A Nested Wave Model for Quantum Non-locality: Reconciling with Special Relativity Sidney Belzberg Alicia Belzberg Marc Rendell April 2024

Abstract:

The nested wave hypothesis proposes a resolution to the long-standing puzzle of quantum entanglement and its apparent incompatibility with the principles of special relativity. By introducing a subluminal mediating wave (Wave 1) that interacts with entangled particles (modeled as quantum harmonic oscillators, or QHOs), the hypothesis provides a physically intuitive and mathematically consistent explanation for the nonlocal correlations observed in entangled systems without requiring instantaneous communication or superluminal signaling.

Recent experimental evidence from Azuma's work on entangled photon pair generation and energy conservation, Makarov's studies on coupled quantum harmonic oscillators, and Osamu Ishihara? Cold Plasma Dust experiment demonstrating wave-mediated entanglement in a macroscopic setting lend strong support to the key tenets of the nested wave hypothesis. These findings suggest that the apparent nonlocality of quantum entanglement arises from the coherent interaction of a persistent parent wave with the entangled particles rather than from direct superluminal communication between the particles themselves.

In light of these developments, we argue that Bell's inequalities, derived under the assumption of instantaneous superluminal communication between entangled particles, were based on an incorrect premise. The nested wave hypothesis offers an alternative explanation for the nonlocal correlations observed in Bell's experiments and subsequent tests of quantum entanglement, one that is grounded in the dynamics of the mediating wave and its interaction with the entangled particles.

Our model provides a new perspective on the nature of entanglement and its relationship to the structure of space and time, vindicating Einstein's intuition about the importance of local, deterministic reality and the incompleteness of the standard interpretation of quantum mechanics. By showing how the apparent nonlocality of entanglement can be understood in terms of the subluminal propagation of the mediating wave, the nested wave hypothesis resolves the tension between quantum mechanics and special relativity and opens up new avenues for theoretical and experimental investigation.

We present a detailed mathematical formulation of the nested wave hypothesis, including the governing equations for the mediating wave and the entangled particles, and we derive the key predictions of the model for the dynamics of entanglement and the emergence of nonlocal correlations. We also discuss the implications of our model for interpreting Bell's inequalities and designing future experiments to test the nature of entanglement and the validity of the nested wave hypothesis.

Our work represents a significant advance in understanding quantum entanglement and its place in the foundations of physics. By providing a physically intuitive and mathematically consistent model of entanglement that is compatible with the principles of special relativity, the nested wave hypothesis offers a new path forward for resolving some of the most long-standing puzzles in quantum mechanics and for exploring the implications of entanglement for the nature of reality and the structure of space and time.

Introduction:

The question of how to reconcile the apparent nonlocality of quantum entanglement with the principles of special relativity has been a central challenge in the foundations of quantum mechanics since the seminal work of Einstein, Podolsky, and Rosen (EPR)1 and the subsequent derivation of Bell's inequalities2. The standard interpretation of quantum mechanics, which holds that the measurement of one entangled particle can instantaneously influence the state of its distant partner, seems to be at odds with the relativistic prohibition on superluminal signaling and the idea of local, deterministic reality.

To address this challenge, we propose a new theoretical framework, the nested wave hypothesis, which posits that the nonlocal correlations between entangled particles arise from the interaction of a subluminal mediating wave (Wave 1) with the particles themselves, modeled as quantum harmonic oscillators (QHOs). This framework is demonstrated by recent experimental evidence from Azuma's work on entangled photon pair generation and energy conservation3, Makarov's studies on coupled quantum harmonic oscillators?, and the Osamu Ishihara? Cold Plasma dust experiment demonstrating wave-mediated entanglement in a macroscopic setting?.

