Topological cones and positively polarizable hyperbolic norms

TL;DR

This paper explores the structure of linear cones over ordered fields, their embeddings into vector spaces, and the properties of hyperbolic norms, revealing connections to Lorentzian geometry and order completeness.

Contribution

It introduces a universal vector space for linear cones over ordered fields and links hyperbolic norms to Lorentzian inner products on these spaces.

Findings

Existence of a unique universal vector space for each linear cone over ordered fields.

Hyperbolic norms induce Lorentzian inner products under certain conditions.

Order-theoretic completeness relates to Wick rotation in Lorentzian geometry.

Abstract

In the first part of this article, we study linear cones over totally ordered fields. We show that for each such cone there uniquely exists a universal vector space (called its spanned vector space) into which it embeds as a generating convex cone. Moreover, we investigate topologies on cones for which the natural cone operations are continuous, and study how these topologies carry over to the spanned vector space. In the second part, we deal with hyperbolic norms which satisfy a polarization identity and are defined on cones over the real numbers. We show that, under reasonable assumptions, such hyperbolic norms induce a Lorentzian inner product on the spanned vector space. Finally, we establish a link between completeness under the Wick rotation of a Lorentzian inner product and order-theoretic completeness.