{\chi}_{\pm} | \phi \rangle$ is zero for the sum. However, adding $\phi_t = \frac{\pi}{2}$ phase shifts to one of the solutions in each pair used in the averaging, there are two separate classes of solutions, designated as $\phi_{\pm}$ corresponding to the $\hat{\chi}_{\pm}$ operator where:

$$ \left| \langle \phi'_{F} | \hat{h}_{s} \cdot \hat{\chi}_{\pm} | \phi_{\pm} + \phi_{t} \rangle \right| = \pm 1 \quad (39) $$

with departures from the average used to characterize noise or transient behavior.

The equation to solve can now be represented more explicitly. Referring to (9) with the $\hat{\chi}_{\pm}$ operator in $\lambda$ remaining after substitution of (36), as well as using (19) for $\phi(x)$ and (39) for the phase projection term, the steady-state equation for the amplitude of $\lambda$ becomes:

$$ \frac{d\lambda}{d\vec{x}} \cdot \hat{v} = \frac{d}{d\vec{x}} (\hat{\chi}_{\pm} \lambda(\phi_e)) \cdot \hat{v} = \mp 2\pi \hat{\chi}_{\pm} \frac{\nu_x}{\nu(x)} \quad (40) $$

$$ \left( \frac{d}{d\vec{x}} \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o(\vec{x}) \right) \cdot \hat{v} - k_e \right) \hat{\chi}_{\pm} \frac{d\lambda}{d\phi_e} = \mp 2\pi \frac{\nu_x}{\hat{\chi}_{\pm} \nu(x)} \quad (41) $$

where the $\hat{\chi}_{\pm}$ operators cancel out due to their identical action.

From (26) and (35), the following is a relationship between constants for $E - m_o c^2 > V_o$, with a particle becoming either weakly or strongly interacting in the Newtonian limit:

$$ \nu_x = \alpha_{\parallel}^{-1} \nu_a = (\hat{k}_c \cdot \hat{x})^2 \nu_a = \frac{\lambda_a}{\lambda_e} \nu_a \quad (42) $$

For the weakly interacting case, $\lambda_e \to \infty$ and $\alpha_{\parallel}^{-1}$ is small, with the particle's path following a spacetime geodesic since no oscillation occurs in a finite amount of time. The phase in (25) is effectively pinned to $\phi_o$.

Without phase evolution, phenomenon such as interference will not happen.

For the strongly interacting case when $\lambda_e \to 0$, $\alpha_{\parallel}^{-1}$ is large. The phase is decoherent in this regime since $\hbar \alpha_{\parallel}^{-1} \nu_a$, which as shown below is equivalent to $\hbar \nu_e$, exceeds the work-energy that can be provided by the background. More is discussed on this for the case of a constant force in the classical limit when $\phi_o$ effectively drifts to track $k_e x$. Without phase coherence, the particle exhibits only classical Newtonian phenomenon as well.

It will now be shown that the phase rotation caused by $\hat{\chi}_{\pm}$ ensures classical correspondence for the velocity. Substituting $\lambda(x)$ from (25) into (41), then solving for $\nu(x)$ using (20), (26), and (42) gives:

$$ \nu(x) = \nu_a \sin^2 \phi_e(x) \quad (43) $$

The velocity of the statistical ensemble of particles then takes the following form with $k_e \equiv \hat{k}_c \cdot \hat{x}$:

$$ \begin{aligned} v(x) &= \hat{\chi}_{\pm} \lambda \nu_{\kappa} \quad (44) \\ &= \lambda_{a} \nu_{a} \kappa \hat{\chi}_{\pm} \sin \phi_{e}(x) \cos \phi_{e}(x) \\ &= \pm \lambda_{a} \nu_{a} \kappa \sin \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_{o} - k_{e} x + \phi_{o} \right) \cos (k_{e} x - \phi_{o}) \end{aligned} \quad (45) $$

The $\hat{\chi}_{\pm}$ operator introduced support for even and odd solutions to (40), and without it the velocity in (45) would represent a pure vibration, such as the case when an impulse force causes the quanta to move perpendicular to the carrier's phase propagation. When the quanta is moving parallel to the carrier's propagation, $\hat{x} \cdot \hat{x}_o = 1$, and the following result is obtained from (23) and (45):

$$ \begin{aligned} v_e &= \bar{v} \cos^2(k_e x - \phi_o) \\ v_{x_o}(x) &= \pm v_e (1 + \sin(k_a x - \phi_a)) \end{aligned} \quad (46) $$

where $\bar{v} = \lambda_a \nu_a$. In Section IV, V, and further below, the behavior of this velocity is discussed for both the coherent and decoherent regimes. Though $\hat{\chi}_{\pm}$ introduced sign ambiguity into the velocity, boundary conditions ultimately choose which sign applies.

The important takeaway from this result is that the frequency and probability are in phase with one another when the quanta is moving in the direction of the carrier's propagation, while being out of phase with the velocity. The probability responds to an increase or decrease in the amplitude and forward velocity of the oscillation, consistent with the probability's dependence on $dt$ and $\nu_e$, as examined further in Section V.

To determine the averages of other quantities, it is first noted that the frequency amplitude $\nu_a$ is identical to $\nu$ in Schrödinger's theory:

$$ h \nu_a = m_o \gamma c^2 \quad (47) $$

Then the velocity amplitude of a particle is cast as a vector, having separate longitudinal and transverse components such that their combined velocity is the speed of
<figure><img src="image_3.png" /><figcaption>FIG. 1. Relationship between the carrier and envelope wavevectors of a free particle.</figcaption></figure>

light:

$$v_a = \bar{v}\hat{x}_o + \sqrt{c^2 - \bar{v}^2}\hat{y}_o \quad (48)$$

which obeys these relationships for two perpendicular wavevectors:

$$k_a \bar{v} = 2\pi \nu_a \qquad k_{\perp} \sqrt{c^2 - \bar{v}^2} = 2\pi \nu_a \quad (49)$$

noting that the transverse vector $\hat{y}_o$ is chosen according to the assumed polarization of the particle's wave component and the quanta's movement in the direction of the carrier's propagation. Statistically speaking, the amplitude of a particle's oscillation is free to point in any direction, such as might be the case for string theory, however the momentum imparted to it invokes phase rotation for a particular direction. Just as for light, it is likely that the particle's transverse oscillation is circularly polarized in general.

This identifies the following assignments with the help of (26), as illustrated in Figure 1:

$$\lambda_c \nu_a = c \quad (50)$$

$$k_c = \frac{2\pi}{\lambda_c} \quad (51)$$

$$k_e = \frac{\bar{v}}{c} k_c \quad (52)$$

$$k_a = \frac{c}{\bar{v}} k_c \quad (53)$$

$$k = \sqrt{k_a^2 + k_{\perp}^2} = k_a \gamma \quad (54)$$

Several important relationships follow from this. First, the classical, spacetime-averaged velocity $\bar{v}$ using the de Broglie or amplitude parameters is preserved. The momenta attributed to the three wavevectors or wavelengths are also readily derived:

$$p_c = \frac{h}{\lambda_c} = \frac{h \nu_a}{c} = m_o \gamma c \quad (55)$$

$$p_e = \frac{h}{\lambda_e} = \frac{\bar{v} h}{c \lambda_c} = m_o \gamma \bar{v} \quad (55)$$

$$p_a = \frac{h}{\lambda_a} = \frac{c h}{\bar{v} \lambda_c} = m_o \gamma \frac{c^2}{\bar{v}}$$

Here, the envelope of the particle's propagation carries its classically-recognized momentum, $p_e$, while $p_a$ seems to support Schrödinger's relativistic calculation for the phase velocity of $\frac{c^2}{\bar{v}}$. However, $p_a$ merely points out that $\lambda_a$ is smaller than the de Broglie wavelength, $\lambda_e$, while the de Broglie frequency $\nu_e$ is smaller than the frequency amplitude $\nu_a$:

$$\nu_e = \frac{\bar{v}}{\lambda_e} = \frac{m_o \gamma \bar{v}^2}{h} = \frac{\bar{v}^2}{c^2} \nu_a = \alpha_{\parallel}^{-1} \nu_a = \nu_x \quad (56)$$

These relationships compensate one another, hence the carrier, envelope, and quanta are actually traversing space at the same velocity, represented by (4), when aligned to the same direction. The combined longitudinal and transverse motion, however, is characterized by a wave-like propagation at the speed of light with wavevector $\bar{k}_c$ pointing in a different direction than the longitudinal momentum of the particle.

This is shown to be consistent with the dynamics of the energy discussed further down. For the particle's carrier and envelope to move at the same velocity, the carrier's wavelength $\lambda_a$ is squeezed along the x axis by the average ratio $\frac{\bar{v}^2}{c^2}$ times the wavelength $\lambda_e$ of the envelope. This is consistent with Snell's law and the observed speed variations of light in various medium. The wavevectors in (51)-(53) and momenta in (55) are constant, corresponding to a momentum eigenstate in Schrödinger's theory, but the spacetime evolution of the frequency, mass, and velocity aren't constant in time. The amplitude envelope grows in size when the particle's frequency of motion slows down and the transverse displacement is increasing, while doing the opposite when the amplitude diminishes, as discussed in Section V.

Such behavior suggests a different interpretation behind the uncertainty principle. Given the variation of the wavelength and velocity with position, the spacetime displacements of the quanta are not uniform, and hence, if a measurement of energy were to be taken at position x this would only determine the particle's average velocity. This highlights the difficulty for pinpointing the velocity and position at the same time, which is consistent with the uncertainty principle.

Regarding the quantum mass, the following is obtained by using the relationships for $\nu_a$ and $\lambda_c$ in (47) and (55):

$$m = \frac{h}{\gamma \alpha_{\parallel} \lambda_a^2 \nu_a \cos^2 \phi_e(x)} \\ m_o = \frac{h}{\gamma \lambda_c^2 \nu_a} = m \cos^2 \phi_e(x) \quad (57)$$

where (26) was used for $\alpha_{\parallel}$.

Using (44) and (57), the momentum $p = m\gamma v$ takes the value:

$$p = m\gamma v = m_o\gamma\bar{v}(1 + \sin(k_a x' - \phi_a)) \quad (58)$$

which has a spacetime average equal to $p_e$ in (55).
<figure><img src="image_2.png" /><figcaption>FIG. 1. Relationship between the carrier and envelope wavevectors of a free particle.</figcaption></figure>

light:

$$v_a = \bar{v}\hat{x}_o + \sqrt{c^2 - \bar{v}^2}\hat{y}_o \quad (48)$$

which obeys these relationships for two perpendicular wavevectors:

$$k_a \bar{v} = 2\pi \nu_a \qquad k_{\perp} \sqrt{c^2 - \bar{v}^2} = 2\pi \nu_a \quad (49)$$

noting that the transverse vector $\hat{y}_o$ is chosen according to the assumed polarization of the particle's wave component and the quanta's movement in the direction of the carrier's propagation. Statistically speaking, the amplitude of a particle's oscillation is free to point in any direction, such as might be the case for string theory, however the momentum imparted to it invokes phase rotation for a particular direction. Just as for light, it is likely that the particle's transverse oscillation is circularly polarized in general.

This identifies the following assignments with the help of (26), as illustrated in Figure 1:

$$\lambda_c \nu_a = c \quad (50)$$

$$k_c = \frac{2\pi}{\lambda_c} \quad (51)$$

$$k_e = \frac{\bar{v}}{c} k_c \quad (52)$$

$$k_a = \frac{c}{\bar{v}} k_c \quad (53)$$

$$k = \sqrt{k_a^2 + k_{\perp}^2} = k_a \gamma \quad (54)$$

Several important relationships follow from this. First, the classical, spacetime-averaged velocity $\bar{v}$ using the de Broglie or amplitude parameters is preserved. The momenta attributed to the three wavevectors or wavelengths are also readily derived:

$$p_c = \frac{h}{\lambda_c} = \frac{h \nu_a}{c} = m_o \gamma c \quad (55)$$

$$p_e = \frac{h}{\lambda_e} = \frac{\bar{v} h}{c \lambda_c} = m_o \gamma \bar{v} \quad (55)$$

$$p_a = \frac{h}{\lambda_a} = \frac{c h}{\bar{v} \lambda_c} = m_o \gamma \frac{c^2}{\bar{v}}$$

Here, the envelope of the particle's propagation carries its classically-recognized momentum, $p_e$, while $p_a$ seems to support Schrödinger's relativistic calculation for the phase velocity of $\frac{c^2}{\bar{v}}$. However, $p_a$ merely points out that $\lambda_a$ is smaller than the de Broglie wavelength, $\lambda_e$, while the de Broglie frequency $\nu_e$ is smaller than the frequency amplitude $\nu_a$:

$$\nu_e = \frac{\bar{v}}{\lambda_e} = \frac{m_o \gamma \bar{v}^2}{h} = \frac{\bar{v}^2}{c^2} \nu_a = \alpha_{\parallel}^{-1} \nu_a = \nu_x \quad (56)$$

These relationships compensate one another, hence the carrier, envelope, and quanta are actually traversing space at the same velocity, represented by (4), when aligned to the same direction. The combined longitudinal and transverse motion, however, is characterized by a wave-like propagation at the speed of light with wavevector $\bar{k}_c$ pointing in a different direction than the longitudinal momentum of the particle.

This is shown to be consistent with the dynamics of the energy discussed further down. For the particle's carrier and envelope to move at the same velocity, the carrier's wavelength $\lambda_a$ is squeezed along the x axis by the average ratio $\frac{\bar{v}^2}{c^2}$ times the wavelength $\lambda_e$ of the envelope. This is consistent with Snell's law and the observed speed variations of light in various medium. The wavevectors in (51)-(53) and momenta in (55) are constant, corresponding to a momentum eigenstate in Schrödinger's theory, but the spacetime evolution of the frequency, mass, and velocity aren't constant in time. The amplitude envelope grows in size when the particle's frequency of motion slows down and the transverse displacement is increasing, while doing the opposite when the amplitude diminishes, as discussed in Section V.

Such behavior suggests a different interpretation behind the uncertainty principle. Given the variation of the wavelength and velocity with position, the spacetime displacements of the quanta are not uniform, and hence, if a measurement of energy were to be taken at position x this would only determine the particle's average velocity. This highlights the difficulty for pinpointing the velocity and position at the same time, which is consistent with the uncertainty principle.

Regarding the quantum mass, the following is obtained by using the relationships for $\nu_a$ and $\lambda_c$ in (47) and (55):

$$m = \frac{h}{\gamma \alpha_{\parallel} \lambda_a^2 \nu_a \cos^2 \phi_e(x)} \\ m_o = \frac{h}{\gamma \lambda_c^2 \nu_a} = m \cos^2 \phi_e(x) \quad (57)$$

where (26) was used for $\alpha_{\parallel}$.

Using (44) and (57), the momentum $p = m\gamma v$ takes the value:

$$p = m\gamma v = m_o\gamma\bar{v}(1 + \sin(k_a x' - \phi_a)) \quad (58)$$

which has a spacetime average equal to $p_e$ in (55).
More specifically, the quantity $v_{\phi} \equiv v_e = \lambda\nu$ represents a spatially varying phase velocity. When the phase is coherent, its value in (46) on average would appear to be $\frac{1}{2}\bar{v}$ and the quantum mass would be twice the rest mass $m_o$, in line with the non-relativistic calculation of the phase velocity using Schrödinger's theory. However, as discussed in Section V, there is a phase discontinuity that ensures a particle's oscillating displacement is equivalent to the classical velocity, on average.