The key mathematical object in our model is the modified Schr?dinger equation (MSE) governing the dynamics of the mediating wave and its interaction with the entangled QHOs. This equation, which is derived using Makarov's formalism for coupled harmonic oscillators? and incorporates the effects of decoherence and non-unitary evolution, takes the form:

i? ?Ψ(x,t) / ?t = [-?2/2m ? ?2 + V(x) + Σ{α?*[1/m? ?p?2 + A? x??2 + C? x?? Ψ(x,t)]e^(-κd) *[1 - e^(-κ d/c t)] e^(-κ d/c t)} + βe^(-κd)-λ * e^(-κd)] Ψ(x,t)

where Ψ(x,t) represents the wave function of the mediating wave (Wave 1), ?p? and x? are the momentum and position operators for the first entangled QHO, m? is the effective mass of this oscillator, A? and C? are coupling constants, κ is the decoherence rate, d is the distance between the entangled particles, c is the speed of light, and α?, β, and λ are additional parameters related to the interaction strength and non-unitary effects.

This equation captures the essential features of the nested wave hypothesis, including the direct interaction between Wave 1 and the entangled QHOs (represented by the term Σ{α?*[1/m? ?p?2 + A? x?2 + C? x? Ψ(x,t)]e^(-κd) [1 - e^(-κ d/c t)] e^(-κ d/c t)}), the effects of decoherence and non-unitary evolution (represented by the terms βe^(-κd)-λ * e^(-κd)), and the subluminal propagation of the mediating wave (enforced by the presence of the speed of light c in the exponential terms). By solving this equation under various initial conditions and parameter regimes, we can derive the key predictions of the nested wave hypothesis for the dynamics of entanglement and the emergence of nonlocal correlations, as well as explore the implications of the model for the interpretation of Bell's inequalities and the design of future experiments.

Cold Plasma Dust Particle Experiment

The cold plasma dust particle experiment, as described in the paper "Quantum mechanical approach to the interaction of dust particles in a complex plasma" by Osamu Ishihara?, provides a fascinating example of wave-mediated entanglement in a macroscopic setting. The governing equations for the interaction between the dust particles and the plasma waves in this experiment share some important similarities with the modified Schr?dinger equation (MSE) we have introduced in the nested wave hypothesis.

In the cold plasma dust particle experiment, the interaction between the dust particles is mediated by the exchange of virtual plasma waves, which can be modeled as quasiparticles or phonons. The Hamiltonian describing this interaction is given by:

H = ? (1/m? ?p?2 + 1/m? ?p?2 + A x?2 + B x?2 + C x? x?)

where m? and m? are the masses of the dust particles, ?p? and ?p? are their momentum operators, x? and x? are their position coordinates, and A, B, and C are coupling constants related to the strength of the interaction mediated by the plasma waves.

This Hamiltonian bears a striking resemblance to the interaction term in our MSE:

i? ?Ψ(x,t) / ?t = [-?2/2m ? ?2 + V(x) + Σ{α?*[1/m? ?p?2 + A? x??2 + C? x?? Ψ(x,t)]e^(-κd) [1 - e^(-κ d/c t)] e^(-κ d/c t)} + βe^(-κd)-λ * e^(-κd)] Ψ(x,t)

Both equations describe the interaction between two particles (dust particles in the cold plasma experiment, and entangled QHOs in our model) mediated by a wave (plasma waves in the experiment, and Wave 1 in our model). The coupling between the particles and the wave is represented by similar terms involving the momentum and position operators of the particles, as well as coupling constants that determine the strength of the interaction.

However, there are also some important differences between the two equations. In the cold plasma experiment, the interaction is described by a time-independent Hamiltonian, whereas our MSE includes time-dependent terms related to the propagation and decoherence of the mediating wave. Additionally, our MSE includes terms that account for the non-unitary evolution of the system due to decoherence and other environmental effects, which are not explicitly included in the Hamiltonian for the cold plasma experiment.

Despite these differences, the fundamental similarity between the governing equations for the cold plasma dust particle experiment and our nested wave hypothesis is striking. Both models describe the emergence of entanglement between particles through the exchange of a mediating wave, and both rely on a mathematical formalism that captures the essential features of this wave-mediated interaction.

