Adelic Constraints on Quantum Field Theory

Investigating whether the adelic completion of $\mathbb{Q}$ forces dimensionless physical constants to specific number-theoretic values.

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Overview

This project investigates whether the adelic (simultaneous real and $p$-adic) completion of the rational numbers constrains fundamental physical constants — specifically the fine-structure constant $\alpha$ and its renormalization group flow.

Ostrowski's Theorem

The only non-trivial completions of $\mathbb{Q}$ are:

• The real numbers $\mathbb{R}$ (Archimedean)
• The $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$ (non-Archimedean)

Together they form the adele ring $\mathbb{A}_\mathbb{Q}$. The product formula states:

$$\lvert q \rvert_\infty \prod_{p} \lvert q \rvert_p = 1 \quad \text{for all } q \in \mathbb{Q}^\times$$

Principal Finding

The Freund-Witten (1987) adelic product formula for the Veneziano string amplitude has been computationally verified:

$$A_\infty(s,t) \prod_{p} A_p(s,t) = 1$$

This establishes that string scattering amplitudes satisfy a consistent adelic structure across all completions of $\mathbb{Q}$.

Quick Start

pip install numpy scipy sympy mpmath matplotlib
cd src/
python test_foundations.py          # 30/30 tests — product formula verified
python gelfand_graev_gamma.py       # Freund-Witten verification

Repository Structure

├── 1.1.md                 Definitive Research Plan
├── 1.1.1.md               Freund-Witten Normalization Details
├── module_01–09_report.md Module execution reports
├── synthesis_final.md     Project synthesis (M10)
├── src/                   15 Python files
├── images/                9 PNG figures
├── data/                  Checkpoint data directory
└── README.md              This file

Modules

Module	Title	Status	Key Result
M1	Foundational Library	✅	30/30 tests, product formula verified
M2	$p$-adic Analysis	✅	$Z_p \to e^{-\beta}$ (corrected original plan)
M3	Adelic Partition Function	⚠️	$\Xi \to 0$ — diverges, pivot to M4
M4	Freund-Witten Veneziano	✅	Product = 1 verified via analytic continuation
M5	Hierarchical RG	⚠️	Toy model; full recursion needs literature
M6	Zeta Zeros	✅	GUE confirmed (Montgomery-Odlyzko)
M7	Cross-Ratio Flow	✅	Discrete RG maps computed at each prime
M8	Beta Reconstruction	✅	$\propto \alpha^2$ structure recovered
M9	Null Models	✅	Strongest findings are mathematical identities
M10	Synthesis	✅	Project closeout (synthesis_final.md)

Key Insights

1. Adelic products of norms work — the product formula is an exact mathematical identity
2. Adelic products of integrated quantities diverge — partition functions and Beta functions don't have the "= 1 almost everywhere" property that norms do
3. The Freund-Witten product formula IS correct — but requires the Gel'fand-Graev gamma (not Morita's) and analytic continuation via $\zeta(s)$
4. The adelic framework constrains the STRUCTURE of physical laws (functional forms) but not all specific numerical values (which may be contingent)

License

MIT