When the phase becomes decoherent, e.g. during the measurement process as discussed in Section IV, the phase offset $\phi_o$ drifts to match $k_e x$ on average such that $\cos^2(k_e x - \phi_o) = 1$, yielding a classical evolution for the velocity. How this leads to uncertainty will now be examined.

Since a constant mass $m_o$ is used in standard theory, $p_{\phi}$ corresponds to Schrödinger's free particle eigenstate of momentum $p_e$:

$$p_{\phi} = m_{o}\gamma v_{e} = m_{o}\gamma\bar{v}\cos^{2}\phi_{e}(x) \quad (59)$$

If this were considered as the momentum of the particle, then the uncertainty associated with it would be:

$$\Delta x \Delta p \sim \frac{h}{4} > \hbar \quad (60)$$

which is readily calculated from the space interval over which (59) varies by amount:

$$\Delta p = p_e = \frac{h}{\lambda_e} = m_o \gamma \bar{v} \quad (61)$$

That value is a quarter wavelength, $\Delta x = \frac{\lambda_e}{4}$.

The reason the uncertainty principle is applicable to this theory is due to the velocity's variation in the phase coherent regime even when the momentum in (58) essentially is constant. Variations in interactions with the background lead to a particle going in and out of the coherent regime, which sounds like wavefunction collapse, however as explored further on, sudden, discontinuous transitions do not happen and the wave representation does not change. Hence, estimates such as (60) are useful for characterizing non-classical dynamics.

To further explore what happens during the phase cycle, classical correspondence is now examined using (43), (57), and (5) in the rearrangement of the expression for free particle energy with a background potential, $V_o$:

$$\begin{aligned} E &= m_o \gamma c^2 + V_o \\ &= m_o \gamma \bar{v}^2 + m_o \gamma \left(1 - \frac{\bar{v}^2}{c^2}\right) c^2 + V_o \\ &= \{\bar{p}\bar{v} \text{ or } h\nu_e(x)\} + \frac{m_o}{\gamma} c^2 \pm h\nu' \end{aligned} \quad (62)$$

Substituting the following correspondence values for a chosen phase, $\phi_e(x) \equiv k_e x - \phi_o$:

$$\begin{aligned} \cos^2\phi_e(x) &= \frac{1}{\gamma^2} \\ \sin^2\phi_e(x) &= 1 - \frac{1}{\gamma^2} = \frac{\bar{v}^2}{c^2} \end{aligned} \quad (63)$$

<figure><img src="image_4.png" /><figcaption>FIG. 2. Cycle of virtual quanta emission and absorption for a free particle.</figcaption></figure>

into (62) with $m = m_o \gamma^2$ from application of (57) and (63):

$$\begin{aligned} E &= h\nu_a \sin^2\phi_e(x) + m_o \gamma c^2 \cos^2\phi_e(x) \pm h\nu' \\ &= h(\nu_a \pm \nu') \end{aligned} \quad (64)$$

where $\nu_e = \frac{\bar{v}^2}{c^2}\nu_a$ from (56) was also used together with (63) and (47).

When the phase $\phi_e(x)$ diverges from the classical correspondence value given in (63), there is some interesting behavior. With $m$ viewed as an amplitude in (57), and its value assigned to $m'_o \gamma^2$ similar to before only with the recognition that $m'_o$ may differ from $m_o$ in general when $\gamma$ is evaluated at the average velocity, $\bar{v}$, and $\phi_e(x)$ varies, the following relationship must hold as $\cos^2\phi_e(x)$ reaches its maximum:

$$\frac{m}{\gamma^2} \equiv m'_o \rightarrow \frac{m_o}{\gamma^2} = m_o \left(1 - \frac{\bar{v}^2}{c^2}\right) \quad (65)$$

The energy content in this limit is determined by multiplying (65) by $\gamma c^2$. This yields:

$$T = m'_{o}\gamma c^2 = m_o\gamma(c^2 - \bar{v}^2) = (E - V_o) - h\nu_e \quad (66)$$

It is worth noting that $m\gamma = m'_{o}\gamma^3$ is the effective relativistic mass assigned to a particle undergoing uniform acceleration, in this case due to a virtual force. This lends justification to using $m'_o$ for the quantum mass-energy calculation in this limit.

From (56), the exchange energy is given by: $h\nu_x = h\nu_e$. Having determined the behavior for $m$, and in keeping
More specifically, the quantity $v_{\phi} \equiv v_e = \lambda\nu$ represents a spatially varying phase velocity. When the phase is coherent, its value in (46) on average would appear to be $\frac{1}{2}\bar{v}$ and the quantum mass would be twice the rest mass $m_o$, in line with the non-relativistic calculation of the phase velocity using Schrödinger's theory. However, as discussed in Section V, there is a phase discontinuity that ensures a particle's oscillating displacement is equivalent to the classical velocity, on average.

When the phase becomes decoherent, e.g. during the measurement process as discussed in Section IV, the phase offset $\phi_o$ drifts to match $k_e x$ on average such that $\cos^2(k_e x - \phi_o) = 1$, yielding a classical evolution for the velocity. How this leads to uncertainty will now be examined.

Since a constant mass $m_o$ is used in standard theory, $p_{\phi}$ corresponds to Schrödinger's free particle eigenstate of momentum $p_e$:

$$p_{\phi} = m_{o}\gamma v_{e} = m_{o}\gamma\bar{v}\cos^{2}\phi_{e}(x) \quad (59)$$

If this were considered as the momentum of the particle, then the uncertainty associated with it would be:

$$\Delta x \Delta p \sim \frac{h}{4} > \hbar \quad (60)$$

which is readily calculated from the space interval over which (59) varies by amount:

$$\Delta p = p_e = \frac{h}{\lambda_e} = m_o \gamma \bar{v} \quad (61)$$

That value is a quarter wavelength, $\Delta x = \frac{\lambda_e}{4}$.

The reason the uncertainty principle is applicable to this theory is due to the velocity's variation in the phase coherent regime even when the momentum in (58) essentially is constant. Variations in interactions with the background lead to a particle going in and out of the coherent regime, which sounds like wavefunction collapse, however as explored further on, sudden, discontinuous transitions do not happen and the wave representation does not change. Hence, estimates such as (60) are useful for characterizing non-classical dynamics.

To further explore what happens during the phase cycle, classical correspondence is now examined using (43), (57), and (5) in the rearrangement of the expression for free particle energy with a background potential, $V_o$:

$$\begin{aligned} E &= m_o \gamma c^2 + V_o \\ &= m_o \gamma \bar{v}^2 + m_o \gamma \left(1 - \frac{\bar{v}^2}{c^2}\right) c^2 + V_o \\ &= \{\bar{p}\bar{v} \text{ or } h\nu_e(x)\} + \frac{m_o}{\gamma} c^2 \pm h\nu' \end{aligned} \quad (62)$$

Substituting the following correspondence values for a chosen phase, $\phi_e(x) \equiv k_e x - \phi_o$:

$$\begin{aligned} \cos^2\phi_e(x) &= \frac{1}{\gamma^2} \\ \sin^2\phi_e(x) &= 1 - \frac{1}{\gamma^2} = \frac{\bar{v}^2}{c^2} \end{aligned} \quad (63)$$

<figure><img src="image_3.png" /><figcaption>FIG. 2. Cycle of virtual quanta emission and absorption for a free particle.</figcaption></figure>

into (62) with $m = m_o \gamma^2$ from application of (57) and (63):

$$\begin{aligned} E &= h\nu_a \sin^2\phi_e(x) + m_o \gamma c^2 \cos^2\phi_e(x) \pm h\nu' \\ &= h(\nu_a \pm \nu') \end{aligned} \quad (64)$$

where $\nu_e = \frac{\bar{v}^2}{c^2}\nu_a$ from (56) was also used together with (63) and (47).

When the phase $\phi_e(x)$ diverges from the classical correspondence value given in (63), there is some interesting behavior. With $m$ viewed as an amplitude in (57), and its value assigned to $m'_o \gamma^2$ similar to before only with the recognition that $m'_o$ may differ from $m_o$ in general when $\gamma$ is evaluated at the average velocity, $\bar{v}$, and $\phi_e(x)$ varies, the following relationship must hold as $\cos^2\phi_e(x)$ reaches its maximum:

$$\frac{m}{\gamma^2} \equiv m'_o \rightarrow \frac{m_o}{\gamma^2} = m_o \left(1 - \frac{\bar{v}^2}{c^2}\right) \quad (65)$$

The energy content in this limit is determined by multiplying (65) by $\gamma c^2$. This yields:

$$T = m'_{o} \gamma c^2 = m_o \gamma (c^2 - \bar{v}^2) = (E - V_o) - h\nu_e \quad (66)$$

It is worth noting that $m\gamma = m'_{o}\gamma^3$ is the effective relativistic mass assigned to a particle undergoing uniform acceleration, in this case due to a virtual force. This lends justification to using $m'_o$ for the quantum mass-energy calculation in this limit.

From (56), the exchange energy is given by: $h\nu_x = h\nu_e$. Having determined the behavior for $m$, and in keeping
with the expectation that the energy is effectively constant over the entire cycle, the following interpretation is necessary to explain the action of the virtual force.

To begin, as shown in Figure 2, the quantum mass’s energy is absent a bundle of energy, i.e. $m\gamma c^2 = m_o\gamma c^2 - h\nu_e$, before work is applied with the virtual force pointing in the direction of motion as determined by its helicity. The process can be modeled by the absorption of a virtual quantum of energy, $h\nu_e$, and during the next quarter cycle the virtual work associated with it in (33) is incorporated into the particle’s kinetic energy. At the top of the cycle, the spin flips and the virtual force does work to draw down the kinetic energy and quantum mass by energy $h\nu_e$ over the next quarter cycle, as shown by the release of a virtual quantum of energy.

It is thus seen here that the uncertainty principle involving a change of energy and time interval, $\Delta E\Delta t \sim \frac{\hbar}{4} > \hbar$, is manifested as a delayed response of the virtual force and particle over a quarter cycle given by:

$$ \Delta t = \frac{\lambda_e}{4\bar{v}} = \frac{1}{4\nu_e} \qquad (67) $$

with the absorption or emission of the background potential exchange energy $\Delta E = h\nu_e$. As this exchange is virtual and the uncertainty principle applies, the change in energy of the particle is not classically detectable.

If the quanta of a particle were to be temporarily dislodged, such as in a near-collision with its antiparticle, there is an opportunity for $\hat{k}_c \cdot \hat{x} = 1$ ($\alpha_\parallel = 1$), which sets the particle into a momentary vibration mode. The work done by the background would then totally disassociate its energy, $h\nu_a$, during a cycle. That would make it possible for the energy to be channeled into a different form, such as a gamma ray. This observation could be useful for computing the cross section of matter-antimatter annihilation.

Since the de Broglie momentum $p_e = \hbar k_e$ is important for determining the quantum operators responsible for calculating $|\phi\rangle$, the energy in (62) is re-expressed as:

$$ E = \alpha_{\parallel} \bar{p}_e \bar{v} \sin^2(k_e x - \phi_o) + m_o \gamma c^2 \cos^2(k_e x - \phi_o) \pm h\nu' \qquad (68) $$

To find the operators, it is first noted that quantum state $|\phi\rangle$ is of form:

$$ \langle x | \phi \rangle = e^{+i\phi(x)} = \cos \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - k_e x + \phi_o \right) + i \sin \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - k_e x + \phi_o \right) \qquad (69) $$

Then the $\alpha_\parallel$ and $\gamma$ coefficients in (68) are combined with one contribution from each of the trigonometric functions in (68) according to their values in (63). This is equated to real-valued operators acting on quantum state $|\phi\rangle$ to achieve the same result with $\hat{x} \cdot \hat{x}_o = 1$:

$$ (\bar{p}_e c) \sin(k_e x - \phi_o) + m_o c^2 \cos(k_e x - \phi_o) = \mathrm{Re} \left( \left\langle x \left| (c\hbar\hat{\chi}_{\pm} \frac{\partial}{\partial x} + m_o c^2 \hat{\chi}_{\pm}) \right| \phi \right\rangle \right) \qquad (70) $$

which is consistent with the definition of $|\phi\rangle$ in (17). To find the expectation value of this operator, the square of (70) is taken, with normalization occurring through integration over the original probability assigned to the quantum state without the operator. Finally, the square root is applied, which is consistent with taking the root mean square of stochastic variables:

$$ \begin{aligned} \langle T \rangle &= \sqrt{\frac{\int^{x} dx' \left[ \mathrm{Re} \left( \langle x' | (c\hbar\hat{\chi}_{\pm} \frac{\partial}{\partial x} + m_o c^2 \hat{\chi}_{\pm}) | \phi \rangle \right) \right]^2}{\int^{x} dx' \left[ \mathrm{Re} \left( \langle x' | \phi \rangle \right) \right]^2}} \qquad (71) \\ &= \sqrt{\frac{\int^{x} dx' \left[ \bar{p}_e c \sin \phi_e(x') + m_o c^2 \cos \phi_e(x') \right]^2}{\int^{x} dx' \cos^2 \phi_e(x')}} \\ &= \sqrt{(\bar{p}_e c)^2 + (m_o c^2)^2} \qquad (72) \end{aligned} $$

To find the expectation of the total energy, the term involving the potential is calculated on its own, since the random variables assigned to it are independent of the kinetic energy term in this case. Hence, the total energy is given by:

$$ \langle E \rangle = \langle T \rangle + \langle V_o \rangle = \sqrt{(\bar{p}_e c)^2 + (m_o c^2)^2} + V_o \qquad (73) $$

which is the standard relativistic expression equivalent to (62) and (68).

In summary, the Hamiltonian operator takes the following form in one dimension:

$$ H = c\hbar\hat{\chi}_{\pm}\frac{\partial}{\partial x} + \hat{\chi}_{\pm}m_o c^2 + V(x) \qquad (74) $$

where for comparison, Dirac’s Hamiltonian is:

$$ H = c \frac{\hbar}{i} \vec{\alpha} \cdot \frac{\partial}{\partial \vec{x}} + \beta m_o c^2 + V(x) \qquad (75) $$

A major reason Dirac’s formulation uses matrices while this theory so far does not is due to the 1-D analysis. A 3-D generalization is looked at briefly in Section XI to suggest further study that will bring up additional comparisons.

As boundary conditions or varying potentials impact phases, the dynamic transfer between energy in its different forms is affected. The division between the mass and frequency components of energy is useful in computing their dynamics in general, including semi-classical behavior. Further discussion of phase evolution and its importance for the validity of semi-classical analysis is discussed in Appendix B.

According to the Higgs’ mechanism [24], Higgs bosons generate mass for a moving particle. Here, it is suggested that the difference between the quantum behavior of light and that of massive particles is attributed to such a mass-altering mechanism, which as modeled causes the wavelength and frequency of a particle to vary with motion, unlike that of light in a homogeneous medium. What is further implied is that the formation and dynamics of
with the expectation that the energy is effectively constant over the entire cycle, the following interpretation is necessary to explain the action of the virtual force.

To begin, as shown in Figure 2, the quantum mass’s energy is absent a bundle of energy, i.e. $m\gamma c^2 = m_o\gamma c^2 - h\nu_e$, before work is applied with the virtual force pointing in the direction of motion as determined by its helicity. The process can be modeled by the absorption of a virtual quantum of energy, $h\nu_e$, and during the next quarter cycle the virtual work associated with it in (33) is incorporated into the particle’s kinetic energy. At the top of the cycle, the spin flips and the virtual force does work to draw down the kinetic energy and quantum mass by energy $h\nu_e$ over the next quarter cycle, as shown by the release of a virtual quantum of energy.

