Adelic Zitterbewegung: p-Adic Regularization of Dirac Trembling Motion and the Product-Formula Cancellation at Macroscopic Scales

In the paper's words

Zitterbewegung (ZBW) — the trembling motion of Dirac wave packets at the Compton scale — is conventionally treated as a purely Archimedean interference effect between the positive- and negative-energy branches of the free Dirac equation. Missing from the standard account is any mechanism by which the non-Archimedean completions of the rationals could enter the dynamics, and any quantitative statement of when such contributions would or would not matter. We propose a conjectural framework in which the free Dirac equation is formulated over the adele ring A = R × ∏_p Q_p, so that the trembling motion acquires simultaneous oscillatory components in the real sector and in every p-adic sector, and we define a sector-resolved velocity expectation together with a cancellation mechanism in which the p-adic contributions to macroscopic motion are suppressed by the adele product formula |x|_∞ ∏_p |x|_p = 1. [...]

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

Numbers it reports

1. At $\ell = \ell_P$: $$\frac{\lambda_C}{\ell_P} = \frac{3.8632 \times 10^{-13}}{1.6165 \times 10^{-35}} = \frac{3.8632}{1.6165} \times 10^{22} = 2.3898 \times 10^{22} \approx 2.39 \times 10^{22}.$$ At the Planck length the conjectured non-Archimedean interference amplitude exceeds its laboratory-scale ($\ell = 1\,\mathrm{m}$) suppression factor by about 35 orders of magnitude: the quotient $2.3898 \times 10^{22} / (3.863 \times 10^{-13}) \approx 6.19 \times 10^{34}$ — the quantitative content of the claim that near the Compton and Planck lengths the non-Archimedean terms dominate.

   from 4. Analysis

2. For a wave packet that is a superposition of positive- and negative-energy branches, the standard cross term produces an oscillatory contribution at the branch energy gap; in the real sector this is the Breit–Schrödinger trembling motion [4], with $$\langle v_\infty \rangle(t) \approx c(1 - 2w) + 2cw\cos(\omega_\infty t), \qquad \omega_\infty = \frac{2m_e c^2}{\hbar},$$ where $w$ is the negative-energy admixture.

   from 3.3 Sector-resolved velocity expectation

3. Under the conjectured mechanism the cross-sector interference amplitude scales as $\lambda_C/\ell$: at $\ell = 1\,\mathrm{m}$ this is $3.863 \times 10^{-13}$ in amplitude ($1.492 \times 10^{-25}$ in intensity-like terms), while at the Planck length the non-Archimedean sector dominates by a factor $\approx 2.39 \times 10^{22}$.

   from Abstract

4. The conjectured suppression law $A_{\mathrm{int}}/A_0 = \lambda_C/\ell$, evaluated at $\ell = 1\,\mathrm{m}$ ($3.863 \times 10^{-13}$ in amplitude, $1.492 \times 10^{-25}$ in intensity-like terms) and at $\ell = \ell_P$ ($2.389 \times 10^{22}$), with the crossover scale $386.3\,\mathrm{nm}$ for a $10^{-6}$ threshold derived as a labeled projection (Section 4, Steps 6–8).

   from 1. Introduction

5. Ratio: $$\frac{\hbar G}{c^3} = \frac{7.04107 \times 10^{-45}}{2.69460 \times 10^{25}} = 2.61313 \times 10^{-70}\,\mathrm{m^2},$$ $$\ell_P = \sqrt{2.61313 \times 10^{-70}}\,\mathrm{m} = 1.6165 \times 10^{-35}\,\mathrm{m}.$$ Step 5: Product formula, explicit verification.

   from 4. Analysis

6. For a stricter threshold $\varepsilon = 10^{-4}$ in a quantity scaling as $1/N^2$ with $N = \ell/\lambda_C$ (an alternative intensity-like convention, in which the relative deviation scales as $(\lambda_C/\ell)^2$ rather than $\lambda_C/\ell$), the crossover would instead lie at $N = (10^{-4})^{-1/2} = 100$, i.e. $\ell = 100\,\lambda_C = 3.8632 \times 10^{-11}\,\mathrm{m}$.

