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Klein--Gordon and Dirac Oscillators with an Apparent Mass Induced by the Momentum-Space Dual of the Fock--Lorentz Transformations

This paper proposes a momentum-space dual of Fock--Lorentz transformations to derive a deformed relativistic mass shell with a time-dependent apparent mass, which is then used to formulate and solve Klein--Gordon and Dirac oscillators, revealing that their energy spectra collapse toward zero as the apparent mass vanishes in the limit of infinite time.

Original authors: Abdelmalek Boumali, N. Jafari, Manizheh Botshekananfard

Published 2026-06-05

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CC0 1.0

Original authors: Abdelmalek Boumali, N. Jafari, Manizheh Botshekananfard

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. Usually, when physicists study how particles move, they assume the "ruler" they use to measure things stays the same size forever. But this paper asks a "what if" question: What if the ruler itself is slowly stretching as time goes on?

The authors propose a new way to look at the rules of motion (relativity) by borrowing a mathematical trick from a theory called Fock–Lorentz transformations. Instead of just changing how space and time look, they flipped the script to see how momentum (the "oomph" of a moving object) behaves under these new rules.

Here is the breakdown of their discovery in everyday terms:

1. The "Shrinking" Mass

The most surprising result of their math is that if you follow a particle along a specific path in the universe, its mass appears to change.

The Analogy: Imagine you are watching a runner on a track. In normal physics, the runner's weight stays the same. In this paper's universe, as time passes, the runner's "apparent mass" slowly gets lighter and lighter, like a balloon slowly deflating.
The Formula: They found a simple rule: The mass at any time $t$ is the original mass divided by $(1 + \text{time} \times \text{constant})$ .
The Catch: This "constant" is tied to the size of the entire universe (the cosmological scale). Because the universe is so huge, this mass change is incredibly slow. For a human lifetime, the change is so tiny it's practically invisible. But over billions of years, the mass would drift significantly.

2. The "Ghost" Ruler

Why does the mass seem to change? The authors suggest it's not that the particle is actually losing weight. Instead, it's an optical illusion caused by the coordinate system.

The Analogy: Imagine you are looking at a painting through a window that is slowly fogging up or stretching. The painting hasn't changed, but your view of it has. The authors argue that if you switch to a different set of "coordinates" (a different way of measuring time and space), the mass looks constant again. The "changing mass" is just a side effect of using the "laboratory" way of measuring things in a universe that is expanding in a specific, mathematical way.

3. The Quantum Oscillators (The Bouncing Balls)

To test what this "shrinking mass" would do to real physics, the authors applied it to two famous models: the Klein–Gordon oscillator and the Dirac oscillator.

The Analogy: Think of these oscillators as a ball attached to a spring, bouncing up and down. In normal physics, the ball has a fixed weight, so it bounces at a steady rhythm.
The Result: In this new model, because the ball's "apparent weight" is slowly dropping, the spring gets weaker relative to the ball.
- The ball doesn't stop bouncing, but the energy levels (the specific heights it can reach) slowly sink toward zero.
- The "rungs" on the ladder of energy levels get closer and closer together.
- Eventually, as time goes on forever, the energy levels collapse into a single flat line at zero. The ball becomes "weightless" in terms of its energy states.

4. Is This Real? (The Reality Check)

The authors are very careful to say this is a theoretical experiment, not a proven fact of nature yet.

The Scale: They assume the "stretching" happens on the scale of the entire universe (the Hubble radius).
The Impact: For any experiment we can do in a lab today (like looking at atoms or electrons), the effect is so small it is completely undetectable. It's like trying to hear a whisper in a hurricane.
The Purpose: The goal isn't to say "mass changes in your kitchen." The goal is to show that if you take these specific mathematical rules seriously, they lead to a consistent, solvable picture where mass appears to drift. It's a "proof of concept" to see how the universe could behave if these rules were true.

Summary

The paper builds a mathematical bridge between a complex theory of space-time and the behavior of quantum particles. It concludes that if the universe follows these specific "Fock–Lorentz" rules, particles will behave as if their mass is slowly evaporating over cosmic time. This causes their energy levels to slowly collapse, but because the universe is so big, this happens so slowly that for all practical human purposes, the laws of physics look exactly the same as they always have.

In short: The paper explores a "what if" scenario where the universe's expansion makes particles look like they are getting lighter, causing their quantum energy levels to slowly fade away, but only over timescales so vast they are essentially invisible to us.

Technical Summary: Klein–Gordon and Dirac Oscillators with an Apparent Mass Induced by the Momentum-Space Dual of the Fock–Lorentz Transformations

Problem Statement
The paper addresses the challenge of deriving relativistic quantum wave equations from a specific deformation of special relativity: the Fock–Lorentz (FL) transformations. While FL transformations introduce a cosmological length scale $R$ and preserve the speed of light via linear-fractional maps, extending this framework to quantum mechanics requires a consistent momentum-space dual. The central problem is to establish a self-consistent route from this dual transformation to the Klein–Gordon (KG) and Dirac equations, specifically analyzing how the resulting time-dependent "apparent mass" affects the spectra of relativistic oscillators. The authors aim to avoid arbitrary time-dependent mass ansätze by deriving the mass scale directly from the geometric structure of the FL conformal factor.