It is thus seen here that the uncertainty principle involving a change of energy and time interval, $\Delta E\Delta t \sim \frac{h}{4} > \hbar$, is manifested as a delayed response of the virtual force and particle over a quarter cycle given by:

$$ \Delta t = \frac{\lambda_e}{4\bar{v}} = \frac{1}{4\nu_e} \qquad (67) $$

with the absorption or emission of the background potential exchange energy $\Delta E = h\nu_e$. As this exchange is virtual and the uncertainty principle applies, the change in energy of the particle is not classically detectable.

If the quanta of a particle were to be temporarily dislodged, such as in a near-collision with its antiparticle, there is an opportunity for $\hat{k}_c \cdot \hat{x} = 1$ ($\alpha_\parallel = 1$), which sets the particle into a momentary vibration mode. The work done by the background would then totally disassociate its energy, $h\nu_a$, during a cycle. That would make it possible for the energy to be channeled into a different form, such as a gamma ray. This observation could be useful for computing the cross section of matter-antimatter annihilation.

Since the de Broglie momentum $p_e = \hbar k_e$ is important for determining the quantum operators responsible for calculating $|\phi\rangle$, the energy in (62) is re-expressed as:

$$ E = \alpha_{\parallel} \bar{p}_e \bar{v} \sin^2(k_e x - \phi_o) + m_o \gamma c^2 \cos^2(k_e x - \phi_o) \pm h\nu' \qquad (68) $$

To find the operators, it is first noted that quantum state $|\phi\rangle$ is of form:

$$ \langle x | \phi \rangle = e^{+i\phi(x)} = \cos \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - k_e x + \phi_o \right) + i \sin \left( \frac{\pi}{2} \hat{x} \cdot \hat{x}_o - k_e x + \phi_o \right) \qquad (69) $$

Then the $\alpha_\parallel$ and $\gamma$ coefficients in (68) are combined with one contribution from each of the trigonometric functions in (68) according to their values in (63). This is equated to real-valued operators acting on quantum state $|\phi\rangle$ to achieve the same result with $\hat{x} \cdot \hat{x}_o = 1$:

$$ (\bar{p}_e c) \sin(k_e x - \phi_o) + m_o c^2 \cos(k_e x - \phi_o) = \mathrm{Re} \left( \left\langle x \left| (c\hbar\hat{\chi}_{\pm} \frac{\partial}{\partial x} + m_o c^2 \hat{\chi}_{\pm}) \right| \phi \right\rangle \right) \qquad (70) $$

which is consistent with the definition of $|\phi\rangle$ in (17). To find the expectation value of this operator, the square of (70) is taken, with normalization occurring through integration over the original probability assigned to the quantum state without the operator. Finally, the square root is applied, which is consistent with taking the root mean square of stochastic variables:

$$ \begin{aligned} \langle T \rangle &= \sqrt{\frac{\int^{x} dx' \left[ \mathrm{Re} \left( \langle x' | (c\hbar\hat{\chi}_{\pm} \frac{\partial}{\partial x} + m_o c^2 \hat{\chi}_{\pm}) | \phi \rangle \right) \right]^2}{\int^{x} dx' \left[ \mathrm{Re} \left( \langle x' | \phi \rangle \right) \right]^2}} \qquad (71) \\ &= \sqrt{\frac{\int^{x} dx' \left[ \bar{p}_e c \sin \phi_e(x') + m_o c^2 \cos \phi_e(x') \right]^2}{\int^{x} dx' \cos^2 \phi_e(x')}} \\ &= \sqrt{(\bar{p}_e c)^2 + (m_o c^2)^2} \qquad (72) \end{aligned} $$

To find the expectation of the total energy, the term involving the potential is calculated on its own, since the random variables assigned to it are independent of the kinetic energy term in this case. Hence, the total energy is given by:

$$ \langle E \rangle = \langle T \rangle + \langle V_o \rangle = \sqrt{(\bar{p}_e c)^2 + (m_o c^2)^2} + V_o \qquad (73) $$

which is the standard relativistic expression equivalent to (62) and (68).

In summary, the Hamiltonian operator takes the following form in one dimension:

$$ H = c\hbar\hat{\chi}_{\pm}\frac{\partial}{\partial x} + \hat{\chi}_{\pm}m_o c^2 + V(x) \qquad (74) $$

where for comparison, Dirac’s Hamiltonian is:

$$ H = c\frac{\hbar}{i}\vec{\alpha} \cdot \frac{\partial}{\partial \vec{x}} + \beta m_o c^2 + V(x) \qquad (75) $$

A major reason Dirac’s formulation uses matrices while this theory so far does not is due to the 1-D analysis. A 3-D generalization is looked at briefly in Section XI to suggest further study that will bring up additional comparisons.

As boundary conditions or varying potentials impact phases, the dynamic transfer between energy in its different forms is affected. The division between the mass and frequency components of energy is useful in computing their dynamics in general, including semi-classical behavior. Further discussion of phase evolution and its importance for the validity of semi-classical analysis is discussed in Appendix B.

According to the Higgs’ mechanism [24], Higgs bosons generate mass for a moving particle. Here, it is suggested that the difference between the quantum behavior of light and that of massive particles is attributed to such a mass-altering mechanism, which as modeled causes the wavelength and frequency of a particle to vary with motion, unlike that of light in a homogeneous medium. What is further implied is that the formation and dynamics of
quantum masses in a system is not independent of their interactions in the system.

Typically, in the analysis of matter in steady-state equilibrium, $\langle\phi'_{F}|\hat{\chi}_{+}|\phi\rangle \ge 0$. Hence, it is tempting to suggest that the opposite charge assigned to anti-matter in its pairing with matter is actually an expression of a negative phase projection, i.e. for a constant electromagnetic force, steady-state equilibrium is achieved with $\phi = \phi'_{F} + \pi$ or $\langle\phi'_{F}|\hat{\chi}_{+}|\phi\rangle \le 0$. This would characterize antimatter's oscillation in spacetime with a 180° phase shift compared to matter traveling in the same direction, thus presenting an alternative to the Feynman-Stueckelberg interpretation of equations for anti-matter [25],[26].

As one final observation, the analysis so far demonstrates that classical concepts can still be used to describe quantum behavior in the absence of extreme limits or measurements, and with phase coherence preserved. Furthermore, the analysis of the next section suggests that, in agreement with experiments so far, transitions between quantum states are not spontaneous.

### III. SUPERPOSITION

To provide an explicit example of how superposition plays out within this framework, the case of a particle in a superposition of two unbounded linearly polarized states will now be considered. To start, as with traditional quantum theory, total state $|\phi\rangle$ is represented by the sum of two steady-state solutions:

$$ \langle x|\phi\rangle = A_{\phi} \left( A_{o}e^{+i\hat{\chi}_{\pm}(k_{o}x-\phi_{o})} + A_{1}e^{+i\hat{\chi}_{\pm}(k_{1}x-\phi_{1})} \right) \quad (76) $$

which involves real-valued amplitudes, $A_o$ and $A_1$. Assuming that the phase offsets are zero and the quanta is moving in the direction of the carrier's propagation, $\hat{x} \cdot \hat{x}_o = 1$ with the probability density written as:

$$ P(x) = N[A_o \sin(k_o x) + A_1 \sin(k_1 x)]^2 \quad (77) $$

where $N = A_{\phi}^2$ is chosen to ensure the total probability of finding the particle within some arbitrary box of length L is 1.

As done previously with (14), the derivative of the probability is taken to find $\lambda(x)$, only now, $\alpha_\parallel$ is assigned separately to each wavevector and as there is no irreducible form for the expression, $\hat{\chi}_\pm$ acts upon every sub-component:

$$ \lambda(x) = 2\pi \frac{A_o \cos(k_o x) + A_1 \cos(k_1 x)}{A_o \alpha_\parallel^o k_o \sin(k_o x) + A_1 \alpha_\parallel^1 k_1 \sin(k_1 x)} \quad (78) $$

Substituting this result into (40) and taking the derivative of the LHS, one gets:

$$ \begin{aligned} \frac{d\lambda}{dx} = & -\frac{2\pi\nu_a}{\nu(x)} A_o \cos(k_o x) \left( \frac{A_o}{\alpha_\parallel^o} \cos(k_o x) + \frac{A_1}{\alpha_\parallel^1} \cos(k_1 x) \right) \\ & -\frac{2\pi\nu_a}{\nu(x)} A_1 \cos(k_1 x) \left( \frac{A_o}{\alpha_\parallel^o} \cos(k_o x) + \frac{A_1}{\alpha_\parallel^1} \cos(k_1 x) \right) \\ & -\frac{2\pi\nu_a}{\nu(x)} \frac{A_o}{\alpha_\parallel^o} \sin(k_o x) (A_o \sin(k_o x) + A_1 \sin(k_1 x)) \\ & -\frac{2\pi\nu_a}{\nu(x)} \frac{A_1}{\alpha_\parallel^1} \sin(k_1 x) (A_o \sin(k_o x) + A_1 \sin(k_1 x)) \end{aligned} \quad (79) $$

where $\nu(x)$ is given by:

$$ \nu(x) = \nu_a (A_o \sin(k_o x) + A_1 \sin(k_1 x))^2 \quad (80) $$

and the following relation was used, justified by the requirement that the correction factor $\alpha_\parallel$ for the action referenced in the probability equation (14) must now account for superposition and be calculable from each wavevector:

$$ \alpha_\parallel^o k_o = \alpha_\parallel^1 k_1 \quad (81) $$

Terms can now be combined and simplified using the following identities, of which the Pythagorean identity is a special case:

$$ \begin{aligned} \cos\alpha\cos\beta &= \frac{1}{2} [\cos(\alpha - \beta) + \cos(\alpha + \beta)] \\ \sin\alpha\sin\beta &= \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)] \end{aligned} \quad (82) $$

Before presenting this result, it will be assumed that the $A_n$ coefficients obey the following normalization condition:

$$ A_o^2 + A_1^2 = 1 \quad (83) $$

As such, the following definition for the expectation value of $\alpha_\parallel$ is introduced:

$$ \frac{1}{\bar{\alpha}_\parallel} = \frac{A_o^2}{\alpha_\parallel^o} + \frac{A_1^2}{\alpha_\parallel^1} = \frac{1}{2} \left( \frac{1}{\alpha_\parallel^o} + \frac{1}{\alpha_\parallel^1} \right) \quad (84) $$

where the average of $\frac{1}{\alpha_\parallel}$ coefficients satisfies the requirement of equivalence when the two wavevectors $k_o$ and $k_1$ are nearly identical.

The RHS combined terms of (79) thus leads to the following final expression of (40) for superimposed states as:

$$ \frac{d\lambda}{dx} = -2\pi \frac{\bar{\alpha}_\parallel^{-1} \nu_a (1 + 2A_o A_1 \cos(k_1 - k_o)x)}{\nu(x)} \quad (85) $$

which confirms the identity of $\nu$ in (80) and establishes the exchange energy as:

$$ h\nu_x = \bar{\alpha}_\parallel^{-1} h\nu_a (1 + 2A_o A_1 \cos(k_1 - k_o)x) \quad (86) $$
quantum masses in a system is not independent of their interactions in the system.

Typically, in the analysis of matter in steady-state equilibrium, $\langle\phi'_F|\hat{\chi}_+|\phi\rangle \ge 0$. Hence, it is tempting to suggest that the opposite charge assigned to anti-matter in its pairing with matter is actually an expression of a negative phase projection, i.e. for a constant electromagnetic force, steady-state equilibrium is achieved with $\phi = \phi'_F + \pi$ or $\langle\phi'_F|\hat{\chi}_+|\phi\rangle \le 0$. This would characterize antimatter's oscillation in spacetime with a 180° phase shift compared to matter traveling in the same direction, thus presenting an alternative to the Feynman-Stueckelberg interpretation of equations for anti-matter [25],[26].

As one final observation, the analysis so far demonstrates that classical concepts can still be used to describe quantum behavior in the absence of extreme limits or measurements, and with phase coherence preserved. Furthermore, the analysis of the next section suggests that, in agreement with experiments so far, transitions between quantum states are not spontaneous.

### III. SUPERPOSITION

To provide an explicit example of how superposition plays out within this framework, the case of a particle in a superposition of two unbounded linearly polarized states will now be considered. To start, as with traditional quantum theory, total state $|\phi\rangle$ is represented by the sum of two steady-state solutions:

$$ \langle x|\phi\rangle = A_{\phi} \left( A_o e^{+i\hat{\chi}_{\pm}(k_o x - \phi_o)} + A_1 e^{+i\hat{\chi}_{\pm}(k_1 x - \phi_1)} \right) \quad (76) $$

which involves real-valued amplitudes, $A_o$ and $A_1$. Assuming that the phase offsets are zero and the quanta is moving in the direction of the carrier's propagation, $\hat{x} \cdot \hat{x}_o = 1$ with the probability density written as:

$$ P(x) = N[A_o \sin(k_o x) + A_1 \sin(k_1 x)]^2 \quad (77) $$

where $N = A_\phi^2$ is chosen to ensure the total probability of finding the particle within some arbitrary box of length L is 1.

As done previously with (14), the derivative of the probability is taken to find $\lambda(x)$, only now, $\alpha_\parallel$ is assigned separately to each wavevector and as there is no irreducible form for the expression, $\hat{\chi}_\pm$ acts upon every sub-component:

$$ \lambda(x) = 2\pi \frac{A_o \cos(k_o x) + A_1 \cos(k_1 x)}{A_o \alpha_\parallel^o k_o \sin(k_o x) + A_1 \alpha_\parallel^1 k_1 \sin(k_1 x)} \quad (78) $$

Substituting this result into (40) and taking the derivative of the LHS, one gets:

$$ \begin{aligned} \frac{d\lambda}{dx} = & -\frac{2\pi\nu_a}{\nu(x)} A_o \cos(k_o x) \left( \frac{A_o}{\alpha_\parallel^o} \cos(k_o x) + \frac{A_1}{\alpha_\parallel^1} \cos(k_1 x) \right) \\ & -\frac{2\pi\nu_a}{\nu(x)} A_1 \cos(k_1 x) \left( \frac{A_o}{\alpha_\parallel^o} \cos(k_o x) + \frac{A_1}{\alpha_\parallel^1} \cos(k_1 x) \right) \\ & -\frac{2\pi\nu_a}{\nu(x)} \frac{A_o}{\alpha_\parallel^o} \sin(k_o x) (A_o \sin(k_o x) + A_1 \sin(k_1 x)) \\ & -\frac{2\pi\nu_a}{\nu(x)} \frac{A_1}{\alpha_\parallel^1} \sin(k_1 x) (A_o \sin(k_o x) + A_1 \sin(k_1 x)) \end{aligned} \quad (79) $$

where $\nu(x)$ is given by:

$$ \nu(x) = \nu_a (A_o \sin(k_o x) + A_1 \sin(k_1 x))^2 \quad (80) $$

and the following relation was used, justified by the requirement that the correction factor $\alpha_\parallel$ for the action referenced in the probability equation (14) must now account for superposition and be calculable from each wavevector:

$$ \alpha_\parallel^o k_o = \alpha_\parallel^1 k_1 \quad (81) $$

Terms can now be combined and simplified using the following identities, of which the Pythagorean identity is a special case:

$$ \begin{aligned} \cos\alpha\cos\beta &= \frac{1}{2} [\cos(\alpha - \beta) + \cos(\alpha + \beta)] \\ \sin\alpha\sin\beta &= \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)] \end{aligned} \quad (82) $$

Before presenting this result, it will be assumed that the $A_n$ coefficients obey the following normalization condition:

$$ A_o^2 + A_1^2 = 1 \quad (83) $$

As such, the following definition for the expectation value of $\alpha_\parallel$ is introduced:

$$ \frac{1}{\bar{\alpha}_\parallel} = \frac{A_o^2}{\alpha_\parallel^o} + \frac{A_1^2}{\alpha_\parallel^1} = \frac{1}{2} \left( \frac{1}{\alpha_\parallel^o} + \frac{1}{\alpha_\parallel^1} \right) \quad (84) $$

where the average of $\frac{1}{\alpha_\parallel}$ coefficients satisfies the requirement of equivalence when the two wavevectors $k_o$ and $k_1$ are nearly identical.

