The universal thickening of the field of real numbers

TL;DR

This paper introduces a universal thickening of the real numbers, connecting concepts from tropical geometry, hyperfields, and p-adic period rings, with implications for quantum physics and operational calculus.

Contribution

It constructs a universal pro-infinitesimal thickening of the real numbers using hyperfield and Witt-like processes, linking tropical geometry and p-adic period algebra analogues.

Findings

Identifies the real hyperfield as a dequantization of the perfection process.

Recovers a field from a hyperfield via archimedean Witt construction.

Provides a spectral analysis of the constructed rings and links to quantum oscillatory integrals.

Abstract

We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the "dequantization" process and yields Viro's tropical real hyperfield. Then we prove that the archimedean Witt construction in the context of hyperfields allows one to recover a field from a hyperfield, and we obtain the universal pro-infinitesimal thickening of the field of real numbers. We provide the real analogues of several algebras used in the construction of the rings of p-adic periods. We supply the canonical decomposition of elements in terms of Teichmuller lifts, we make the link with the Mikusinski field of operational calculus and compute the Gelfand spectrum of the archimedean counterparts of the rings of p-adic periods. In the second part of the paper we discuss the complex case and its relation with the theory of oscillatory integrals in quantum physics.