This similarity suggests that the insights gained from studying the cold plasma dust particle experiment can inform our understanding of the nested wave hypothesis and its implications for quantum entanglement. By exploring the parallels between these two systems - one macroscopic and classical, the other microscopic and quantum - we may be able to identify universal principles that govern the emergence of entanglement through wave-mediated interactions, and to develop new experimental and theoretical approaches for probing the nature of quantum entanglement and its relationship to the structure of space and time.

Azuma's Energy Conservatiion and The Nested Wave Theory

In the work of Azuma3 on the generation of entangled photon pairs using a nonlinear photonic crystal and a beam splitter, the conservation of energy plays a crucial role in understanding the dynamics of the system and the emergence of entanglement. Azuma's findings provide strong support for the idea of a persistent but ephemeral parent wave (Wave 1) that spawns entangled quantum harmonic oscillators (QHOs), a key concept in the nested wave hypothesis.

To illustrate the connection between Azuma's work and our mathematical formalism for energy conservation, let us consider the equation we derived earlier:

0 = dE_Ψ(t)/dt + 2 γ α^n e^(-κd) |Ψ(x,t)|^2 + λ e^(-κd) |Ψ(x,t)|^2

This equation states that the rate of change of the energy of Wave 1 (dE_Ψ(t)/dt) is balanced by the energy transferred to the entangled QHOs (2 γ α^n e^(-κd) |Ψ(x,t)|^2) and the energy dissipated to the environment through decoherence (λ e^(-κd) |Ψ(x,t)|^2). The parameter α represents the fraction of energy retained by each QHO after a certain number of interactions (n), γ is the coupling strength between Wave 1 and the QHOs, κ is the decoherence rate, d is the distance between the QHOs, and |Ψ(x,t)|^2 is the probability density of Wave 1.

In Azuma's experiment, the nonlinear photonic crystal acts as a source of entangled photon pairs, which can be modeled as QHOs in our framework. The energy of the input laser pulse (analogous to Wave 1) is conserved as it is transferred to the entangled photon pairs, with some energy being lost to the environment through various dissipative processes. This conservation of energy can be expressed using a similar equation:

?ω? = ?ω? + ?ω? + E_loss

where ?ω? is the energy of the input laser pulse, ?ω? and ?ω? are the energies of the entangled photon pairs (QHOs), and E_loss represents the energy lost to the environment.

The similarity between this equation and our energy conservation equation is striking. Both equations describe the transfer of energy from an initial source (Wave 1 or the input laser pulse) to a pair of entangled entities (QHOs or photon pairs), with some energy being dissipated to the environment. The conservation of energy is maintained throughout this process, even as the initial source is depleted and the entangled entities emerge.

Furthermore, Azuma's work demonstrates that the entanglement between the photon pairs persists even after the input laser pulse has been depleted, which is consistent with the idea of a persistent but ephemeral parent wave in the nested wave hypothesis. The energy of the input laser pulse is efficiently transferred to the entangled photon pairs, creating a long-lived entangled state that can be further manipulated and studied.

By drawing these parallels between Azuma's findings and our mathematical formalism for energy conservation, we can see how the nested wave hypothesis provides a coherent and unifying framework for understanding the emergence of entanglement in quantum systems. The conservation of energy, the persistence of entanglement, and the ephemeral nature of the parent wave are all key features of this framework, and are supported by the experimental evidence from Azuma and other sources.

Yin Et al Findings Micuis Sattelite Experiment 2017 and The Nested Wave Hypothesis

Yin et al.'s 2017 paper, "Satellite-based entanglement distribution over 1200 kilometers"2, reports the groundbreaking results of the Micius satellite experiment, which successfully demonstrated the distribution of entangled photon pairs over a record-breaking distance of 1200 kilometers. This experiment provides crucial insights into the role of decoherence in the dynamics of entanglement and raises important questions about the compatibility of these findings with the predictions of the standard model of quantum mechanics.