The RHS combined terms of (79) thus leads to the following final expression of (40) for superimposed states as:

$$ \frac{d\lambda}{dx} = -2\pi \frac{\bar{\alpha}_\parallel^{-1} \nu_a (1 + 2A_o A_1 \cos(k_1 - k_o)x)}{\nu(x)} \quad (85) $$

which confirms the identity of $\nu$ in (80) and establishes the exchange energy as:

$$ h\nu_x = \bar{\alpha}_\parallel^{-1} h\nu_a (1 + 2A_o A_1 \cos(k_1 - k_o)x) \quad (86) $$
It's useful to recognize that the linear superposition of individual states in (76) and (77) is preserved in the expression of $\nu(x)$. Given the dependence of energy on $\nu(x)$, further justification of (85) can be sought by checking to see if the energy is consistent with (86). Here, unlike the single-state case, the exchange energy varies with a beat frequency determined by the difference in wavevectors.

First, $\lambda(x)$ in (78) is rewritten based on the results so far:

$$ \lambda(x) = \bar{\lambda} \frac{A_o \cos(k_o x) + A_1 \cos(k_1 x)}{A_o \sin(k_o x) + A_1 \sin(k_1 x)} \quad (87) $$

where the definition of $\bar{\lambda}$ is as follows:

$$ \bar{\lambda} = \frac{2\pi}{\alpha_{\parallel}^o k_o} = \frac{2\pi}{\alpha_{\parallel}^1 k_1} \quad (88) $$

Taking another look at (57), the following result is readily derived from the expression for $\lambda(x)$ and $\nu(x)$:

$$ m_o = m (A_o \cos(k_o x) + A_1 \cos(k_1 x))^2 \quad (89) $$

with the enforcement of the requirement:

$$ m_o = \frac{h}{\gamma \bar{\alpha}_{\parallel} \bar{\lambda}^2 \nu_a} \quad (90) $$

Using the same analysis as for the single state case, the expression for the energy of the particle then becomes:

$$ \begin{aligned} E &= h\nu_a (A_o \sin(k_o x) + A_1 \sin(k_1 x))^2 \\ & \quad + m_o \gamma c^2 (A_o \cos(k_o x) + A_1 \cos(k_1 x))^2 \pm h\nu' \\ &= h\nu_a (1 + 2A_o A_1 \cos(k_1 - k_o)x) \pm h\nu' \end{aligned} \quad (91) $$

where again the normalization of coefficients was used, as well as (82). It is thus seen that the difference between (86) and (91) is the $\bar{\alpha}_{\parallel}^{-1}$ coefficient.

This coefficient represents the conversion between frequencies $\nu_a$ and $\nu_e$ but due to their respective dependencies on different velocities, $\bar{\alpha}_{\parallel}^{-1}$ also represents a conversion between the total energy scale and kinetic energy scale. Recalling the analysis for a single particle state, on average the virtual force does work $h\bar{\nu}_e$, representing a change in kinetic energy that transitions with the cosine function between the two energy levels of the superimposed states. Thus, not only is the virtual force influencing the propagation of the superimposed states, it is assisting in the transfer of energy between them, consistent with the uncertainty principle.

Unlike classical or standard quantum theory, the particle's energy is not constant with superposition and the reason is that the net work done by the background isn't zero with each cycle of the particle's oscillation. However, over a longer spacetime cycle, it is. This means that the particle is in quasi-equilibrium and the oscillation of its energy will continue until a state transition occurs. In order for this calculation to be consistent, then, both of the original single-state energies must be expressed in (91).

<figure><img src="image_5.png" /><figcaption>FIG. 3. Energy levels and expectation values for a superposition of states in which either absorption or emission of a quantum of energy occurs.</figcaption></figure>

The only way for this to happen is to place additional requirements on the coefficients $A_n$ along with (83) such that the following is obeyed:

$$ \bar{E} = \frac{1}{2}(E_1 + E_o) \quad (92) $$

$$ 2A_o A_1 \bar{E} = \frac{1}{2} (E_1 - E_o) \equiv \frac{\Delta E}{2} \quad (93) $$

Here, as shown in Figure 3, the energy of the particle oscillates about the average of the two single-state energies but as seen further below, the expectation value depends on whether emission or absorption is favored. The exchange energy is half the difference that will either boost the particle to the higher energy, $E_1$, or reduce it to the lower energy, $E_o$, hence the impact of the sign variation of the cosine function is represented by $\Delta E$. The coefficients that solve these equations are given by:

$$ A_o^2, A_1^2 = \frac{1}{2} \mp \frac{1}{2} \left[ 1 - \left( \frac{\Delta E}{2\bar{E}} \right)^2 \right]^{\frac{1}{2}} \quad (94) $$

where the assignment of $\mp$ to which coefficient depends on whether or not the higher or lower energy state is more favored, which impacts the resultant expectation value for the energy:

$$ \langle E \rangle = A_o^2 E_o + A_1^2 E_1 = \bar{E} \pm \frac{1}{2} \Delta E \left[ 1 - \left( \frac{\Delta E}{2\bar{E}} \right)^2 \right]^{\frac{1}{2}} \quad (95) $$

In summary, unlike standard quantum theory, the superposition of states is not determined by probability factors, a.k.a. $A_n$ coefficients, that are arbitrary. Hence, one can say superposition in this framework is classical and deterministic. However, one could treat the coefficients as unknowns if history is not tracked for a particle, or if multiple particles are entangled and indistinguishability applies. Then (95) expresses an uncertainty ($\Delta E$) for the expectation value of energy, which again is unlike standard quantum theory's calculation since that does not include uncertainty in its expression, though a standard deviation can be computed separately.
It's useful to recognize that the linear superposition of individual states in (76) and (77) is preserved in the expression of $\nu(x)$. Given the dependence of energy on $\nu(x)$, further justification of (85) can be sought by checking to see if the energy is consistent with (86). Here, unlike the single-state case, the exchange energy varies with a beat frequency determined by the difference in wavevectors.

First, $\lambda(x)$ in (78) is rewritten based on the results so far:

$$ \lambda(x) = \bar{\lambda} \frac{A_o \cos(k_o x) + A_1 \cos(k_1 x)}{A_o \sin(k_o x) + A_1 \sin(k_1 x)} \quad (87) $$

where the definition of $\bar{\lambda}$ is as follows:

$$ \bar{\lambda} = \frac{2\pi}{\alpha_{\parallel}^o k_o} = \frac{2\pi}{\alpha_{\parallel}^1 k_1} \quad (88) $$

Taking another look at (57), the following result is readily derived from the expression for $\lambda(x)$ and $\nu(x)$:

$$ m_o = m (A_o \cos(k_o x) + A_1 \cos(k_1 x))^2 \quad (89) $$

with the enforcement of the requirement:

$$ m_o = \frac{h}{\gamma \bar{\alpha}_{\parallel} \bar{\lambda}^2 \nu_a} \quad (90) $$

Using the same analysis as for the single state case, the expression for the energy of the particle then becomes:

$$ \begin{aligned} E &= h\nu_a (A_o \sin(k_o x) + A_1 \sin(k_1 x))^2 \\ & \quad + m_o \gamma c^2 (A_o \cos(k_o x) + A_1 \cos(k_1 x))^2 \pm h\nu' \\ &= h\nu_a (1 + 2A_o A_1 \cos(k_1 - k_o)x) \pm h\nu' \end{aligned} \quad (91) $$

where again the normalization of coefficients was used, as well as (82). It is thus seen that the difference between (86) and (91) is the $\bar{\alpha}_{\parallel}^{-1}$ coefficient.

This coefficient represents the conversion between frequencies $\nu_a$ and $\nu_e$ but due to their respective dependencies on different velocities, $\bar{\alpha}_{\parallel}^{-1}$ also represents a conversion between the total energy scale and kinetic energy scale. Recalling the analysis for a single particle state, on average the virtual force does work $h\bar{\nu}_e$, representing a change in kinetic energy that transitions with the cosine function between the two energy levels of the superimposed states. Thus, not only is the virtual force influencing the propagation of the superimposed states, it is assisting in the transfer of energy between them, consistent with the uncertainty principle.

Unlike classical or standard quantum theory, the particle's energy is not constant with superposition and the reason is that the net work done by the background isn't zero with each cycle of the particle's oscillation. However, over a longer spacetime cycle, it is. This means that the particle is in quasi-equilibrium and the oscillation of its energy will continue until a state transition occurs. In order for this calculation to be consistent, then, both of the original single-state energies must be expressed in (91).

<figure><img src="image_4.png" /><figcaption>FIG. 3. Energy levels and expectation values for a superposition of states in which either absorption or emission of a quantum of energy occurs.</figcaption></figure>

The only way for this to happen is to place additional requirements on the coefficients $A_n$ along with (83) such that the following is obeyed:

$$ \bar{E} = \frac{1}{2}(E_1 + E_o) \quad (92) $$

$$ 2A_o A_1 \bar{E} = \frac{1}{2} (E_1 - E_o) \equiv \frac{\Delta E}{2} \quad (93) $$

Here, as shown in Figure 3, the energy of the particle oscillates about the average of the two single-state energies but as seen further below, the expectation value depends on whether emission or absorption is favored. The exchange energy is half the difference that will either boost the particle to the higher energy, $E_1$, or reduce it to the lower energy, $E_o$, hence the impact of the sign variation of the cosine function is represented by $\Delta E$. The coefficients that solve these equations are given by:

$$ A_o^2, A_1^2 = \frac{1}{2} \mp \frac{1}{2} \left[ 1 - \left( \frac{\Delta E}{2\bar{E}} \right)^2 \right]^{\frac{1}{2}} \quad (94) $$

where the assignment of $\mp$ to which coefficient depends on whether or not the higher or lower energy state is more favored, which impacts the resultant expectation value for the energy:

$$ \langle E \rangle = A_o^2 E_o + A_1^2 E_1 = \bar{E} \pm \frac{1}{2} \Delta E \left[ 1 - \left( \frac{\Delta E}{2\bar{E}} \right)^2 \right]^{\frac{1}{2}} \quad (95) $$

In summary, unlike standard quantum theory, the superposition of states is not determined by probability factors, a.k.a. $A_n$ coefficients, that are arbitrary. Hence, one can say superposition in this framework is classical and deterministic. However, one could treat the coefficients as unknowns if history is not tracked for a particle, or if multiple particles are entangled and indistinguishability applies. Then (95) expresses an uncertainty ($\Delta E$) for the expectation value of energy, which again is unlike standard quantum theory's calculation since that does not include uncertainty in its expression, though a standard deviation can be computed separately.
Regardless of how the quantum state is represented, either with definite assignments to the $A_n$ coefficients or a set of assignments, quantum collapse is depicted here as an orderly transition that reduces the uncertainty to zero over time. It does not require the loss of phase coherence or any sudden, discontinuous transition to a final state, which has been the subject of experiments and study by spontaneous localization theories [4]. As to how superposition takes place, one can consider the particle-in-a-box problem discussed in a following section.

For here, such particles will have phase ($\phi_t$ and $\phi_x$) noise before equilibrium is established, which, depending on depth of penetration, may result in an inelastic collision with the barrier. This happens if $\phi$ is evolving, reducing the strength of the barrier's force, as a particle penetrates a barrier and then reflects such that the net work done at the barrier boundary is nonzero. Such a mechanism introduces a quasi-stable oscillation of energy into the system, a source of uncertainty, and as such, the particle's state must be represented as a superposition of free-particle solutions.

As noted previously, in this framework such states are identical to those inside a quantum well, discussed in Section VIII, whereby the phase offset $\phi_o$ adjusts based on interactions with the barrier. Regarding which equilibrium solutions will be triggered for a change in the particle's energy such that a stable state is manifested within the well, the $A_n$ coefficients can represent a probability just as for Schrödinger's theory, since phase evolution and interactions with the background are untracked.

## IV. NEWTON'S LAW REVISITED WITH A CLASSICALLY CONSTANT APPLIED FORCE

Given the dynamic properties of a particle in uniform motion, it is natural to ask what it means to have a constant applied force. Is it, in fact, constant? To find out, a similar approach for analysis will be undertaken as before, only the classical concept of force will be denoted as $\bar{F}$, with its corresponding representation in this framework given by:

$$ \mathbf{F} = F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle + F_{v} \langle \phi'_{F} | \hat{h}_{s} \cdot \hat{\chi}_{\pm} | \phi \rangle \quad (96) $$

where $F_v$ is the virtual force amplitude assigned to a constant background potential.

From the prior analysis in Section II, the virtual force's phase projection term was statistically averaged to $\pm 1$ because it was assumed that the particles acquired a constant velocity $v_o$ in the presence of a background potential. The statistical average over phase offsets $\phi_t$ is not in place for the constant force in this analysis because just as forces and phases can be summed, the quantum state $|\phi\rangle$ can be represented as the sum of two phase states:

$$ |\phi\rangle = |\phi_F\rangle + |\phi_{\pm} + \phi_t\rangle \quad (97) $$

where phase noise $\phi_t$ resides with the steady-state solutions assigned to a virtual force.

While the $\hat{h}_s \cdot \hat{\chi}_{\pm}$ operator provides sign compensation for the correct pairing of states summed in the $\phi_t$ averaging process, leading to $\pm 1$ for the virtual force projection term, the $\phi_F$ states, implicitly dependent on the $\phi_{\pm}$ phase subtracted from them, do not have the phase noise aligning them constructively. Instead, the $\hat{h}_s \cdot \hat{\chi}_{\pm}$ operator forces the paired-up states $\phi_F$ with $\phi_{\pm}$ to combine out of phase and vanish. Likewise, the superposition of states generating $\phi_{\pm}$ do not have sign compensation provided by the $\hat{h}_s$ operator for the applied force projection term. Hence:

$$ \begin{gathered} \left| \langle \phi'_{F} | \hat{h}_{s} \cdot \hat{\chi}_{\pm} | \phi_{F} \rangle \right| = 0 \\ \left| \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi_{\pm} + \phi_{t} \rangle \right| = 0 \end{gathered} \quad (98) $$

Before proceeding, it is worth noting that the analysis of this paper ultimately describes motion, which as discussed can be expressed in terms of the carrier's periodicity $\lambda_a$ and $\nu_a$, or the envelope's horizontal periodicity with $\lambda_e$ and $\nu_e$. Either description is valid because the equations introduce cancellations for the scale factor conversions between these parameters. For a fully relativistic calculation it may be convenient to start with the amplitudes, such as for the hydrogen atom, while for non-relativistic calculations such as this problem, the envelope parameters are preferred due to their dependence on $\bar{v}$ as $\bar{v} \to 0$.

