Zitterbewegung as an Archimedean–2-adic Adelic Mixing Signature: Spectral-Gap Formalization, Numerical Anchors, and a Falsifiable Simulation Program

In the paper's words

Zitterbewegung (ZBW) — the trembling motion of a Dirac wave packet at angular frequency ω = 2mc²/ℏ — is conventionally treated as an interference artifact of the Dirac Hamiltonian. We propose and partially formalize a stronger claim: ZBW is the observable trace of mixing between the Archimedean place R (denoted ∞) and the 2-adic place Q₂ of the rationals, in the framework of adelic quantum mechanics. We (i) prove a spectral-gap proposition showing that any operator coupling an ∞-localized sector to a p-adic sector forces an oscillation at Δ E/ℏ, with the exact two-level frequency Ω = √(Δ E)² + 4|g|²/ℏ; (ii) derive the electron ZBW frequency ω_ZBW = 1.55268 × 10²1 mathrmrad s^-1 and reduced Compton wavelength 3.8616 × 10^-13 m with full arithmetic; (iii) argue that the prime 2 is structurally distinguished because the Bruhat–Tits tree of Q₂ is 3-regular, the smallest degree any prime admits; and (iv) specify a trapped-ion Dirac simulation test, labeled throughout as a projection, in which the oscillation amplitude is regressed [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Under the stated assumptions $|g|(n) = g_0\,2^{-n}$ with $g_0 = 10^{-3}\,\Delta E$ and weak coupling $|g| \ll |\Delta E|$, the projected oscillation amplitude is $A_{\mathrm{pop}}(n) = 4 \times 10^{-6} \times 4^{-n}$: $n$ $D_2(n) = 2^{-n}$ $A_{\mathrm{pop}}(n)$ (projection) 0 $1$ $4.0 \times 10^{-6}$ 1 $0.5$ $1.0 \times 10^{-6}$ 2 $0.25$ $2.5 \times 10^{-7}$ 3 $0.125$ $6.25 \times 10^{-8}$ 10 $9.77 \times 10^{-4}$ $3.81 \times 10^{-12}$ Uncertainty on these projections is dominated by the unknown simulator constant $g_0$; the factor-of-$4$ suppression per step is independent of $g_0$ within the assumed linear scaling.

   from 5. Results

2. Conversely, observation of the projected factor-of-$4$ suppression would not prove adelic mixing — it would only show consistency; the frequency channel is inaccessible ($\delta\omega \approx 3.11 \times 10^{15}\,\mathrm{rad\,s^{-1}}$ relative to $\omega_{\mathrm{ZBW}} \approx 1.55 \times 10^{21}\,\mathrm{rad\,s^{-1}}$ is a $2 \times 10^{-6}$ relative shift, and absolute free-electron spectroscopy at this resolution is unavailable).

   from 6. Discussion

3. We formalized Zitterbewegung as a candidate observable signature of mixing between the Archimedean place and the 2-adic place: a spectral-gap Proposition shows any such coupling forces an oscillation at $\Omega = \sqrt{(\Delta E)^2 + 4|g|^2}/\hbar$; explicit derivations give $\omega_{\mathrm{ZBW}} = 1.55268 \times 10^{21}\,\mathrm{rad\,s^{-1}}$ and $\lambda_C = 3.8616 \times 10^{-13}\,\mathrm{m}$; the prime 2 is structurally minimal via its 3-regular Bruhat–Tits tree; and a trapped-ion protocol with projected factor-of-$4$ amplitude suppression per confinement step makes the claim falsifiable.

   from 7. Conclusion

How it is done

1. A simulation protocol (Sections 3 and 5), labeled as a projection: a trapped-ion Dirac simulator in the style named in the source research idea, in which the Compton-scale oscillation amplitude is regressed against an engineered 2-adic delocalization measure.

   from 1. Introduction

2. Reference [1] reviews the emergence of $p$-adic mathematical physics from attempts to find a non-Archimedean approach to spacetime and string dynamics at the Planck scale, and describes the successful formulation of $p$-adic and adelic quantum mechanics with complex-valued wave functions of $p$-adic and adelic arguments respectively.

   from 2. Background and Related Work

3. Let $\mathbb{A}_{\mathbb{Q}}$ be the adele ring of $\mathbb{Q}$, with places $\infty$ (Archimedean, $\mathbb{R}$) and primes $p$ ($\mathbb{Q}_p$).

   from 3.1 Adelic setup

4. The ball of radius $2^{-n}$ is $2^n\mathbb{Z}_2$, and $\mathbb{Z}_2 / 2^n\mathbb{Z}_2$ has exactly $2^n$ cosets, so $$ \mu_2(2^n \mathbb{Z}_2) = 2^{-n}. $$ We define the 2-adic delocalization measure of a packet whose 2-adic support is confined to the ball $2^n\mathbb{Z}_2$ as $D_2(n) = 2^{-n}$: tighter 2-adic confinement (larger $n$) means smaller delocalization measure.

   from 3.3 The 2-adic delocalization measure

What would falsify it

1. We propose and partially formalize a stronger claim: ZBW is the observable trace of mixing between the Archimedean place $\mathbb{R}$ (denoted $\infty$) and the 2-adic place $\mathbb{Q}_2$ of the rationals, in the framework of adelic quantum mechanics.

   from Abstract

2. Under this reading, a wave packet localized at an Archimedean position $x_{\infty}$ necessarily carries Fourier components that are delocalized on $\mathbb{Q}_2$, and the Compton-scale oscillation at $\omega = 2E/\hbar$ is the observable trace of the $\infty \leftrightarrow 2$ channel.

   from 1. Introduction

3. It is also falsifiable: if a $p \geq 3$ sector reproduces the observed amplitude scaling, or if the oscillation amplitude in a controlled Dirac simulator fails to track a 2-adic delocalization measure, the claim fails.

   from 1. Introduction

Where the claim stops

1. The prime-selection argument via Bruhat–Tits valence is structural, not dynamical: since $\Omega$ is independent of $p$ in the Proposition, prime choice can only manifest through the coupling amplitude $g$, and we have no derivation that $g$ actually depends on the delocalization measure $D_2(n)$ as assumed in Derivation 7.

   from 6. Discussion

2. The program fails if (i) a simulator with a $p \geq 3$ sector reproduces the same amplitude scaling, contradicting the 2-adic specificity; (ii) the measured amplitude $A(n)$ in the trapped-ion protocol does not scale as $4^{-n}$ under the engineered delocalization parameter; or (iii) the frequency $\omega_{\mathrm{sim}} = 2\Delta_{\mathrm{eff}}/\hbar$ changes when the coupling geometry is varied, contradicting the Proposition's $p$-independence.

   from 6. Discussion

3. We report no new experimental data; every quantitative result is computed here from stated constants or explicitly flagged as a projection with stated assumptions.

   from Abstract

Where to read

1. Abstract205 words · 1 min
2. 1. Introduction396 words · 2 min
3. 2. Background and Related Work947 words · 4 min
4. 3. Methods730 words · 3 min
5. 4. Analysis820 words · 4 min
6. 5. Results273 words · 1 min
7. 6. Discussion322 words · 1 min
8. 7. Conclusion89 words · 1 min
9. Appendix A. Divergence report27 words · 1 min
10. Appendix B. Claim attribution93 words · 1 min