https://www.mdpi.com/2673-3161/6/2/37
the relativistic momentum p = γmv can be related to the average beat frequency.....the frequency ωD of the driving field does not appear explicitly in the
beat formula, but only as a reference level in the definition of relative detuning. It acts
virtually in the background. From a field-theoretic point of view, this background can
be regarded as a vacuum field that sets the reference level against which the rest energy
is defined. Lorentz invariance of the vacuum ensures that a particle interacts with the
same field regardless of the particle’s speed.
The present models replace de Broglie’s
purely kinematic argumentation with a dynamical approach. In short, phase modulation
represents a nonlinear dynamical process that is consistent with relativistic dispersion and
is equally capable of maintaining synchrony. However, nonlinearity comes at a price. It
is more difficult to build up wave packets in a nonlinear model. This is why we have
concentrated on the particle aspect and on modulating the particle momentum, to preserve
synchrony between the internal periodic processes of the propagating particle and the
associated phase wave.
However, this is only a preliminary outline of why the phase model of synchronization
and its interpretation in terms of propagation driven by elliptical gears can serve as an
analogue of relativistic particle motion. A full appreciation of this new example of quasi-
classical dynamics as a quantum analogue requires further in-depth theoretical analysis.
Appl. Mech. 2026, 7(3), 55; https://doi.org/10.3390/applmech7030055
As self-oscillations can be considered clock-like self-sustaining processes, we establish the quantum correspondences of the model by linking the nonlinear dynamics of
phase modulation to de Broglie’s seminal idea that particles behave like tiny clocks. ...De Broglie proved that the phase of a clock [particle] moving with the velocity 𝑣 = 𝛽𝑐 stays in synchrony with the phase wave, provided that the latter propagates with superluminal speed 𝑣 = 𝑐/𝛽. He coined the term ‘law of phase harmony’ ...
The wave concept of group velocity highlights the clash between the classical particle ontology and the quantum interpretation. In the classical framework, the state of an individual ‘phase particle’ in phase space is described by a well-defined momentum and position. Superposition of states is impossible. This is the reason why the present ZBW explanation in a two-center model differs from the standard QM interpretation, which is based on interference between positive and negative frequency modes in localized wave packets. In contrast, quantum representations depend on the superposition of continuous
distributions of momentum states. The product of the standard deviations of momentum and position is limited by the uncertainty relation ∆𝑥∆𝑝 ≥ ℏ/2. Individual phases are not observable; only phase differences can be measured.
https://www.mdpi.com/2673-3161/7/3/55
, the temporal evolution of a quantum harmonic oscillator is identical to that of a classical rotator. Consequently, the quantum phase can be considered a hidden ontological variable. While the individual phases of the quantum states contributing to the ensemble are veiled by quantum uncertainty, the emerging phase of the ensemble has a real status, just like the classical phase.
We propose that the present phase model is another example of a dual relationship
between classical and quantum phase. It establishes coherence among coupled or driven
classical oscillators and coherent quantum states, which most closely conform to a classical description. They minimize the uncertainty between complementary variables, such
as momentum and position, or amplitude and phase. Their temporal evolution remains
maximally localized, and their expectation values evolve in exactly the same way as position and momentum in a classical oscillator.
Thus, the assumption of a dual relationship between classical and quantum coherence appears self-evident.
A freely propagating photon field has no rest mass because it propagates at the speed of
light. However, when the field is confined, it acquires mass because accelerating the
containment, which is considered massless, requires force [25]. In a way, mass behaves like canned energy.
25 = Van der Mark, M.B.; ‘t Hooft, G.W. Light is heavy. arXiv 2015,
Rest mass refers to the center of mass of a closed system considered
completely at rest. However, this state is an idealization, as it does not account for the
internal degrees of freedom that oscillate at the speed of light, giving rise to ZBW. Ac-
cording to this view, the mass and spin of an electron both result from effects residing in
the near field.
The precession frequency is proportional to the energy gap between the low- and
high-energy states, given by ∆𝐸 = ℏ𝜔 = (ℏ𝑒/𝑚)𝐵. The energy difference defines the
resonance frequency, which is at the core of many important magnetic resonance tech-
nologies. These range from electron spin resonance (EPR) and nuclear magnetic reso-
nance (NMR) to magnetic resonance imaging (MRI). MRI is an important, noninvasive
technique used in medical diagnostics.
spin as a rotation in spacetime that transcends our common-sense understanding of ordinary rotations. Thus, it conveys as far as possible a tangible experience of spin, although it is an abstract, inherently relativistic quantum property. Furthermore, it draws attention to
another important fact. Contrary to the common view that relativity only applies to ob-
jects moving at relativistic velocities, it highlights the relativistic nature of spin, which is
equally dominant in the non-relativistic domain. This includes the description of spin-
ning particles at rest or moving at small, non-relativistic velocities.
Some theorists even doubt the physical
reality of ZBW, considering it a mathematical artifact arising from attempts to impose a
single-particle description [36].
At present, direct observation of ZBW with electrons appears to be unfeasible due
to the exceedingly high frequency (~10ଶଵ Hz) and the small amplitude (~10ି
ଵଷ m),
though experimental indications have been reported based on channeling experiments
[37]. Successful quantum simulations of ZBW in different physical platforms have been
carried out, for instance using trapped ions [38], Bose–Einstein condensates [39], and
photonic systems [40]. ZBW and its effects on electrons in condensed matter systems are
reported in [41,42], based on the analogy between the band structure of narrow-band
semiconductors and the Dirac equation.
Nonlinear phase dynamics provides a self-regulating mechanism between energy and
momentum that aligns with relativistic dispersion in a propagating system. A third rel-
evant feature is the reduction to a two-center description, which exhibits spin-like be-
havior. The latter is represented by the Viviani curve and its various topologically
equivalent embeddings.
synchronization exhibits equivalent symmetries expressed by negative frequencies. These appear much less mysterious than negative energies. Complete phase locking occurs within a frequency gap of
width ∆𝜔 = 2𝜔.
a connection is established between the principles of sync-based
self-organization in classical oscillators and a phenomenological model of particle crea-
tion and their relativistic dynamics. This opens up a coherent perspective on the over-
arching role of self-structuring processes and the emergence of new properties across all
scales, from the very large to the very small....
the strangeness and abstractness of the quantum world, and of all its applications,
which transcend many of our classical intuitions, are deeply rooted in relativity, both
special and general. The present approach is based on semiclassical dynamic models that
incorporate geometric and topological representations as well as fundamental symmetry
principles. It transfers principles of synchronization-based self-organization and emergence to the realm of particle physics.
......