To begin, in reference to (1), an equivalent equation to (8) in terms of $\tilde{k}$ is used in this analysis for convenience:

$$ F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle \mp F_{v} = \frac{d}{dt} \left( \hbar \hat{\chi}_{\pm} \tilde{k} \right) = \hbar \hat{\chi}_{\pm} \frac{d\tilde{k}}{dt} = \hbar \hat{\chi}_{\pm} \frac{d\tilde{k}}{d\bar{x}} \cdot \bar{v} \quad (99) $$

where it assumed the motion of the particle's quanta and force are aligned in the same direction, and from (41) the $\mp$ sign is attached to $F_v$. With $\tilde{k}v = 2\pi\nu$, this gives:

$$ F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle \mp F_{v} = \hat{\chi}_{\pm} \frac{1}{\tilde{k}} \left( \frac{d\tilde{k}}{d\bar{x}} \cdot \hat{v} \right) h\nu \quad (100) $$

where it is noted that the application of $\hat{\chi}_{\pm}$ is in the order shown from left to right.

As discussed for virtual forces, there are two components that solve separate expressions of this equation. One pertains to characterizing carrier evolution, represented by wavevector $k_a$, and the other for the horizontal extent of the envelope, represented by wavevector $k_e$. It will be shown here how one only needs to solve explicitly for the envelope, while using properties of the carrier's dynamics already explored for a free particle to characterize its solution when a force is applied. This can be summarized by stating that the carrier dynamics in 1-D is only a function of energy, with the carrier's amplitude envelope shaped by forces. Hence, it is expected that (26) still applies with $k_a = \frac{c^2}{\bar{v}^2} k_e$ and $\nu_a = \frac{c^2}{\bar{v}^2} \nu_e$, where $\bar{v}$ can be determined from classical analysis.

Expressing $\tilde{k}$ and $\nu$ as phase-dependent functions in general for the accelerating particle with an amplitude
Regardless of how the quantum state is represented, either with definite assignments to the $A_n$ coefficients or a set of assignments, quantum collapse is depicted here as an orderly transition that reduces the uncertainty to zero over time. It does not require the loss of phase coherence or any sudden, discontinuous transition to a final state, which has been the subject of experiments and study by spontaneous localization theories [4]. As to how superposition takes place, one can consider the particle-in-a-box problem discussed in a following section.

For here, such particles will have phase ($\phi_t$ and $\phi_x$) noise before equilibrium is established, which, depending on depth of penetration, may result in an inelastic collision with the barrier. This happens if $\phi$ is evolving, reducing the strength of the barrier's force, as a particle penetrates a barrier and then reflects such that the net work done at the barrier boundary is nonzero. Such a mechanism introduces a quasi-stable oscillation of energy into the system, a source of uncertainty, and as such, the particle's state must be represented as a superposition of free-particle solutions.

As noted previously, in this framework such states are identical to those inside a quantum well, discussed in Section VIII, whereby the phase offset $\phi_o$ adjusts based on interactions with the barrier. Regarding which equilibrium solutions will be triggered for a change in the particle's energy such that a stable state is manifested within the well, the $A_n$ coefficients can represent a probability just as for Schrödinger's theory, since phase evolution and interactions with the background are untracked.

## IV. NEWTON'S LAW REVISITED WITH A CLASSICALLY CONSTANT APPLIED FORCE

Given the dynamic properties of a particle in uniform motion, it is natural to ask what it means to have a constant applied force. Is it, in fact, constant? To find out, a similar approach for analysis will be undertaken as before, only the classical concept of force will be denoted as $\bar{F}$, with its corresponding representation in this framework given by:

$$ \mathbf{F} = F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle + F_{v} \langle \phi'_{F} | \hat{h}_{s} \cdot \hat{\chi}_{\pm} | \phi \rangle \quad (96) $$

where $F_v$ is the virtual force amplitude assigned to a constant background potential.

From the prior analysis in Section II, the virtual force's phase projection term was statistically averaged to $\pm 1$ because it was assumed that the particles acquired a constant velocity $v_o$ in the presence of a background potential. The statistical average over phase offsets $\phi_t$ is not in place for the constant force in this analysis because just as forces and phases can be summed, the quantum state $|\phi\rangle$ can be represented as the sum of two phase states:

$$ |\phi\rangle = |\phi_F\rangle + |\phi_{\pm} + \phi_t\rangle \quad (97) $$

where phase noise $\phi_t$ resides with the steady-state solutions assigned to a virtual force.

While the $\hat{h}_s \cdot \hat{\chi}_{\pm}$ operator provides sign compensation for the correct pairing of states summed in the $\phi_t$ averaging process, leading to $\pm 1$ for the virtual force projection term, the $\phi_F$ states, implicitly dependent on the $\phi_{\pm}$ phase subtracted from them, do not have the phase noise aligning them constructively. Instead, the $\hat{h}_s \cdot \hat{\chi}_{\pm}$ operator forces the paired-up states $\phi_F$ with $\phi_{\pm}$ to combine out of phase and vanish. Likewise, the superposition of states generating $\phi_{\pm}$ do not have sign compensation provided by the $\hat{h}_s$ operator for the applied force projection term. Hence:

$$ \begin{gathered} \left| \langle \phi'_{F} | \hat{h}_{s} \cdot \hat{\chi}_{\pm} | \phi_{F} \rangle \right| = 0 \\ \left| \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi_{\pm} + \phi_{t} \rangle \right| = 0 \end{gathered} \quad (98) $$

Before proceeding, it is worth noting that the analysis of this paper ultimately describes motion, which as discussed can be expressed in terms of the carrier's periodicity $\lambda_a$ and $\nu_a$, or the envelope's horizontal periodicity with $\lambda_e$ and $\nu_e$. Either description is valid because the equations introduce cancellations for the scale factor conversions between these parameters. For a fully relativistic calculation it may be convenient to start with the amplitudes, such as for the hydrogen atom, while for non-relativistic calculations such as this problem, the envelope parameters are preferred due to their dependence on $\bar{v}$ as $\bar{v} \to 0$.

To begin, in reference to (1), an equivalent equation to (8) in terms of $\tilde{k}$ is used in this analysis for convenience:

$$ F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle \mp F_{v} = \frac{d}{dt} \left( \hbar \hat{\chi}_{\pm} \tilde{k} \right) = \hbar \hat{\chi}_{\pm} \frac{d\tilde{k}}{dt} = \hbar \hat{\chi}_{\pm} \frac{d\tilde{k}}{d\bar{x}} \cdot \bar{v} \quad (99) $$

where it assumed the motion of the particle's quanta and force are aligned in the same direction, and from (41) the $\mp$ sign is attached to $F_v$. With $\tilde{k}v = 2\pi\nu$, this gives:

$$ F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle \mp F_{v} = \hat{\chi}_{\pm} \frac{1}{\tilde{k}} \left( \frac{d\tilde{k}}{d\bar{x}} \cdot \hat{v} \right) h\nu \quad (100) $$

where it is noted that the application of $\hat{\chi}_{\pm}$ is in the order shown from left to right.

As discussed for virtual forces, there are two components that solve separate expressions of this equation. One pertains to characterizing carrier evolution, represented by wavevector $k_a$, and the other for the horizontal extent of the envelope, represented by wavevector $k_e$. It will be shown here how one only needs to solve explicitly for the envelope, while using properties of the carrier's dynamics already explored for a free particle to characterize its solution when a force is applied. This can be summarized by stating that the carrier dynamics in 1-D is only a function of energy, with the carrier's amplitude envelope shaped by forces. Hence, it is expected that (26) still applies with $k_a = \frac{c^2}{\bar{v}^2} k_e$ and $\nu_a = \frac{c^2}{\bar{v}^2} \nu_e$, where $\bar{v}$ can be determined from classical analysis.

Expressing $\tilde{k}$ and $\nu$ as phase-dependent functions in general for the accelerating particle with an amplitude
and envelope phase term that are dependent on $x$, consistent with the solution to (16):

$$ \mathrm{Re} \left( \tilde{k}(x) \right) = k_e^a(x) k_e^{\phi}(x) \quad (101) $$

$$ \nu_e(x) = \nu_e^a(x) \nu_e^{\phi}(x) \quad (102) $$

The real part of (100) then takes the following form:

$$ F \langle \phi_F' | \hat{\chi}_{\pm} | \phi \rangle = \frac{h \nu_e^a}{k_e^a} \frac{dk_e^a}{dx} (\hat{\chi}_{\pm} \nu_e^{\phi}) + \nu_e^a \nu_e^{\phi} \left( \hat{\chi}_{\pm} \frac{h}{k_e^{\phi}} \frac{dk_e^{\phi}}{dx} \right) \quad (103) $$

Classically, a particle with an initial velocity $v_o$ will undergo constant acceleration in response to $\bar{F}$ with the following non-relativistic velocity as a function of displacement $x$:

$$ \bar{v} \equiv v_e^a(x) = \sqrt{2 \frac{\bar{F}}{m_o} x + v_o^2} \quad (104) $$

which is obtained by solving the quadratic equation that expresses displacement as a function of time, $x = \frac{1}{2} \frac{\bar{F}}{m_o} t^2 + v_o t$, and velocity as a function of time, $v = v_o + \frac{\bar{F}}{m_o} t$.

Using the classical correspondence limit for momentum, it is expected that the amplitude for the wavevector of a particle $k_e^a(x)$ will take the following form in the non-relativistic limit:

$$ p_a(x) = m_o v_e^a(x) = \hbar k_e^a(x) \quad (105) $$

$$ k_e^a(x) = \frac{m_o}{\hbar} \sqrt{2 \frac{\bar{F}}{m_o} x + v_o^2} \quad (106) $$

In line with the prior analysis for free particles, the non-relativistic correspondence limit for energy is obtained by equating the work done to the frequency as a function of $x$. To do this calculation, the following relationship is generated in the non-relativistic limit using (5):

$$ T = \frac{1}{2} m_o v^2 = \frac{1}{2} m_o \lambda_e^2 \nu_e^2 = \frac{1}{2} h \nu_e \quad (107) $$

with $\gamma \to 1$. This leads to the following result:

$$ \begin{aligned} \frac{1}{2} h \nu_e^a(x) &= \bar{F} x + \frac{1}{2} m_o v_o^2 \\ h \nu_e^a(x) &= 2 \bar{F} x + m_o v_o^2 \end{aligned} \quad (108) $$

The first term of (103) absent the $\nu_e^{\phi}(x)$ term then reduces to the following, using (104) and (106):

$$ \frac{h \nu_e^a(x)}{k_e^a(x)} \frac{dk_e^a}{dx} = \frac{2 \bar{F} x + m_o v_o^2}{\frac{m_o}{\hbar} \left( 2 \frac{\bar{F}}{m_o} x + v_o^2 \right)} \frac{\bar{F}}{\hbar} = \bar{F} \quad (109) $$

where it is seen that the classical notion of force, $\bar{F}$, can be reinterpreted as an operator on $k_e^a$:

$$ \bar{F} k_e^a(x) = h \nu_e^a \frac{dk_e^a}{dx} \quad (110) $$

As a verification that the analysis could be done with momentum amplitude $p_a(x)$ in place of $p_e(x)$, the conversion $k_a^a = \frac{c^2}{\bar{v}^2} k_e^a$ can be used to verify that $\frac{1}{k_a^a} \frac{dk_a^a}{dx}$ yields the same result as (109).

For consideration of $F$'s influence on the phase $k_e^{\phi}$, the solution to (103) consistent with (17) and (24) in the prior analysis for $k$ with a constant potential is the following:

$$ k_e^{\phi}(x) = \tan \left( \int^{x} k_e^{a}(x') dx' - \phi_o \right) \quad (111) $$

Here, (103) is recognized as a generalization of the prior constant potential analysis with a straightforward extension of that solution to account for a space-dependent transverse amplitude. Evaluating part of the second term of (103), one obtains:

$$ \begin{aligned} \hat{\chi}_{\pm} \frac{h}{k_e^{\phi}} \frac{dk_e^{\phi}}{dx} &= h k_e^a(x) \hat{\chi}_{\pm} \left[ \frac{\cos^{-2} \left( \int^x k_e^a(x') dx' - \phi_o \right)}{\tan \left( \int^x k_e^a(x') dx' - \phi_o \right)} \right] \\ &= - \frac{h k_e^a(x)}{\hat{\chi}_{\pm} \cos \left( \int^x k_e^a(x') dx' - \phi_o \right) \sin \left( \int^x k_e^a(x') dx' - \phi_o \right)} \\ &= \mp \frac{h k_e^a(x)}{\sin^2 \left( \int^x k_e^a(x') dx' - \phi_o \right)} = \mp \frac{h k_e^a(x)}{\nu_e^{\phi}(x)} \end{aligned} \quad (112) $$

which matches the prior constant potential analysis (43) and (56) with $\bar{F}$ absent and $\gamma \to 1$ where, using (108), the frequency takes the following form:

$$ \begin{aligned} h \nu_e(x) &= h \nu_e^a(x) \nu_e^{\phi}(x) \\ &= (2 \bar{F} x + m_o v_o^2) \sin^2 \left( \int^x k_e^a(x') dx' - \phi_o \right) \end{aligned} \quad (113) $$

Altogether, then, (103) reduces to the following:

$$ F \langle \phi_F' | \hat{\chi}_{\pm} | \phi \rangle \mp F_v = \bar{F} \hat{\chi}_{\pm} \nu_e^{\phi}(x) \mp h k_e^a(x) \nu_e^a(x) \quad (114) $$

which yields the following assignments:

$$ \begin{aligned} F &= \bar{F} \\ \langle \phi_F' | \hat{\chi}_{\pm} | \phi \rangle &= \cos^2 \left( \int^x k_e^a(x') dx' - \phi_o \right) \quad (115) \\ F_v &= h k_e^a(x) \nu_e^a(x) \end{aligned} $$

where $F_v$ matches the analysis for a constant potential in (34). Also, $\nu_x = \nu_e = \nu_e^a$ when $\bar{F} = 0$ in (108).

Just like for a virtual force, the phase assigned to a classical force and that of a particle with a spin-correction factor are in lock step with one another:

$$ \langle \phi_F' | x \rangle = \langle x | \hat{\chi}_{\pm} | \phi \rangle = \cos \left( \int^x k_e^a(x') dx' - \phi_o \right) \quad (116) $$

which is absent of noise with the assumption that phase mediation has established a steady-state equilibrium.
and envelope phase term that are dependent on $x$, consistent with the solution to (16):

$$ \mathrm{Re} \left( \tilde{k}(x) \right) = k_e^a(x) k_e^{\phi}(x) \quad (101) $$

$$ \nu_e(x) = \nu_e^a(x) \nu_e^{\phi}(x) \quad (102) $$

The real part of (100) then takes the following form:

$$ F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle = \frac{h \nu_{e}^{a} dk_{e}^{a}}{k_{e}^{a} dx} (\hat{\chi}_{\pm} \nu_{e}^{\phi}) + \nu_{e}^{a} \nu_{e}^{\phi} \left( \hat{\chi}_{\pm} \frac{h}{k_{e}^{\phi}} \frac{dk_{e}^{\phi}}{dx} \right) \quad (103) $$

Classically, a particle with an initial velocity $v_o$ will undergo constant acceleration in response to $\bar{F}$ with the following non-relativistic velocity as a function of displacement $x$:

$$ \bar{v} \equiv v_e^a(x) = \sqrt{2 \frac{\bar{F}}{m_o} x + v_o^2} \quad (104) $$

which is obtained by solving the quadratic equation that expresses displacement as a function of time, $x = \frac{1}{2} \frac{\bar{F}}{m_o} t^2 + v_o t$, and velocity as a function of time, $v = v_o + \frac{\bar{F}}{m_o} t$.

