An Axiomatization Proposal and a Global Existence Theorem for Strong Emergence Between Parameterized Lagrangian Field Theories

TL;DR

This paper introduces an axiomatization for strong emergence in parameterized Lagrangian field theories and proves a global existence theorem, demonstrating how certain theories can emerge from polynomial evaluations of differential operators.

Contribution

It provides a formal axiomatic framework and a theorem establishing conditions for strong emergence between parameterized field theories, including gravity and noncommutativity.

Findings

Emergence of gravity from noncommutative scalar field theories.

Conditions under which polynomial theories emerge from parameterized Lagrangians.

Potential reciprocal emergence of noncommutativity from gravity.

Abstract

In this paper we propose an axiomatization for the notion of strong emergence phenomenon between field theories depending on additional parameters, which we call parameterized field theories. We present sufficient conditions ensuring the existence of such phenomena between two given Lagrangian theories. More precisely, we prove that a Lagrangian field theory depending linearly on an additional parameter emerges from every multivariate polynomial theory evaluated at differential operators which have well-defined Green functions (or, more generally, that has a right-inverse in some extended sense). As a motivating example, we show that the phenomenon of gravity emerging from noncommutativy, in the context of a real or complex scalar field theory, can be recovered from our emergence theorem. We also show that, in the same context, we could also expect the reciprocal, i.e., that noncommutativity could emerge from gravity. Some other particular cases are analyzed.