Zitterbewegung as a p-Adic Observable: Ultrametric Readout and Intrinsic Topological Protection for Majorana Qubits

In the paper's words

The Zitterbewegung (ZBW) of a Dirac fermion — the rapid oscillatory motion at the Compton frequency arising from interference between positive- and negative-energy states — has resisted direct experimental observation for nearly a century. The standard interpretation attributes this to the Compton wavelength (λ_C ≈ 3.86 × 10^-13 m for electrons) falling below the resolution of any imaging technology [established]. We propose a different explanation: ZBW is not a signal in the Archimedean (real-number) topology but a p-adic observable whose information content resides in the ultrametric connectivity of momentum-spin eigenstates, not in the amplitude of real-valued oscillations. [...]

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

Numbers it reports

1. Property Value Interpretation Nodes 240 Discrete momentum-spin basis ZBW edges (threshold $e^{-1}$) 5,460 Dense local connectivity Graph density 0.190 Moderately connected MST edges 239 Tree projection of the graph Gromov $\delta$ (avg) 0.000 MST is a true tree Gromov $\delta$ (max) 0.000 No deviation from tree structure Ultrametric violations (/500) 154 (31%) Full graph is NOT ultrametric Isosceles triangle ratio 0.064 Strongly non-ultrametric Unique distance values 52 Continuous spectrum

   from 3.2 Results

2. Procedure: Prepare a ZBW-coupled state $\psi(0)$ with known momentum p and spin projection Perform a "fuzzy" measurement that only resolves the state to 2-bit precision (4 possible outcomes) From the 2-bit outcome, attempt to predict the full 8-bit state using: (a) Standard quantum state tomography (Archimedean — requires $\sim 2^{8}$ measurements) (b) Hensel lifting (p-adic — if the state is p-adic, 2 bits should lift to 8 bits) Compare prediction accuracy vs. number of additional measurements Decision rule: If (a) requires exponentially more measurements than (b) to achieve equivalent fidelity, Hensel lifting is operating — the ZBW state is p-adic.

   from 6b.2 Protocol B: Hensel Lifting of ZBW States

3. Protocol B (1-3 years): Hensel lifting requires advances in trapped-ion state preparation at the fidelity needed for 2-bit → 8-bit lifting.

   from 8b.1 Priority Roadmap

4. The 31% ultrametric violation rate in the full graph represents the "Archimedean shadow" of an underlying p-adic structure.

   from 3.3 Interpretation

5. The analysis in §3 established that the ZBW transition graph for a Dirac fermion has a tree-like MST ($\delta$ = 0) but 31% ultrametric violations in the full graph.

   from 4b.1 Motivation

6. The Majorana constraint reduces edges by 57% (to 43% of Dirac) while preserving full connectivity (both graphs have 239 MST edges = 240 nodes).

   from 4b.4 Interpretation

How it is done

1. We construct a graph $\mathcal{G} = (V, E)$ where: Nodes $V$ are momentum-spin eigenstates $|p_x, p_y, s\rangle$ of the Dirac Hamiltonian on a discretized momentum grid ($\Delta p = 0.5\,m$, $11 \times 11$ grid, 240 nodes including spin) Edges $E$ connect states with non-zero ZBW coupling: $w_{ij} = |\langle \psi_i | \hat{v}_x | \psi_j \rangle|$ with exponential momentum-locality cutoff

   from 3.1 Construction

2. No Archimedean perturbation — regardless of energy scale — can resolve a p-adic fixed point, because the Archimedean and p-adic metrics on $\mathbb{Q}$ are mutually singular (Ostrowski's theorem [established] ).

   from 4.3 Intrinsic Protection Without QEC

3. Protocol: Measure the distribution of waiting times between ZBW-induced spin flips in a Majorana nanowire system.

   from 5.1 Spin Noise Spectroscopy for Ultrametric Clustering

4. Protocol: In p-adic analysis, Hensel's lemma guarantees that an approximate root (known to order $p^k$) can be uniquely lifted to an exact root.

   from 5.2 Hensel Lifting Experiment

What would falsify it

1. We propose a different explanation: ZBW is not a signal in the Archimedean (real-number) topology but a p-adic observable whose information content resides in the ultrametric connectivity of momentum-spin eigenstates, not in the amplitude of real-valued oscillations.

   from Abstract

2. Keywords: Zitterbewegung, Majorana fermion, p-adic observable, ultrametric topology, Bruhat-Tits tree, topological quantum computing, intrinsic dynamical decoupling

   from Abstract

3. 2010), graphene, and photonic microcavities [established] The Majorana connection: The question of whether a Majorana fermion ($\psi = \psi^c$, its own antiparticle) exhibits ZBW, and if so in what form, has received surprisingly little theoretical attention [my conjecture about the gap] This paper addresses the third point and proposes a novel framework: that ZBW is best understood as an ultrametric (p-adic) observable whose tree-like structure provides intrinsic topological protection when the fermion satisfies the Majorana condition.

   from 1. Introduction

Where the claim stops

1. The Hensel lifting proposal requires p-adic measurement theory that does not yet exist.

   from 6.3 Limitations and Open Questions

2. ZBW as an oscillatory kinematic phenomenon does not distinguish Dirac from Majorana fermions.

   from 2.2 Majorana ZBW — Computational Verification

3. However, braiding is challenging to implement and thermal errors still require some QEC.

   from 4.3 Intrinsic Protection Without QEC

Where to read

1. Abstract274 words · 1 min
2. 1. Introduction351 words · 2 min
3. 2. ZBW in Dirac vs. Majorana Fields310 words · 1 min
4. 3. Bruhat-Tits Tree of ZBW Eigenstates243 words · 1 min
5. 4. Ultrametric Protection Mechanism302 words · 1 min
6. 5. Experimental Proposals240 words · 1 min
7. 6. Discussion268 words · 1 min
8. 7. Conclusions155 words · 1 min
9. 4b. Dirac vs Majorana Bruhat-Tits Comparison398 words · 2 min
10. 5b. p-Adic Valuation of the Mass Parameter334 words · 1 min
11. 6b. Expanded Experimental Protocols493 words · 2 min
12. 7b. Connection to p-Adic Anyons and Adelic QEC227 words · 1 min