Tate's Thesis as a Template for Adelic Quantum Mechanics: Local-Global Structure and the Emergence of Archimedean Artifacts

In the paper's words

We propose that Tates 1950 thesis provides an explicit structural template for constructing adelic quantum mechanics. The Dragovich programme has realized this template for free theories; Huang, Stoica, and Zhong (2022) provide an independent proof-of-concept in CFT. We interpret the Langlands program as adelic physics, reframe the measurement problem as a completion problem, and classify measure-theoretic artifacts of the archimedean place.

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

Numbers it reports

1. This paper proposes that Tate's 1950 thesis, which reformulated Hecke $L$-functions via adelic analysis with local zeta integrals at each completion integrated over the idele class group to yield a global functional equation, provides an explicit structural template for constructing adelic quantum mechanics.

   from Abstract

2. We have proposed that Tate's 1950 thesis provides an explicit structural template for constructing adelic quantum mechanics, with four correspondences: local zeta integrals $\to$ local quantum dynamics, idele class group integration $\to$ global coherence via the product formula, the functional equation $\to$ path integral consistency, and the archimedean factor $\to$ the distinguished role of measurement at the $\infty$-place.

   from 10. Conclusion

How it is done

1. Ostrowski's theorem classifies all completions of the rational numbers as the real numbers $\mathbb{R} = \mathbb{Q}_\infty$ and the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$.

   from Abstract

2. But the rational numbers $\mathbb{Q}$ admit infinitely many completions, and Ostrowski's theorem [1] establishes that every non-trivial absolute value on $\mathbb{Q}$ is equivalent to either the standard archimedean absolute value $|\cdot|_\infty$ or a $p$-adic absolute value $|\cdot|_p$ for some prime $p$.

   from 1. Introduction

3. Tate's main theorem is that $Z_\mathbb{A}$ satisfies a functional equation : $$Z_\mathbb{A}(f, \chi, s) = Z_\mathbb{A}(\widehat{f}, \chi^{-1}, 1-s)$$ where $\widehat{f}$ is the adelic Fourier transform of $f$.

   from 2.4 Global Adelic Integration and the Functional Equation

4. In Tate's thesis, the archimedean factor differs qualitatively from the $p$-adic factors: it involves the classical Gamma function $\Gamma(s)$ rather than the geometric series $(1 - p^{-s})^{-1}$ of the $p$-adic factors. $[\text{speculative}]$ In the quantum template, this reflects the qualitative difference between archimedean and non-archimedean completions: the archimedean place supports continuous parameters, differentiable paths, and connected state spaces—all properties that are absent at $p$-adic places.

   from 3.4 The Archimedean Factor $\to$ Distinguished Measurement Role

What would falsify it

1. The claim that "Langlands is adelic physics" is a structural interpretation of this evidence, not a prediction of new experiments in the near term.

   from 6.3 Constraints and Caveats

2. $[\text{not yet falsifiable}]$ This reframing is currently not experimentally testable.

   from 7.3 Falsifiability Assessment

3. We can identify what would constitute a test: If we could construct a measurement apparatus that couples to a $p$-adic completion (e.g., a device whose relevant degrees of freedom are defined over $\mathbb{Q}_p$ rather than $\mathbb{R}$), the reframing predicts that such an apparatus would "measure" the $p$-adic component of the adelic state, yielding different statistics than an archimedean measurement of the "same" observable.

   from 7.3 Falsifiability Assessment

Where the claim stops

1. The physical interpretation depends on an unproven mathematical conjecture—this does not invalidate the interpretation but constrains its scope to what is already proven ($\text{GL}(2)$, some cases of $\text{GL}(n)$).

   from 6.3 Constraints and Caveats

2. The standard dictionary—Riemann surfaces $\leftrightarrow$ function fields, which are one-dimensional objects over finite fields—does not directly generalize to number fields.

   from 6.3 Constraints and Caveats

3. The extension to interacting quantum field theories is conjectural and represents a major open problem.

   from 10.2 Limitations

Where to read

1. Abstract267 words · 1 min
2. 1. Introduction573 words · 2 min
3. 2. Tate's Thesis Recalled559 words · 2 min
4. 3. The Template: From Tate's Thesis to Adelic Quantum Mechanics562 words · 2 min
5. 4. The Dragovich Programme as Realization407 words · 2 min
6. 5. Huang-Stoica-Zhong: Number Theory from Physics288 words · 1 min
7. 6. The Langlands Program: Adelic Physics Discovered Before Its Time456 words · 2 min
8. 7. The Measurement Problem as a Completion Problem373 words · 2 min
9. 8. UV and IR Divergences as Archimedean Artifacts505 words · 2 min
10. 9. Measure-Theoretic Artifacts: A Classification371 words · 2 min
11. 10. Conclusion506 words · 2 min
12. Declarations169 words · 1 min