Adelic Factorization of Quantum Dynamics Across All Completions of ℚ

In the paper's words

We explore the conjecture that the conventional choice of the Archimedean completion ℝ as the underlying number field of quantum theory is a matter of convention rather than necessity. By invoking Ostrowski’s theorem, which partitions the completions of ℚ into the unique Archimedean place ∞ and the family of non‑Archimedean ℚₚ for each prime p, we propose a framework in which quantum dynamics factorizes into local contributions at every place. Following the proposal in [9] that Tate’s 1950 thesis provides an explicit structural template for constructing adelic quantum mechanics, we adopt the local–global product structure as a working analogy and formulate consistency conditions that enforce a global adelic product structure on quantum amplitudes. [...]

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

Numbers it reports

1. To make the proposal concrete, we (i) derive a formal adelic path‑integral expression whose functional equation constrains physical amplitudes, (ii) construct elementary p‑adic quantum models formally, with the reproduction of standard real‑valued quantum mechanics in the limit where non‑Archimedean contributions are suppressed remaining a conjectured goal rather than a verified result, and (iii) perform a numerical illustration using fidelity data from a state‑of‑the‑art superconducting quantum processor (105 qubits, single‑qubit fidelity 99.90 %, two‑qubit fidelity 99.56 %, readout fidelity 98.7 %).

   from Abstract

2. The product fidelity $F_{\text{prod}} = \prod_i f_i$, where each $f_i$ is the decimal representation of a fidelity (e.g., 99.90 % → 0.9990).

   from 3. Methods

3. This demonstrates that adelic measures can control distribution properties across all completions, a principle we import to quantum amplitudes. [2] Tianyan: Cloud services with quantum advantage describes a cloud‑accessible superconducting quantum processor (Tianyan‑287) featuring 105 qubits and operational fidelities of 99.90 % (single‑qubit), 99.56 % (two‑qubit), and 98.7 % (readout).

   from 2. Background and Related Work

4. The arithmetic of these fidelities yields an exact mean fidelity of $\bar{F}=0.993866\overline{6}$ (i.e. $99.3867\%$, rounded to $99.39\%$) and an exact product fidelity of $F_{\text{prod}}=0.9816745428$ (i.e. $98.16745428\%$, rounded to $98.17\%$), illustrating how local‑place metrics can be aggregated adelically.

   from Abstract

5. This observation motivates the hypothesis that quantum dynamics might admit an analogous adelic factorization: a global amplitude 𝔸 should decompose as $$ \mathcal{A} \;=\; \prod_{v\in\{\infty,2,3,5,\dots\}} \mathcal{A}_v, $$ where each factor $\mathcal{A}_v$ lives in the local field corresponding to place $v$.

   from 1. Introduction

6. Third, we provide a numerical illustration using fidelity statistics from the Tianyan‑287 superconducting quantum processor (see [2]), showing how local‑place performance metrics can be combined adelically.

   from 1. Introduction

How it is done

1. By invoking Ostrowski’s theorem, which partitions the completions of $\mathbb{Q}$ into the unique Archimedean place $\infty$ and the family of non‑Archimedean $\mathbb{Q}_{p}$ for each prime p, we propose a framework in which quantum dynamics factorizes into local contributions at every place.

   from Abstract

2. Ostrowski’s theorem, however, tells us that $\mathbb{Q}$ admits infinitely many non‑Archimedean completions $\mathbb{Q}_p$, one for each prime $p$, which are mutually singular with respect to the usual topology on $\mathbb{R}$.

   from 1. Introduction

3. The literature on adelic methods in arithmetic dynamics and mathematical physics provides several building blocks for our proposal. [1] Quasi‑adelic measures and equidistribution on $\mathbb{P}^1$ reports that Baker‑Rumely and Favre‑Rivera‑Letelier independently proved an arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure on $\mathbb{P}^1$.

   from 2. Background and Related Work

4. Our approach proceeds in three parallel tracks.

   from 3. Methods

What would falsify it

1. Can the adelic product predict observable corrections (e.g., tiny deviations in interference fringes) that are within reach of near‑term quantum hardware?

   from 6.3 Open Questions

2. Our analysis suggests that the emergence of $\mathbb{R}$ in physics may be understood as a global consistency artifact, while the non‑Archimedean factors predict subtle structural signatures that could be probed in future quantum‑information experiments.

   from Abstract

3. Falsifiability: If future experiments on p‑adic‑inspired quantum simulators reveal deviations from the predicted adelic product (e.g., the global amplitude does not factorize), the conjecture would be falsified.

   from 6.2 Potential Failure Modes

Where the claim stops

1. The conjecture that the Archimedean real numbers arise as a global consistency artifact rather than a fundamental necessity remains open, inviting further mathematical and experimental scrutiny.

   from 7. Conclusion

2. Ostrowski’s theorem, however, tells us that $\mathbb{Q}$ admits infinitely many non‑Archimedean completions $\mathbb{Q}_p$, one for each prime $p$, which are mutually singular with respect to the usual topology on $\mathbb{R}$.

   from 1. Introduction

3. Our contribution is primarily conceptual and illustrative; we do not claim a complete physical derivation of adelic quantum mechanics.

   from 1. Introduction

Where to read

1. Abstract264 words · 1 min
2. 1. Introduction301 words · 1 min
3. 2. Background and Related Work660 words · 3 min
4. 3. Methods269 words · 1 min
5. 4. Analysis297 words · 1 min
6. 5. Results114 words · 1 min
7. 6. Discussion314 words · 1 min
8. 7. Conclusion117 words · 1 min
9. Appendix A. Divergence report30 words · 1 min
10. Appendix B. Claim attribution112 words · 1 min