Using the classical correspondence limit for momentum, it is expected that the amplitude for the wavevector of a particle $k_e^a(x)$ will take the following form in the non-relativistic limit:

$$ p_a(x) = m_o v_e^a(x) = \hbar k_e^a(x) \quad (105) $$

$$ k_e^a(x) = \frac{m_o}{\hbar} \sqrt{2 \frac{\bar{F}}{m_o} x + v_o^2} \quad (106) $$

In line with the prior analysis for free particles, the non-relativistic correspondence limit for energy is obtained by equating the work done to the frequency as a function of $x$. To do this calculation, the following relationship is generated in the non-relativistic limit using (5):

$$ T = \frac{1}{2} m_o v^2 = \frac{1}{2} m_o \lambda_e^2 \nu_e^2 = \frac{1}{2} h \nu_e \quad (107) $$

with $\gamma \to 1$. This leads to the following result:

$$ \begin{aligned} \frac{1}{2} h \nu_e^a(x) &= \bar{F}x + \frac{1}{2} m_o v_o^2 \\ h \nu_e^a(x) &= 2\bar{F}x + m_o v_o^2 \end{aligned} \quad (108) $$

The first term of (103) absent the $\nu_e^\phi(x)$ term then reduces to the following, using (104) and (106):

$$ \frac{h \nu_e^a(x)}{k_e^a(x)} \frac{dk_e^a}{dx} = \frac{2 \bar{F} x + m_o v_o^2}{\frac{m_o}{\hbar} \left( 2 \frac{\bar{F}}{m_o} x + v_o^2 \right)} \frac{\bar{F}}{\hbar} = \bar{F} \quad (109) $$

where it is seen that the classical notion of force, $\bar{F}$, can be reinterpreted as an operator on $k_e^a$:

$$ \bar{F} k_e^a(x) = h \nu_e^a \frac{dk_e^a}{dx} \quad (110) $$

As a verification that the analysis could be done with momentum amplitude $p_a(x)$ in place of $p_e(x)$, the conversion $k_a^a = \frac{c^2}{\bar{v}^2} k_e^a$ can be used to verify that $\frac{1}{k_a^a} \frac{dk_a^a}{dx}$ yields the same result as (109).

For consideration of $F$'s influence on the phase $k_e^\phi$, the solution to (103) consistent with (17) and (24) in the prior analysis for $k$ with a constant potential is the following:

$$ k_e^\phi(x) = \tan \left( \int^x k_e^a(x') dx' - \phi_o \right) \quad (111) $$

Here, (103) is recognized as a generalization of the prior constant potential analysis with a straightforward extension of that solution to account for a space-dependent transverse amplitude. Evaluating part of the second term of (103), one obtains:

$$ \begin{aligned} \hat{\chi}_{\pm} \frac{h}{k_e^{\phi}} \frac{dk_e^{\phi}}{dx} &= h k_e^a(x) \hat{\chi}_{\pm} \left[ \frac{\cos^{-2} \left( \int^x k_e^a(x') dx' - \phi_o \right)}{\tan \left( \int^x k_e^a(x') dx' - \phi_o \right)} \right] \\ &= - \frac{h k_e^a(x)}{\hat{\chi}_{\pm} \cos \left( \int^x k_e^a(x') dx' - \phi_o \right) \sin \left( \int^x k_e^a(x') dx' - \phi_o \right)} \\ &= \mp \frac{h k_e^a(x)}{\sin^2 \left( \int^x k_e^a(x') dx' - \phi_o \right)} = \mp \frac{h k_e^a(x)}{\nu_e^{\phi}(x)} \end{aligned} \quad (112) $$

which matches the prior constant potential analysis (43) and (56) with $\bar{F}$ absent and $\gamma \to 1$ where, using (108), the frequency takes the following form:

$$ \begin{aligned} h \nu_e(x) &= h \nu_e^a(x) \nu_e^{\phi}(x) \\ &= (2 \bar{F} x + m_o v_o^2) \sin^2 \left( \int^x k_e^a(x') dx' - \phi_o \right) \end{aligned} \quad (113) $$

Altogether, then, (103) reduces to the following:

$$ F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle \mp F_v = \bar{F} \hat{\chi}_{\pm} \nu_e^{\phi}(x) \mp h k_e^a(x) \nu_e^a(x) \quad (114) $$

which yields the following assignments:

$$ \begin{aligned} F &= \bar{F} \\ \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle &= \cos^2 \left( \int^x k_e^a(x') dx' - \phi_o \right) \quad (115) \\ F_v &= h k_e^a(x) \nu_e^a(x) \end{aligned} $$

where $F_v$ matches the analysis for a constant potential in (34). Also, $\nu_x = \nu_e = \nu_e^a$ when $\bar{F} = 0$ in (108).

Just like for a virtual force, the phase assigned to a classical force and that of a particle with a spin-correction factor are in lock step with one another:

$$ \langle \phi'_{F} | x \rangle = \langle x | \hat{\chi}_{\pm} | \phi \rangle = \cos \left( \int^x k_e^a(x') dx' - \phi_o \right) \quad (116) $$

which is absent of noise with the assumption that phase mediation has established a steady-state equilibrium.
Having a quantum mass definition that incorporates phase as follows:

$$m_{\phi} \equiv \frac{m_o}{\cos^2\left(\int^x k_e^a(x')dx' - \phi_o\right)} \quad (117)$$

the expression matches (57) when only a virtual force is applied and the phase has a linear dependence on $x$.

Given the sine-squared frequency dependence in (113) with a matching cotangent form for the wavelength corresponding to the inverse of (111), a virtual force in this analysis produces the same velocity as the prior analysis, expressed as the product of cosine and sine functions. Since $\hat{\chi}_{\pm}$ is already accounted for in the definition of $m_{\phi}$, the product $m_{\phi}v$ where $v = \sin(\int^x dx' \phi_e)\cos(\int^x dx' \phi_e)$ matches (111) for $p = \hbar\chi_p k_e^a(x)k_e^{\phi}(x)$. If $\hat{\chi}_{\pm}$ is not included, then $m_{\phi}$ would be represented by a sine squared function and the operator, when applied to $m_{\phi}v$ according to (1), with the rules of (2), would again reduce to the same expression.

With the definition of $m_{\phi}$ in (117), the following relation holds:

$$\frac{F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle}{m_o} = \frac{\bar{F}}{m_{\phi}} \quad (118)$$

which states that when accounting for the quantum mass and associated phase, the acceleration obtained for a particle in this framework and that of classical physics is not identical unless additional observations are made.

First, the dynamics in this study can be characterized in one of two ways. Either the force is constant, i.e. $\bar{F}$ is used with mass $m_{\phi}$ causing variable acceleration, or the force is variable with projection term $\langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle$ and the mass $m = m_o$ is constant. Either picture calculates the same acceleration.

Consistent with the discussion presented throughout, different views are convenient, i.e. quantum mass is treated as a constant for certain problems involving forces and as such quantum effects are accounted for by the phase projection term. However, as discussed for a free particle's virtual quantum exchanges of $\hbar\nu_e$ with the background, and noting that the phase dependencies for velocity and frequency consistent with (6) suggests $m_{\phi}$ must be used, as well as for problems such as the hydrogen atom, it is convenient to view quantum mass as a variable quantity.

The question then becomes, how does the classical result of constant acceleration with $\gamma \approx 1$ for a constant force come about? In noting that the phase term in (117) does not support simple harmonic behavior, it is not possible for a spacetime average over the phase to lead to this classical limit. Instead, the theory states that a background exchange process involving energy $\hbar\nu_x$ is responsible for introducing a coherent phase. As a particle accelerates, the virtual force acting on it and required exchange energy increases. At some point $\hbar\nu_x$ exceeds what the background potential is able to provide and as such the phase becomes decoherent. This can be expressed by

letting $\phi_o$ drift to match the evolving phase component involving the wavevector, thus suggesting the following relation in the decoherence limit, on average:

$$\int^x k_e^a(x')dx' - \phi_o = 0 \quad (119)$$

This sets $m_{\phi} = m_o$ with the forces and mass supporting classical Newtonian behavior.

As mentioned, the dynamics of the carrier with wavelength $\lambda_a$ can be determined independently of the envelope. To do this, it will first be argued that $\vec{k}_c$ does not depend directly upon an applied force. The reason is that the speed attributed to wavelength $\lambda_c$ and frequency amplitude $\nu_a$ is already set to the speed of light in (50). Furthermore, both this theory and standard quantum analysis specifies that the frequency amplitude $\nu_a$ depends only on energy. As classical correspondence is required to hold in the decoherence limit, the relationships between the spacetime averages of state variables and classical quantities are well-defined.

This means that semi-classical analysis involving the dynamics of the wavevector's amplitude and its relationship to force is applicable, while the underlying evolution of $\vec{k}_c \cdot \hat{x}$, which characterizes the ratio of $\lambda_a$ to $\lambda_c$, can be determined only from the energy, or the Hamiltonian's value, if $\hat{x}_o \parallel \hat{x}$. In consideration of what has been presented so far, it can be seen how this theoretical framework both coincides with and diverges from standard quantum theory.

It will now be shown that as a result of the arguments just given and the ones that follow, the constant potential analysis used to determine relationships (48) - (54) is still applicable to the case of a constant applied force, only with $\bar{v}$ given by (104). For in the absence of a force term, the same constant-potential solution for $\vec{k}_c$ applies to (100), i.e. there is a direction for $\vec{k}_c$ for which the derivative is zero, suggesting $|\vec{k}_c|$ is a constant. If the particle were going at the speed of light, the direction of $\vec{k}_c$ must be parallel to the velocity based on the requirement that the wave behavior of the particle has no envelope separate from its carrier. Hence the derivative of $\vec{k}_c$ in terms of $x$ is zero. In contrast, in a Galilean reference frame traveling with the particle momentarily, the direction of $\vec{k}_c$ must be perpendicular to the velocity to force the dot product of (100) to zero. These are precisely the properties of the free-particle solution with $k_a$ and $k_{\perp}$ components, only now using $\bar{v}$ calculated in (104) for the carrier described by (23).

The velocity of the quanta moving in the $\hat{x}$ direction with the phase moving along the $\hat{x}_o$ direction is thus given by an expression consistent with (44):

$$v = \frac{2\pi\kappa(x_o)}{k_e^a(x)} \hat{\chi}_{\pm} \frac{\nu_e(x)}{k_e^{\phi}(x)} \quad (120)$$
Having a quantum mass definition that incorporates phase as follows:

$$m_{\phi} \equiv \frac{m_o}{\cos^2\left(\int^x k_e^a(x')dx' - \phi_o\right)} \quad (117)$$

the expression matches (57) when only a virtual force is applied and the phase has a linear dependence on $x$.

Given the sine-squared frequency dependence in (113) with a matching cotangent form for the wavelength corresponding to the inverse of (111), a virtual force in this analysis produces the same velocity as the prior analysis, expressed as the product of cosine and sine functions. Since $\hat{\chi}_{\pm}$ is already accounted for in the definition of $m_{\phi}$, the product $m_{\phi}v$ where $v = \sin(\int^x dx' \phi_e)\cos(\int^x dx' \phi_e)$ matches (111) for $p = \hbar\chi_p k_e^a(x)k_e^{\phi}(x)$. If $\hat{\chi}_{\pm}$ is not included, then $m_{\phi}$ would be represented by a sine squared function and the operator, when applied to $m_{\phi}v$ according to (1), with the rules of (2), would again reduce to the same expression.

With the definition of $m_{\phi}$ in (117), the following relation holds:

$$\frac{F \langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle}{m_o} = \frac{\bar{F}}{m_{\phi}} \quad (118)$$

which states that when accounting for the quantum mass and associated phase, the acceleration obtained for a particle in this framework and that of classical physics is not identical unless additional observations are made.

First, the dynamics in this study can be characterized in one of two ways. Either the force is constant, i.e. $\bar{F}$ is used with mass $m_{\phi}$ causing variable acceleration, or the force is variable with projection term $\langle \phi'_{F} | \hat{\chi}_{\pm} | \phi \rangle$ and the mass $m = m_o$ is constant. Either picture calculates the same acceleration.

Consistent with the discussion presented throughout, different views are convenient, i.e. quantum mass is treated as a constant for certain problems involving forces and as such quantum effects are accounted for by the phase projection term. However, as discussed for a free particle's virtual quantum exchanges of $h\nu_e$ with the background, and noting that the phase dependencies for velocity and frequency consistent with (6) suggests $m_{\phi}$ must be used, as well as for problems such as the hydrogen atom, it is convenient to view quantum mass as a variable quantity.

The question then becomes, how does the classical result of constant acceleration with $\gamma \approx 1$ for a constant force come about? In noting that the phase term in (117) does not support simple harmonic behavior, it is not possible for a spacetime average over the phase to lead to this classical limit. Instead, the theory states that a background exchange process involving energy $h\nu_x$ is responsible for introducing a coherent phase. As a particle accelerates, the virtual force acting on it and required exchange energy increases. At some point $h\nu_x$ exceeds what the background potential is able to provide and as such the phase becomes decoherent. This can be expressed by

letting $\phi_o$ drift to match the evolving phase component involving the wavevector, thus suggesting the following relation in the decoherence limit, on average:

$$\int^x k_e^a(x')dx' - \phi_o = 0 \quad (119)$$

This sets $m_{\phi} = m_o$ with the forces and mass supporting classical Newtonian behavior.

As mentioned, the dynamics of the carrier with wavelength $\lambda_a$ can be determined independently of the envelope. To do this, it will first be argued that $\vec{k}_c$ does not depend directly upon an applied force. The reason is that the speed attributed to wavelength $\lambda_c$ and frequency amplitude $\nu_a$ is already set to the speed of light in (50). Furthermore, both this theory and standard quantum analysis specifies that the frequency amplitude $\nu_a$ depends only on energy. As classical correspondence is required to hold in the decoherence limit, the relationships between the spacetime averages of state variables and classical quantities are well-defined.

This means that semi-classical analysis involving the dynamics of the wavevector's amplitude and its relationship to force is applicable, while the underlying evolution of $\vec{k}_c \cdot \hat{x}$, which characterizes the ratio of $\lambda_a$ to $\lambda_c$, can be determined only from the energy, or the Hamiltonian's value, if $\hat{x}_o \parallel \hat{x}$. In consideration of what has been presented so far, it can be seen how this theoretical framework both coincides with and diverges from standard quantum theory.

It will now be shown that as a result of the arguments just given and the ones that follow, the constant potential analysis used to determine relationships (48) - (54) is still applicable to the case of a constant applied force, only with $\bar{v}$ given by (104). For in the absence of a force term, the same constant-potential solution for $\vec{k}_c$ applies to (100), i.e. there is a direction for $\vec{k}_c$ for which the derivative is zero, suggesting $|\vec{k}_c|$ is a constant. If the particle were going at the speed of light, the direction of $\vec{k}_c$ must be parallel to the velocity based on the requirement that the wave behavior of the particle has no envelope separate from its carrier. Hence the derivative of $\vec{k}_c$ in terms of $x$ is zero. In contrast, in a Galilean reference frame traveling with the particle momentarily, the direction of $\vec{k}_c$ must be perpendicular to the velocity to force the dot product of (100) to zero. These are precisely the properties of the free-particle solution with $k_a$ and $k_{\perp}$ components, only now using $\bar{v}$ calculated in (104) for the carrier described by (23).

