adelic-qft

Adelic Constraints on Quantum Field Theory

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

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