Zitterbewegung: From Archimedean Puzzle to Adelic Observable

In the paper's words

The structural underdetermination of the position operator in relativistic quantum mechanics arises from a single representation-theoretic origin: the Silent Parameter Principle. SU(2) and Z2^x share identical fusion rules at matching levels. The 2-adic ZBW frequency deviates from the Archimedean value by sqrt(2) (41.4%).

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

Numbers it reports

1. An experimental timeline based on single trapped-ion Dirac simulation is presented, predicting measurable deviation at $\sqrt{2}$ within a 3--5 year, sub- $ 500K program.

   from Abstract

2. The representation $V_j$ has dimension $2j+1$.

   from 2.1 $\mathrm{SU}(2)$ Representation Theory

3. Phase Description Timeline Status A ZBW simulator with Bruhat-Tits encoding 2027--2028 Technology readiness level 7 B Bruhat-Tits readout demonstration 2028--2029 Protocol exists (P3, https://papers.qnfo.org/papers/zbw-majorana-tqc-p3-bruhat-tits-readout/) C $d=1$ frequency measurement 2029--2030 Core prediction test D Full adelic ZBW measurement 2030--2032 Archimedean + 2-adic in same apparatus Resource estimate: $\lt $ $ 500K incremental equipment, 3--5 full-time equivalents, executable within a single PhD thesis timeline (3--4 years).

   from 6.3 Timeline and Resources

4. Experimental Timeline (Question 5): A minimal viable experiment using a single trapped ion with Bruhat-Tits state encoding can test the $\sqrt{2}$ prediction within a 3--5 year, sub- $ 500K program.

   from 7. Conclusions

How it is done

1. We prove that the truncated representation ring $R(\mathrm{SU}(2))_k$ is isomorphic to the character ring $R(G_{n+1})$ of the finite quotient $G_{n+1} = (\mathbb{Z}/2^{n+1}\mathbb{Z})^\times$ for matching levels, establishing $\mathrm{SU}(2) \cong \mathbb{Z}_2^\times$ at the fusion-ring level (Theorems 1--3).

   from Abstract

2. By Ostrowski's theorem, the field $\mathbb{Q}$ of rational numbers admits exactly two types of completions: the Archimedean completion $\mathbb{R}$ and the non-Archimedean completions $\mathbb{Q}_p$ for each prime $p$.

   from 1. Introduction

3. Theorem 1 (Fusion Ring Isomorphism).

   from 2.3 The Fusion Ring Isomorphism

4. Theorem 4 (Satake/Langlands Correspondence).

   from 2.4 The Langlands Connection

What would falsify it

1. The structural underdetermination of the position operator in relativistic quantum mechanics -- manifested at the Archimedean place as zitterbewegung (the Newton-Wigner no-go) and at p-adic places as a discrete Bruhat-Tits observable -- arises from a single representation-theoretic origin: the Silent Parameter Principle.

   from Abstract

2. All three falsifiability conditions are satisfied, and the framework resolves zitterbewegung as a completion-relative phenomenon of the fusion ring $\mathcal{R}(k)$, invariant across all completions of $\mathbb{Q}$, with the basis labeling determining which physical observable is extracted.

   from Abstract

3. The parameter selecting the basis is "silent" because it does not affect the fusion rules; it only affects the basis-dependent observable.

   from 1. Introduction

Where the claim stops

1. The parameter selecting the basis is "silent" because it does not affect the fusion rules; it only affects the basis-dependent observable.

   from 1. Introduction

2. Boundary regime ($d = 0$): ZBW is an essential (non-removable) feature -- the FW transformation does not converge.

   from 5.2 p-Adic Foldy-Wouthuysen Transformation

Where to read

1. Abstract249 words · 1 min
2. 1. Introduction476 words · 2 min
3. 2. The Silent Parameter Principle546 words · 2 min
4. 3. p-Adic ZBW Frequency215 words · 1 min
5. 4. The ZBW as a Quantum Clock225 words · 1 min
6. 5. Interacting QFT and Foldy-Wouthuysen at p-Adic Places298 words · 1 min
7. 6. Experimental Feasibility204 words · 1 min
8. 7. Conclusions381 words · 2 min