The velocity of the quanta moving in the $\hat{x}$ direction with the phase moving along the $\hat{x}_o$ direction is thus given by an expression consistent with (44):

$$v = \frac{2\pi\kappa(x_o)}{k_e^a(x)} \hat{\chi}_{\pm} \frac{\nu_e(x)}{k_e^{\phi}(x)} \quad (120)$$
V. TRAJECTORY OF PARTICLE MOTION

As mentioned, a particle's trajectory is not represented explicitly in the derivation of (40), hence the equation's steady-state solution corresponds to a momentum eigen-state in Schrödinger's theory. With the velocity given as a function of $\vec{x}$, the displacement is $\vec{x}$ at that point, however to find the trajectory, i.e. $\vec{x}$ as a function of time, an integration still must be performed.

The case of a free particle moving in the positive $\hat{x}_o$ direction will now be examined, starting with the velocity as written in (46):

$$ 
\begin{aligned}
v_{x_o}(x) = \frac{dx}{dt} &\approx v_e = \bar{v} \cos^2(k_e x - \phi_o) \quad (121) \\
k_e &= \frac{\bar{v}}{c} k_c
\end{aligned}
$$

Here, a spacetime average is performed over several cycles of $\lambda_a$ and $\frac{\lambda_a}{\lambda_e} \ll 1$ with $\sin(k_a x - \phi_a)$ averaging to zero. The distance $x$ is measured along the $\hat{x}_o$ direction. The velocity in the $\hat{y}_o$ direction in contrast is given by (48) when $\hat{x} \cdot \hat{x}_o = 0$, with $\lambda_c \nu_a = c$:

$$ 
\begin{aligned}
v_{y_o}(y) = \frac{dy_a}{dt} &\approx \frac{c}{\gamma} \cos(k_e x - \phi'_o) \sin(k_e x - \phi'_o) \\
&= \frac{\lambda_c \nu_a}{2\gamma} \sin(2k_e x - 2\phi'_o) \quad (122)
\end{aligned}
$$

where the same spacetime averaging is performed. Also, as noted in the discussion about (22), $\phi'_o$ is different from $\phi_o$ by the factor $\frac{\pi}{2}$.

Both velocities can be used to determine the extent of the waveform in both the $\hat{x}_o$ and $\hat{y}_o$ direction. For the $\hat{x}_o$ direction, the position and time dependence of (121) can be separated, leading to the following derivation:

$$ 
\begin{aligned}
v_{x_o} \int^{t} dt' &= \int^{x} \frac{dx'}{\cos^2(k_e x' - \phi_o)} \\
k_e v_{x_o} t &= \tan(k_e x - \phi_o) \\
x(t) &= \lambda_e \left( \frac{\tan^{-1}(\omega_e t) + \phi_o}{2\pi} \right) \quad (123)
\end{aligned}
$$

Likewise, for the $\hat{y}_o$ direction, the amplitude $y_a$ can be similarly determined with $\phi'_o$ set to zero:

$$ 
\begin{aligned}
\int^{y_a} dy'_a &\approx \frac{\lambda_c \nu_a}{2\gamma} \int^{t} dt' \sin(2\tan^{-1}(\omega_e t')) \\
y_a &= \frac{\lambda_c \nu_a}{2\gamma} \frac{\ln(\omega_e^2 t^2 + 1)}{\omega_e} \quad (124)
\end{aligned}
$$

The $y$ coordinate as a function of $x(t)$ is then obtained by including the carrier phase component $\sin(k_a x(t) - \phi_a)$, in which any constant offset is ignored, thus giving:

$$ y(t) \approx \lambda_e \frac{\ln(\omega_e^2 t^2 + 1)}{4\pi\gamma\bar{v}/c} \sin(k_a x(t) - \phi_a) \quad (125) $$

<figure><img src="image_6.png" /><figcaption>FIG. 4. Waveform for a relativistic particle characterized with space and spacetime displacements. In space, the carrier waveform is traveling uniformly, however in time, the waveform is slower in the region near the envelope's peak. This is consistent with the probability's classical dependence on the time spent at position $x$, which approaches zero at the locations with zero transverse amplitude. The oscillation of the velocity apart from its average gives rise to a phase discontinuity at the probability peaks due to the behavior of the inverse tangent function. This phase discontinuity is traversed by a bubble region in space that contains the energy of the particle.</figcaption></figure>

where $\omega_e = 2\pi\nu_e$ and $\bar{v} = \lambda_e \nu_e$ was used. Figure 4 plots out the displacement $y$ versus $x$ as well as time $t$. The graphs are consistent with the free particle probability ($\propto \sin^2\phi_e$) for detection, which reaches zero when the velocity ($\propto \cos^2\phi_e$) approaches its maximum, corresponding to zero transverse oscillation.

Equation (123) exposes the impossibility for the quanta's displacement to reach a phase that will continuously increase and wrap around itself every $2\pi$ radians as the particle travels through space, due to the inverse tangent function. To resolve the paradox, it is postulated that the applicability of phase must be extended over a region of space that contains the energy of the waveform, rather than residing with a singularity. This region is determined by the distance of separation in the calculation of $x(t)$ on both sides of a $\frac{\pi}{2}$ value in phase, as shown in Figure 4, that permits the particle to continue with an average velocity matching the constant, classical veloc-
V. TRAJECTORY OF PARTICLE MOTION

As mentioned, a particle's trajectory is not represented explicitly in the derivation of (40), hence the equation's steady-state solution corresponds to a momentum eigen-state in Schrödinger's theory. With the velocity given as a function of $\vec{x}$, the displacement is $\vec{x}$ at that point, however to find the trajectory, i.e. $\vec{x}$ as a function of time, an integration still must be performed.

The case of a free particle moving in the positive $\hat{x}_o$ direction will now be examined, starting with the velocity as written in (46):

$$v_{x_o}(x) = \frac{dx}{dt} \approx v_e = \bar{v} \cos^2(k_e x - \phi_o) \quad (121)$$
$$k_e = \frac{\bar{v}}{c} k_c$$

Here, a spacetime average is performed over several cycles of $\lambda_a$ and $\frac{\lambda_a}{\lambda_e} \ll 1$ with $\sin(k_a x - \phi_a)$ averaging to zero. The distance $x$ is measured along the $\hat{x}_o$ direction. The velocity in the $\hat{y}_o$ direction in contrast is given by (48) when $\hat{x} \cdot \hat{x}_o = 0$, with $\lambda_c \nu_a = c$:

$$v_{y_o}(y) = \frac{dy_a}{dt} \approx \frac{c}{\gamma} \cos(k_e x - \phi'_o) \sin(k_e x - \phi'_o) \\ = \frac{\lambda_c \nu_a}{2\gamma} \sin(2k_e x - 2\phi'_o) \quad (122)$$

where the same spacetime averaging is performed. Also, as noted in the discussion about (22), $\phi'_o$ is different from $\phi_o$ by the factor $\frac{\pi}{2}$.

Both velocities can be used to determine the extent of the waveform in both the $\hat{x}_o$ and $\hat{y}_o$ direction. For the $\hat{x}_o$ direction, the position and time dependence of (121) can be separated, leading to the following derivation:

$$v_{x_o} \int^{t} dt' = \int^{x} \frac{dx'}{\cos^2(k_e x' - \phi_o)}$$
$$k_e v_{x_o} t = \tan(k_e x - \phi_o)$$
$$x(t) = \lambda_e \left( \frac{\tan^{-1}(\omega_e t) + \phi_o}{2\pi} \right) \quad (123)$$

Likewise, for the $\hat{y}_o$ direction, the amplitude $y_a$ can be similarly determined with $\phi'_o$ set to zero:

$$\int^{y_a} dy'_a \approx \frac{\lambda_c \nu_a}{2\gamma} \int^{t} dt' \sin(2\tan^{-1}(\omega_e t'))$$
$$y_a = \frac{\lambda_c \nu_a}{2\gamma} \frac{\ln(\omega_e^2 t^2 + 1)}{\omega_e} \quad (124)$$

The $y$ coordinate as a function of $x(t)$ is then obtained by including the carrier phase component $\sin(k_a x(t) - \phi_a)$, in which any constant offset is ignored, thus giving:

$$y(t) \approx \lambda_e \frac{\ln(\omega_e^2 t^2 + 1)}{4\pi\gamma\bar{v}/c} \sin(k_a x(t) - \phi_a) \quad (125)$$

<figure><img src="image_5.png" /><figcaption>FIG. 4. Waveform for a relativistic particle characterized with space and spacetime displacements. In space, the carrier waveform is traveling uniformly, however in time, the waveform is slower in the region near the envelope's peak. This is consistent with the probability's classical dependence on the time spent at position $x$, which approaches zero at the locations with zero transverse amplitude. The oscillation of the velocity apart from its average gives rise to a phase discontinuity at the probability peaks due to the behavior of the inverse tangent function. This phase discontinuity is traversed by a bubble region in space that contains the energy of the particle.</figcaption></figure>

where $\omega_e = 2\pi\nu_e$ and $\bar{v} = \lambda_e \nu_e$ was used. Figure 4 plots out the displacement $y$ versus $x$ as well as time $t$. The graphs are consistent with the free particle probability ($\propto \sin^2\phi_e$) for detection, which reaches zero when the velocity ($\propto \cos^2\phi_e$) approaches its maximum, corresponding to zero transverse oscillation.

Equation (123) exposes the impossibility for the quanta's displacement to reach a phase that will continuously increase and wrap around itself every $2\pi$ radians as the particle travels through space, due to the inverse tangent function. To resolve the paradox, it is postulated that the applicability of phase must be extended over a region of space that contains the energy of the waveform, rather than residing with a singularity. This region is determined by the distance of separation in the calculation of $x(t)$ on both sides of a $\frac{\pi}{2}$ value in phase, as shown in Figure 4, that permits the particle to continue with an average velocity matching the constant, classical veloc-
ity. The equation of the phase bubble is then a modified version of (123):

$$x_b(t) = \lambda_e \left( \frac{\tan^{-1}(\omega_e t) + \chi_b \omega_e t + \phi_o}{2\pi} \right) \quad (126)$$

where the $\chi_b$ coefficient and term permits the bubble to travel through itself in a fashion that reaches across the phase discontinuity. The value found empirically for $\chi_b$ is:

$$\chi_b \approx \frac{1}{5.11} \quad (127)$$

In the second graph of Figure 4, the oscillation frequency appears to increase with increasing velocity of the carrier towards the region with zero transverse amplitude, however this is an artifact of the steady-state solution's dependence on position $x$. Here, the region of space traversed increases with increasing velocity, exposing more oscillation of the carrier compared to the region where the velocity approaches zero in the phase discontinuity region, and the envelope frequency approaches its maximum. Due to the slowdown, the time span assigned to the waveform oscillation is at its maximum there.

## VI. NON-LOCALITY AND THE ABSENCE OF COLLISIONS BETWEEN FUNDAMENTAL PARTICLES

Collisions between leptons such as electrons have not been observed [27] and the suggestion so far has been that fundamental particles are singularities. The phase bubble concept suggests they are not. To examine what happens when two beams of identical particles are directed towards each other, for simplicity only those pairs that move along an imaginary line that connects them is considered and it will be assumed that the distances of interaction are close enough such that the finite speed of light is not a factor in establishing steady-state phase mediation between particles.

Up until now, the dynamics of phase mediation has not been discussed. In this problem it will be seen how relative phases between a particle, another particle, and the background are established. To begin, the phase $\phi'_F(x)$ for a field of a particle varies independently of the field's variation though the strength of phase mediation is governed by a coupling constant $\Gamma$ that depends on a force field amplitude's strength relative to the background virtual force. The equation governing this mediation process for each particle is postulated to take the following form:

$$\frac{d\phi'_F}{d\phi} = \frac{\pi}{2} \Gamma \langle \phi'_F | \hat{\chi}_{\pm} | \phi \rangle \quad (128)$$

which has the following solution:

$$\phi'_F - \phi_c(t) = \frac{\pi}{2} \int^{\phi(t)} \Gamma \langle \phi'_F | \phi' \rangle \langle \phi' | \hat{\chi}_{\pm} | \phi \rangle d\phi' \quad (129)$$

where the phase of the background and measuring equipment is referenced as 0, which has been assumed in steady-state problems throughout the body of this work, while $\phi_c(t)$ represents the particle's phase deviation from it, or just as appropriately, the background's phase relative to the particle, which is a function of time.

This is a transcendental equation that determines the response of an initially non-interacting particle entering an interacting partner's force field. As the body of this work has stated, the sign of the force must ultimately change sign with the phase of the particle it interacts with to retain classical correspondence. It will become evident that this can be achieved if the field's response is a step function centered about a phase defined relative to the particle's phase offset $\phi_c$, using Heaviside functions $\Theta$, further broken down into a Fourier decomposition:

$$\begin{aligned} \langle \phi'_F(t=0) | \phi' \rangle &= (\Theta(\phi' - \phi_c) - \Theta(-\phi' + \phi_c)) \\ &= \frac{4}{\pi} \sum_{n=odd} b_n \sin((2n-1)\phi' - \phi_{cn}) \end{aligned} \quad (130)$$

where $b_n = \frac{1}{2n-1}$ and $\phi_{cn} = \phi_c$ initially.

Regarding the particle's spin-adjusted quantum state in the calculation of $\langle \phi' | \hat{\chi}_{\pm} | \phi \rangle$, it is assumed to already have a steady-state dependency established with the background while the other particle it is interacting with does not:

$$\langle \phi' | \hat{\chi}_{\pm} | \phi \rangle = \sin \phi' = \cos \phi_c \quad (131)$$

where $\phi'$ represents the integration variable in (129). This is equivalent to solutions discussed previously that refer to $\cos\phi_c$ where the dependency on $\phi_c$ assumes the quanta is moving parallel to the carrier's direction of propagation, $\hat{x}_o$. For a free particle, $\phi_c$ depends linearly on $x$, while as discussed for forces in section IV, integration over the wavevector is involved.

After one cycle of the particle's oscillation has been processed, the solution (129) for new coefficients $b'_n$ becomes:

$$\begin{aligned} & \sum_{n=odd} 2\pi b'_n \\ &= \sum_{n=odd} 2b_n \int^{2\pi+\delta} \Gamma \sin((2n-1)\phi' - \phi_{cn}(\phi')) \sin\phi' d\phi' \\ & \approx \sum_{n=odd} 2b_n \bar{\Gamma} \cos\bar{\phi}_{cn} \int^{2\pi+\delta} \sin((2n-1)\phi') \sin\phi' d\phi' \end{aligned} \quad (132)$$

where the following trigonometric identity was used:

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \quad (133)$$

Here, $\alpha = (2n-1)\phi'$ and $\beta = \phi_{cn}$, with the approximation that the second term of (133) is negligible when integrated in (132). An additional approximation is introduced in pulling out the $\cos\phi_{cn}$ term utilizing an average for the phase offsets, $\bar{\phi}_{cn}$. The use of $\delta$ compensates
ity. The equation of the phase bubble is then a modified version of (123):

$$x_b(t) = \lambda_e \left( \frac{\tan^{-1}(\omega_e t) + \chi_b \omega_e t + \phi_o}{2\pi} \right) \quad (126)$$

where the $\chi_b$ coefficient and term permits the bubble to travel through itself in a fashion that reaches across the phase discontinuity. The value found empirically for $\chi_b$ is:

$$\chi_b \approx \frac{1}{5.11} \quad (127)$$

In the second graph of Figure 4, the oscillation frequency appears to increase with increasing velocity of the carrier towards the region with zero transverse amplitude, however this is an artifact of the steady-state solution's dependence on position $x$. Here, the region of space traversed increases with increasing velocity, exposing more oscillation of the carrier compared to the region where the velocity approaches zero in the phase discontinuity region, and the envelope frequency approaches its maximum. Due to the slowdown, the time span assigned to the waveform oscillation is at its maximum there.

