Complexity in Tame Quantum Theories

TL;DR

This paper introduces a framework for quantifying the complexity of physical systems using o-minimal structures, linking logical information to quantum field theories and their UV completions.

Contribution

It proposes a novel approach to measure complexity in quantum theories through o-minimal structures, especially Pfaffian and sharply o-minimal structures.

Findings

Many physical systems can be formulated with Pfaffian o-minimal structures.

The complexity of physical observables can be quantified within this framework.

O-minimality is linked to the tameness and UV properties of quantum field theories.

Abstract

Inspired by the notion that physical systems can contain only a finite amount of information or complexity, we introduce a framework that allows for quantifying the amount of logical information needed to specify a function or set. We then apply this methodology to a variety of physical systems and derive the complexity of parameter-dependent physical observables and coupling functions appearing in effective Lagrangians. In order to implement these ideas, it is essential to consider physical theories that can be defined in an o-minimal structure. O-minimality, a concept from mathematical logic, encapsulates a tameness principle. It was recently argued that this property is inherent to many known quantum field theories and is linked to the UV completion of the theory. To assign a complexity to each statement in these theories one has to further constrain the allowed o-minimal structures. To exemplify this, we show that many physical systems can be formulated using Pfaffian o-minimal structures, which have a well-established notion of complexity. More generally, we propose adopting sharply o-minimal structures, recently introduced by Binyamini and Novikov, as an overarching framework to measure complexity in quantum theories.