   from 4. Analysis

How it is done

1. We propose a conjectural framework in which the free Dirac equation is formulated over the adele ring $\mathbb{A} = \mathbb{R} \times \prod_p \mathbb{Q}_p$, so that the trembling motion acquires simultaneous oscillatory components in the real sector and in every $p$-adic sector, and we define a sector-resolved velocity expectation together with a cancellation mechanism in which the $p$-adic contributions to macroscopic motion are suppressed by the adele product formula $|x|_\infty \prod_p |x|_p = 1$.

   from Abstract

2. The foundational review [1] explains that p-adic mathematical physics emerged in 1987 from efforts to find a non-Archimedean approach to spacetime and string dynamics at the Planck scale, and that among its main achievements is p-adic and adelic quantum mechanics.

   from 2. Background and Related Work

3. By the Ostrowski theorem, every non-trivial absolute value on $\mathbb{Q}$ is equivalent to the usual Archimedean one $|\cdot|_\infty$ or to a p-adic norm $|\cdot|_p$ for some prime $p$.

   from 3.1 The adele ring and the product formula

4. To falsify the conjecture we propose the following computational steps: Construct explicit $p$-adic gamma matrices $\boldsymbol{\alpha}_p, \beta_p$ satisfying the Clifford algebra, respecting the p-adic Hilbert-space geometry and the equiangular-line bound of [8].

   from 3.5 Simulation program (specified, not executed)

What would falsify it

1. Not vacuum electron physicists: the ZB of a free relativistic electron in a vacuum "is probably not directly observable for year to come and one has to recourse to its simulations" [3].

   from 1. Introduction

2. If a result held that tied the observable trembling motion to the arithmetic of $\mathbb{Q}$ across all of its completions, it would change what ZBW experiments are testing — from probing band structure alone to probing a putative adelic structure of quantum dynamics — and it would supply a concrete regularization mechanism at short distances, potentially softening the singular short-distance behavior of the Dirac propagator and informing quantum-gravity models that require a minimal length.

   from 1. Introduction

3. We then conjecture that the cross-sector interference amplitude in the observable position is proportional to the product of $p$-adic norms of the dimensionless scale ratio $r = \ell/\lambda_C$, which by the product formula equals $\lambda_C/\ell$.

   from 1. Introduction

Where the claim stops

1. (i) The adelic Dirac equation of Section 3.2 is postulated, not derived; with complex-valued sector wave functions one cannot construct a direct analogue of the Schrödinger equation with a p-adic Heisenberg algebra [1], so no standard route to the sector dynamics exists.

   from 6. Discussion

2. (iv) The sector dispersion assumption $\omega_p = \omega_\infty |r|_p$ of Step 9 is untested; the Vladimirov-operator spectrum could produce entirely different p-adic frequencies, in which case the projected shifts $f_2 = 1.2351 \times 10^{20}\,\mathrm{Hz}$ and $f_5 = 4.9404 \times 10^{19}\,\mathrm{Hz}$ would fail.

   from 6. Discussion

3. The central conjecture is falsified if the simulation program of Section 3.5, executed on $\mathbb{Q}_p$ with the Vladimirov operator, yields p-adic sector frequencies that do not scale as $|r|_p$ — this is the single calculation whose outcome decides the fate of the suppression law, because the law's only free content is the proportionality to $\prod_p |r|_p$.

   from 6. Discussion

Where to read

1. Abstract289 words · 1 min
2. 1. Introduction768 words · 3 min
3. 2. Background and Related Work1,417 words · 6 min
4. 3. Methods873 words · 4 min
5. 4. Analysis832 words · 4 min
6. 5. Results247 words · 1 min
7. 6. Discussion612 words · 3 min
8. 7. Practical Implications and Engineering Pathways704 words · 3 min
9. 8. Conclusion212 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report168 words · 1 min
12. Appendix B. Claim attribution144 words · 1 min