## VI. NON-LOCALITY AND THE ABSENCE OF COLLISIONS BETWEEN FUNDAMENTAL PARTICLES

Collisions between leptons such as electrons have not been observed [27] and the suggestion so far has been that fundamental particles are singularities. The phase bubble concept suggests they are not. To examine what happens when two beams of identical particles are directed towards each other, for simplicity only those pairs that move along an imaginary line that connects them is considered and it will be assumed that the distances of interaction are close enough such that the finite speed of light is not a factor in establishing steady-state phase mediation between particles.

Up until now, the dynamics of phase mediation has not been discussed. In this problem it will be seen how relative phases between a particle, another particle, and the background are established. To begin, the phase $\phi'_F(x)$ for a field of a particle varies independently of the field's variation though the strength of phase mediation is governed by a coupling constant $\Gamma$ that depends on a force field amplitude's strength relative to the background virtual force. The equation governing this mediation process for each particle is postulated to take the following form:

$$\frac{d\phi'_F}{d\phi} = \frac{\pi}{2} \Gamma \langle \phi'_F | \hat{\chi}_{\pm} | \phi \rangle \quad (128)$$

which has the following solution:

$$\phi'_F - \phi_c(t) = \frac{\pi}{2} \int^{\phi(t)} \Gamma \langle \phi'_F | \phi' \rangle \langle \phi' | \hat{\chi}_{\pm} | \phi \rangle d\phi' \quad (129)$$

where the phase of the background and measuring equipment is referenced as 0, which has been assumed in steady-state problems throughout the body of this work, while $\phi_c(t)$ represents the particle's phase deviation from it, or just as appropriately, the background's phase relative to the particle, which is a function of time.

This is a transcendental equation that determines the response of an initially non-interacting particle entering an interacting partner's force field. As the body of this work has stated, the sign of the force must ultimately change sign with the phase of the particle it interacts with to retain classical correspondence. It will become evident that this can be achieved if the field's response is a step function centered about a phase defined relative to the particle's phase offset $\phi_c$, using Heaviside functions $\Theta$, further broken down into a Fourier decomposition:

$$\begin{aligned} \langle \phi'_F(t=0) | \phi' \rangle &= (\Theta(\phi' - \phi_c) - \Theta(-\phi' + \phi_c)) \\ &= \frac{4}{\pi} \sum_{n=odd} b_n \sin((2n-1)\phi' - \phi_{cn}) \end{aligned} \quad (130)$$

where $b_n = \frac{1}{2n-1}$ and $\phi_{cn} = \phi_c$ initially.

Regarding the particle's spin-adjusted quantum state in the calculation of $\langle \phi' | \hat{\chi}_{\pm} | \phi \rangle$, it is assumed to already have a steady-state dependency established with the background while the other particle it is interacting with does not:

$$\langle \phi' | \hat{\chi}_{\pm} | \phi \rangle = \sin \phi' = \cos \phi_c \quad (131)$$

where $\phi'$ represents the integration variable in (129). This is equivalent to solutions discussed previously that refer to $\cos \phi_c$ where the dependency on $\phi_c$ assumes the quanta is moving parallel to the carrier's direction of propagation, $\hat{x}_o$. For a free particle, $\phi_c$ depends linearly on $x$, while as discussed for forces in section IV, integration over the wavevector is involved.

After one cycle of the particle's oscillation has been processed, the solution (129) for new coefficients $b'_n$ becomes:

$$\begin{aligned} & \sum_{n=odd} 2\pi b'_n \\ &= \sum_{n=odd} 2b_n \int^{2\pi+\delta} \Gamma \sin((2n-1)\phi' - \phi_{cn}(\phi')) \sin\phi' d\phi' \\ & \approx \sum_{n=odd} 2b_n \bar{\Gamma} \cos \bar{\phi}_{cn} \int^{2\pi+\delta} \sin((2n-1)\phi') \sin\phi' d\phi' \end{aligned} \quad (132)$$

where the following trigonometric identity was used:

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \quad (133)$$

Here, $\alpha = (2n-1)\phi'$ and $\beta = \phi_{cn}$, with the approximation that the second term of (133) is negligible when integrated in (132). An additional approximation is introduced in pulling out the $\cos\phi_{cn}$ term utilizing an average for the phase offsets, $\bar{\phi}_{cn}$. The use of $\delta$ compensates
for the changing value of $\phi_{cn}$, which requires an integration over a shorter span than $2\pi$ to account for the change in the complete cycle of a Fourier component's oscillation.

Given the orthogonality of different Fourier components, the lowest order frequency coefficient is dominant, with:

$$2\pi b'_{1} \approx (2\pi + \delta)b_{1}\bar{\Gamma}cos\bar{\phi}_{c1} \quad (134)$$

Since the solution in (129) involves the calculation of phase, Fourier decomposition and the subsequent integration give an updated phase offset for each Fourier component after each cycle has passed. With the requirement that the step function response of the force field remain unchanged, its Fourier coefficients are fixed. Hence, looking at the lowest order coefficient, the change represented by $b'_{1}$ can be re-expressed in terms of the change in phase offset $\phi'_{c1} - \phi_{c1}$ associated with a complete cycle of integration:

$$2\pi + \phi'_{c1} - \phi_{c1} \approx (2\pi + \delta)\bar{\Gamma}cos\bar{\phi}_{c1} \quad (135)$$

Using the definition of $\delta$ and representing the average $\bar{\phi}_{c1}$ in terms of the old and new phase offset, the following approximations hold for a positive $\phi_{c1}$ and negative $\delta$:

$$\delta \approx \bar{\phi}_{c1} - \phi_{c1} \\ \bar{\phi}_{c1} \approx \frac{\phi_{c1} + \phi'_{c1}}{2} \quad (136)$$

Substituting these into (135), the following relationship is obtained:

$$\phi'_{c1} = \phi_{c1} + \frac{\epsilon - 1}{2 - \epsilon} 4\pi \quad (137)$$

where:

$$\epsilon \equiv \bar{\Gamma}cos\bar{\phi}_{c1} \quad (138)$$

When $\epsilon = 0$, $\phi'_{c1} = \phi_{c1} - 2\pi$, i.e. no effective change in phase offset is observed. This could happen if $\phi_{c1} = \pm \frac{\pi}{2}$, however given interactions with the background, such an offset won't be maintained for long. To ensure that phase mediation takes place, a lower bound needs to be set for $\epsilon$ at $\phi'_{c1} = -\phi_{c1}$. Using this value for $\phi'_{c1}$, the following is derived from (137):

$$\begin{aligned} -\phi_{c1} &\le \phi'_{c1} \le \phi_{c1} \\ \epsilon_o &\le \epsilon \le 1 \\ \epsilon_o &= \frac{2(\pi - \phi_{c1})}{2\pi - \phi_{c1}} \quad (139) \end{aligned}$$

where for consistency in the derivation of (135), $\phi_{c1}$ is restricted to the following range:

$$0 \le \phi_{c1} \le \pi \quad (140)$$

With enough oscillations, $\phi_{c1} \to 0$ provided $\epsilon \ne \{\epsilon_o, 1\}$. If $\epsilon = 1$, then equilibrium with the background

<figure><img src="image_7.png" /><figcaption>FIG. 5. Depiction of the Great Smokey Dragon with the evolution of phase offset $\phi_p$ that leaves and eventually returns to a classical phase offset $\phi_c$ associated with a massive system.</figcaption></figure>

has already been established. Likewise, the case $\epsilon = \epsilon_o$ suggests that either equilibrium was established or if $\epsilon_o \ne 0$, then the phase would become chaotic with a change of sign after every cycle. It can also be shown that $b'_n \to 0$ for $n > 1$, or equivalently, the phase offsets for other Fourier components drift such that no higher order oscillations occur after mediation takes place.

Hence, the steady-state alignment between phases $\phi'_F$ and $\phi$ are in agreement with the steady-state analysis in Section IV. What is different, however, is the specification of $\phi'_F$ as a field quantity that has boundary conditions due to the reversibility requirement for (129). This is seen by restating (129) in terms of position, starting at the origin:

$$\phi'_{F}(x(t)) - \phi_{c}(t) = \frac{\pi}{2} \int_{0}^{x(t)} \Gamma \langle \phi'_{F} | x' \rangle \langle x' | \hat{\chi}_{\pm} | \phi \rangle \frac{d\phi(x')}{dx'} dx' \to 0 \quad (141)$$

The particles have a relative phase offset of $\phi_c$ at the beginning of their interaction when the two are together, but as the distance grows, the phase offsets of the two particles change to match each other. This introduces an entanglement that is irreversible, due to the phase offset's change with $t$, not $\phi(x(t))$. On the other hand, (141) can be reversed to become:

$$\begin{aligned} \phi'_{F}(x=0) &= \frac{\pi}{2} \int_{x(t)}^{0} \Gamma \langle \phi'_{F} | x' \rangle \langle x' | \hat{\chi}_{\pm} | \phi \rangle \frac{d\phi(x')}{dx'} dx' \\ &= \phi_{c}(t) = 0 \end{aligned} \quad (142)$$

where the integral in this case must satisfy the constraint that the phase offset $\phi_c$ changes to zero due to the entanglement that happens as the particles approach one another.

This behavior characterizes the non-locality of the theory. For multiple particles, it is straightforward to show that the same process occurs, i.e. it is possible for one particle to become entangled with another via an intermediary particle, even though they can't entangle directly between themselves. This mechanism is important for Bose and Fermi statistics, discussed in Appendix C.
for the changing value of $\phi_{cn}$, which requires an integration over a shorter span than $2\pi$ to account for the change in the complete cycle of a Fourier component's oscillation.

Given the orthogonality of different Fourier components, the lowest order frequency coefficient is dominant, with:

$$2\pi b'_{1} \approx (2\pi + \delta)b_{1}\bar{\Gamma}cos\bar{\phi}_{c1} \quad (134)$$

Since the solution in (129) involves the calculation of phase, Fourier decomposition and the subsequent integration give an updated phase offset for each Fourier component after each cycle has passed. With the requirement that the step function response of the force field remain unchanged, its Fourier coefficients are fixed. Hence, looking at the lowest order coefficient, the change represented by $b'_{1}$ can be re-expressed in terms of the change in phase offset $\phi'_{c1} - \phi_{c1}$ associated with a complete cycle of integration:

$$2\pi + \phi'_{c1} - \phi_{c1} \approx (2\pi + \delta)\bar{\Gamma}cos\bar{\phi}_{c1} \quad (135)$$

Using the definition of $\delta$ and representing the average $\bar{\phi}_{c1}$ in terms of the old and new phase offset, the following approximations hold for a positive $\phi_{c1}$ and negative $\delta$:

$$\delta \approx \bar{\phi}_{c1} - \phi_{c1} \\ \bar{\phi}_{c1} \approx \frac{\phi_{c1} + \phi'_{c1}}{2} \quad (136)$$

Substituting these into (135), the following relationship is obtained:

$$\phi'_{c1} = \phi_{c1} + \frac{\epsilon - 1}{2 - \epsilon}4\pi \quad (137)$$

where:

$$\epsilon \equiv \bar{\Gamma}cos\bar{\phi}_{c1} \quad (138)$$

When $\epsilon = 0$, $\phi'_{c1} = \phi_{c1} - 2\pi$, i.e. no effective change in phase offset is observed. This could happen if $\phi_{c1} = \pm\frac{\pi}{2}$, however given interactions with the background, such an offset won't be maintained for long. To ensure that phase mediation takes place, a lower bound needs to be set for $\epsilon$ at $\phi'_{c1} = -\phi_{c1}$. Using this value for $\phi'_{c1}$, the following is derived from (137):

$$\begin{aligned} -\phi_{c1} &\le \phi'_{c1} \le \phi_{c1} \\ \epsilon_o &\le \epsilon \le 1 \\ \epsilon_o &= \frac{2(\pi - \phi_{c1})}{2\pi - \phi_{c1}} \quad (139) \end{aligned}$$

where for consistency in the derivation of (135), $\phi_{c1}$ is restricted to the following range:

$$0 \le \phi_{c1} \le \pi \quad (140)$$

With enough oscillations, $\phi_{c1} \to 0$ provided $\epsilon \ne \{\epsilon_o, 1\}$. If $\epsilon = 1$, then equilibrium with the background

<figure><img src="image_6.png" /><figcaption>FIG. 5. Depiction of the Great Smokey Dragon with the evolution of phase offset $\phi_p$ that leaves and eventually returns to a classical phase offset $\phi_c$ associated with a massive system.</figcaption></figure>

has already been established. Likewise, the case $\epsilon = \epsilon_o$ suggests that either equilibrium was established or if $\epsilon_o \ne 0$, then the phase would become chaotic with a change of sign after every cycle. It can also be shown that $b'_n \to 0$ for $n > 1$, or equivalently, the phase offsets for other Fourier components drift such that no higher order oscillations occur after mediation takes place.

Hence, the steady-state alignment between phases $\phi'_F$ and $\phi$ are in agreement with the steady-state analysis in Section IV. What is different, however, is the specification of $\phi'_F$ as a field quantity that has boundary conditions due to the reversibility requirement for (129). This is seen by restating (129) in terms of position, starting at the origin:

$$\phi'_{F}(x(t)) - \phi_{c}(t) = \frac{\pi}{2} \int_{0}^{x(t)} \Gamma \langle \phi'_{F} | x' \rangle \langle x' | \hat{\chi}_{\pm} | \phi \rangle \frac{d\phi(x')}{dx'} dx' \to 0 \quad (141)$$

The particles have a relative phase offset of $\phi_c$ at the beginning of their interaction when the two are together, but as the distance grows, the phase offsets of the two particles change to match each other. This introduces an entanglement that is irreversible, due to the phase offset's change with $t$, not $\phi(x(t))$. On the other hand, (141) can be reversed to become:

$$\begin{aligned} \phi'_{F}(x=0) &= \frac{\pi}{2} \int_{x(t)}^{0} \Gamma \langle \phi'_{F} | x' \rangle \langle x' | \hat{\chi}_{\pm} | \phi \rangle \frac{d\phi(x')}{dx'} dx' \\ &= \phi_{c}(t) = 0 \end{aligned} \quad (142)$$

where the integral in this case must satisfy the constraint that the phase offset $\phi_c$ changes to zero due to the entanglement that happens as the particles approach one another.

This behavior characterizes the non-locality of the theory. For multiple particles, it is straightforward to show that the same process occurs, i.e. it is possible for one particle to become entangled with another via an intermediary particle, even though they can't entangle directly between themselves. This mechanism is important for Bose and Fermi statistics, discussed in Appendix C.