Methodology
The authors employ a "momentum-space dual" approach, defined as a formal interchange of spacetime coordinates $(ct, \mathbf{x})$ with four-momentum components $(E/c, \mathbf{p})$ within the FL transformation laws.

Dual Transformation: The standard FL transformations are mapped to momentum space. Crucially, the conformal denominator $D(t, x; v)$ is retained as a function of the spacetime point where the relation is evaluated, effectively restricting the analysis to a specific observer's world line ( $x=0$ ) in a preferred cosmological frame.
Casimir Invariant: A quadratic invariant is derived for these dual transformations. By combining the rescaled momentum norm with the transformation rule of the FL conformal factor, the authors derive a deformed dispersion relation:
$(E^2 - p^2c^2)\left(1 + \frac{ct}{R}\right)^2 = m_0^2c^4$
This is recast as a standard relativistic mass shell with a time-dependent apparent mass: $m_{\text{app}}(t) = m_0 / (1 + ct/R)$ .
Quantization: Canonical quantization is applied to this dispersion relation to generate modified KG and Dirac equations. The Dirac equation is constructed such that squaring it reproduces the KG operator, modulo first-order corrections proportional to $\dot{m}_{\text{app}}$ .
Oscillator Models: The formalism is applied to one-dimensional KG and Dirac oscillators. Given that the cosmological time scale $R/c$ vastly exceeds microscopic oscillator periods, the adiabatic approximation is utilized. The authors derive instantaneous spectra and estimate non-adiabatic corrections using WKB and Bogoliubov techniques.

Key Contributions and Results

Derivation of Apparent Mass: The paper establishes that restricting the FL conformal factor to a cosmological world line induces a mass scale $m_{\text{app}}(t)$ that decays monotonically as $t \to \infty$ . For $|t| \ll R/c$ , this recovers the standard mass $m_0$ with linear corrections.
Consistency of Wave Equations: The modified Dirac equation is shown to be consistent with the modified KG equation. The discrepancy between the squared Dirac operator and the KG operator is of order $\bar{\lambda}_C/R$ (the ratio of the reduced Compton wavelength to the FL scale), which is negligible ( $\sim 10^{-39}$ for electrons) in the adiabatic regime.
Instantaneous Spectra:
- KG Oscillator: The instantaneous energy spectrum is derived as $E_n^2(t) = m_{\text{app}}^2(t) + 2m_{\text{app}}(t)\omega(n+1)$ . As $t \to \infty$ , the energy levels and level spacings collapse toward zero, though the wavefunctions remain normalizable at any finite time.
- Dirac Oscillator: The calculation is performed in component form. The authors explicitly demonstrate how the upper and lower spinor components form coupled ladders that must be paired to form a physical spectrum. The resulting physical spectrum is $E_N^2(t) = m_{\text{app}}^2(t) + 2m_{\text{app}}(t)\omega N$ .
Adiabatic Validity: The validity of the approximation is governed by the dimensionless parameter $\epsilon = c/(R\omega)$ . For cosmological values of $R$ and realistic oscillator frequencies, $\epsilon \ll 1$ . Consequently, non-adiabatic corrections (particle production) are exponentially suppressed, and the system is effectively adiabatically invariant.
Geometric Interpretation: The time dependence of the mass is interpreted not as an intrinsic change in particle properties, but as a consequence of viewing the physics in laboratory coordinates rather than "comoving" conformal coordinates $(T, \mathbf{X})$ .

Significance and Claims
The authors position this work as a "proof-of-principle" analysis rather than a complete, covariant phase-space theory. The significance lies in:

Analytic Tractability: Providing a controlled ansatz that yields closed-form expressions for the spectra of relativistic oscillators under FL deformation, avoiding the complexities of curved momentum-space geometries or Hopf algebras.
Spectral Consequences: Demonstrating that the FL conformal factor induces a specific, calculable spectral drift where energy levels collapse toward zero as $t \to \infty$ , distinct from standard special relativity.
Phenomenological Modesty: The paper explicitly states that for cosmological values of $R$ , the effects are too small to be observed in current laboratory settings. The fractional mass drift is of order $H_0$ , making it difficult to distinguish from unit redefinitions or standard cosmological redshift.
Future Directions: The work identifies specific ingredients required for a more complete theory, including alternative dualizations, fully covariant formulations, three-dimensional oscillators, and quantum field theoretic treatments of particle production.

The paper concludes that while the model does not predict immediately observable signals, it successfully isolates the spectral consequences of the FL conformal factor and clarifies the mathematical structure required to test such deformations via dimensionless observables (e.g., ratios of transition frequencies).