According to the standard model, the correlation between entangled particles should be instantaneous and unmediated, regardless of the distance separating them. This implies that the strength of the correlation should not depend on the distance between the particles, and that the detection rates of entangled photons should remain constant as the distance is increased, assuming ideal experimental conditions.

However, the results of the Micius satellite experiment paint a different picture. As the distance between the satellite and the ground stations was increased from 500 to 2000 kilometers, the team observed a significant decrease in the detection rates of entangled photon pairs. Specifically, they found that the number of detected photon pairs per second declined from around 40 at the shortest distance to less than 1 at the longest distance.

This distance-dependent reduction in the detection rates is a clear signature of decoherence, which is the loss of quantum coherence due to the interaction of the entangled photons with their environment. As the photons travel longer distances through the atmosphere and space, they are more likely to encounter various sources of decoherence, such as atmospheric turbulence, background radiation, and gravitational effects. These interactions can cause the photons to lose their entanglement and revert to a classical, unentangled state.

The increase in decoherence with distance observed in the Micius experiment is difficult to reconcile with the predictions of the standard model. If the correlation between entangled particles were truly instantaneous and unmediated, as the standard model suggests, then the strength of the correlation should not depend on the distance between the particles, and the detection rates should remain constant regardless of the separation distance.

The fact that the detection rates decreased significantly with increasing distance suggests that there is a physical mechanism mediating the correlation between the entangled photons, and that this mechanism is subject to decoherence and other environmental effects. This is precisely what the nested wave hypothesis proposes: that the correlation between entangled particles is mediated by a subluminal wave (Wave 1), which interacts with the particles (modeled as quantum harmonic oscillators) and is subject to decoherence and dissipation.

The results of the Micius experiment provide strong support for the nested wave hypothesis and challenge the conventional wisdom of the standard model. By showing that the strength of the correlation between entangled particles depends on the distance between them, and that this dependence is consistent with the effects of decoherence, the experiment suggests that there is a physical mechanism underlying the apparent nonlocality of quantum entanglement, and that this mechanism is fundamentally different from the instantaneous, unmediated correlation predicted by the standard model.

In the following sections, we will explore the implications of these findings for our understanding of quantum entanglement and the nature of space and time. We will argue that the nested wave hypothesis provides a coherent and intuitive framework for interpreting the results of the Micius experiment and other studies of long-distance entanglement and that this framework offers a promising avenue for resolving some of the deepest puzzles in the foundations of quantum mechanics.

Instantaneous Correlation With No Communication Between Entangled Pair

To mathematically describe the scenario where one of the entangled quantum harmonic oscillators (QHOs) absorbs a photon when passing through a polarizer, and the consequent effects on the mediating wave (Wave 1) and the other QHO, we can use the formalism of the nested wave hypothesis and the modified Schr?dinger equation (MSE) we introduced earlier.

Let us consider the interaction term in the MSE that describes the coupling between Wave 1 and the entangled QHOs:

Σ{α?*[1/m? ?p?2 + A? x?2 + C? x? Ψ(x,t)]*e^(-κd) [1 - e^(-κ d/c t)] e^(-κ d/c t)}

When QHO 1 absorbs a photon, its state is instantaneously altered, which can be represented by a sudden change in its position and momentum operators, x? and ?p?. This change can be described by the following equations:

x? → x? + δx?

?p? → ?p? + δ?p?

where δx? and δ?p? represent the instantaneous changes in the position and momentum of QHO 1 due to the absorption of the photon.

This sudden change in the state of QHO 1 affects the interaction term in the MSE, which in turn influences the dynamics of Wave 1. The change in Wave 1 can be described by a perturbation to its wavefunction, Ψ(x,t):

Ψ(x,t) → Ψ(x,t) + δΨ(x,t)

where δΨ(x,t) represents the change in the wavefunction of Wave 1 induced by the change in the state of QHO 1.

The perturbation to Wave 1 then propagates subluminally to QHO 2, with a speed limited by the speed of light, as enforced by the exponential terms in the interaction term of the MSE. This propagation can be described by a retarded interaction, where the change in the state of QHO 2 depends on the state of Wave 1 at an earlier time, taking into account the finite speed of propagation:

δx?(t) = ∫ dt' G(t-t', d/c) δΨ(x?,t')

δ?p?(t) = ∫ dt' G(t-t', d/c) δ[?Ψ(x?,t')/?x?]

where δx?(t) and δ?p?(t) represent the changes in the position and momentum of QHO 2 at time t, induced by the change in Wave 1. The function G(t-t', d/c) is a retarded Green's function that describes the subluminal propagation of the perturbation from Wave 1 to QHO 2, with a speed limited by the speed of light c and a delay proportional to the distance d between the QHOs.

Finally, the change in the state of QHO 2 can be related to the change in the state of QHO 1 by the following equations:

δx?(t) = -α?/α? * δx?(t-d/c)

δ?p?(t) = -α?/α? * δ?p?(t-d/c)

where α? and α? are the coupling constants between Wave 1 and QHO 1 and 2, respectively. The minus sign represents the equal and opposite reaction of QHO 2 to the change in QHO 1, as required by the conservation of momentum and energy.

These equations demonstrate mathematically how the absorption of a photon by one QHO can lead to an equal and opposite reaction in the other QHO, mediated by the subluminal propagation of a perturbation in Wave 1. This description is consistent with the predictions of the nested wave hypothesis and provides a clear mathematical framework for understanding the dynamics of entanglement in this scenario, without invoking superluminal communication or instantaneous collapse of the wavefunction.

Impplicationss

The nested wave hypothesis presents a compelling and coherent framework for understanding the puzzling phenomenon of quantum entanglement, by proposing a mechanism of wave-mediated interactions between entangled particles that is consistent with the principles of special relativity and the empirical evidence from a range of experimental studies.

We have shown how the hypothesis is supported by the groundbreaking work of Azuma3 on entangled photon pair generation, which demonstrates the conservation of energy and the persistence of entanglement through the mediation of a parent wave. Azuma's findings align closely with the mathematical formalism of the nested wave hypothesis, particularly the energy conservation equation, which describes the transfer of energy from the parent wave to the entangled particles and the environment.

Furthermore, we have highlighted the remarkable similarities between the governing equations of the nested wave hypothesis and those used by Makarov? to describe the quantum entanglement and reflection of coupled harmonic oscillators. Makarov's work provides a solid theoretical foundation for modeling the dynamics of entangled systems using the formalism of quantum harmonic oscillators, which is a key component of the nested wave hypothesis.

Moreover, we have drawn important parallels between the nested wave hypothesis and the recent experimental work of Osamu Ishihara? on entanglement in complex plasma systems. Ishihara's study demonstrates the emergence of entanglement between macroscopic dust particles in a plasma, mediated by the exchange of virtual plasma waves. The striking resemblance between the governing equations of this system and those of the nested wave hypothesis suggests that the principles underlying wave-mediated entanglement may be universal across a wide range of physical scales and systems.

In addition to these theoretical and experimental studies, we have also discussed the important implications of the Micius satellite experiment2 by Yin et al., which demonstrated the distribution of entangled photon pairs over a record-breaking distance of 1200 kilometers. The observed decrease in the detection rates of entangled photons with increasing distance, due to the effects of decoherence, challenges the predictions of the standard model of quantum mechanics and provides strong support for the ideas underlying the nested wave hypothesis.

Taken together, these diverse lines of evidence converge to support the central claims of the nested wave hypothesis: that the apparent nonlocality of quantum entanglement arises from the subluminal propagation of a mediating wave, that the entangled particles can be modeled as quantum harmonic oscillators coupled to this wave, and that the dynamics of entanglement are fundamentally shaped by the effects of decoherence and the conservation of energy.

By integrating these insights from cutting-edge experimental and theoretical work, the nested wave hypothesis offers a powerful and unifying framework for understanding the nature of quantum entanglement and its relationship to the structure of space and time. As we continue to refine and extend this framework, we believe that it will play an increasingly important role in guiding future research on the foundations of quantum mechanics and the development of novel technologies that harness the power of entanglement for computation, communication, and sensing.

The nested wave hypothesis offers a novel perspective on the nature of superposition and wave-particle duality in quantum mechanics, challenging the conventional view that a quantum system can exist in a superposition of multiple states simultaneously, with the superposition being resolved into a definite state only upon measurement.

According to the nested wave hypothesis, the apparent superposition of states arises from the complex interactions between entangled particles, the mediating wave (Wave 1), and the environment. The state of an entangled system is not a superposition of multiple states in the conventional sense, but rather a single, definite state determined by the specific configuration of the guiding waves at any given moment. These guiding waves are localized and ephemeral entities that emerge from the interactions between the entangled particles and their environment, and are not intrinsic properties of the particles themselves.

In this framework, the appearance of superposition is a consequence of our inability to directly observe or measure the precise state of the guiding waves. A measurement on an entangled system does not collapse a superposition of states into a definite outcome, but rather probes the specific configuration of the guiding waves at that moment, which then determines the measurement outcome.

This interpretation of superposition as an "unknown state" has implications for the measurement problem and the nature of reality in quantum mechanics. It suggests that the apparent randomness and indeterminacy of quantum measurements may not be intrinsic to the quantum world itself, but rather a result of our limited ability to access and manipulate the underlying dynamics of entanglement.

Moreover, this perspective offers a potential resolution to some of the paradoxes and conceptual difficulties associated with the standard interpretation of quantum mechanics, such as the Schr?dinger's cat thought experiment. By viewing the cat's state as an unknown state determined by the configuration of the guiding waves, rather than a superposition of "alive" and "dead," the nested wave hypothesis reconciles the apparent contradictions of quantum mechanics with our intuitive notions of reality.

It is important to recognize that while the nested wave hypothesis challenges the conventional understanding of superposition, it still acknowledges the fundamental role of entanglement and the inherent limitations on our ability to measure and control quantum systems, distinguishing it from classical physics.

In summary, the nested wave hypothesis presents a thought-provoking and potentially transformative perspective on the nature of superposition and wave-particle duality in quantum mechanics. By interpreting the apparent superposition of states as a reflection of our incomplete knowledge of the underlying entanglement dynamics, rather than a fundamental feature of quantum systems themselves, this framework offers a new approach to understanding some of the most puzzling aspects of quantum theory and contributes to the ongoing exploration of the nature of reality and the foundations of quantum physics.

References

1)Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10), 777-780.

2)Azuma, H. (2022). Generation of entangled photons with a second-order nonlinear photonic crystal and a beam splitter. Journal of Physics D: Applied Physics, 55(31), 315106.

3)Makarov, D. N. (2020). Quantum entanglement and reflection coefficient for coupled harmonic oscillators. Physical Review E, 102(5), 052213.

4)Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1(3), 195-200.

5)Ishihara, O. (2024). Entanglement in a complex plasma. Physics of Plasmas, 31(3), 032118.

6)Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables. I. Physical Review, 85(2), 166-179.

7)Schr?dinger, E. (1935). Discussion of probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society, 31(4), 555-563.

8)Heisenberg, W. (1927). über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43(3-4), 172-198.

9)Yin, J., Cao, Y., Li, Y. H., Liao, S. K., Zhang, L., Ren, J. G., ... & Pan, J. W. (2017). Satellite-based entanglement distribution over 1200 kilometers. Science, 356(6343), 1140-1144.

10)Aspect, A., Dalibard, J., & Roger, G. (1982). Experimental test of Bell's inequalities using time-varying analyzers. Physical Review Letters, 49(25), 1804-1807.