Relationalism

TL;DR

This paper explores different philosophical and mathematical interpretations of relationalism and background independence in physics, contrasting Leibniz--Mach--Barbour and Rovelli--Crane approaches, and discusses implications for quantum gravity theories.

Contribution

It provides a detailed comparison of relational concepts in physics, introduces new expansions on configuration space, and applies these ideas to quantum gravity and related theories.

Findings

Categorization is a formal prerequisite for quantization.

Perspecting is a categorical operation relevant to relational approaches.

Propositioning leads to the use of topoi in understanding propositions about systems.

Abstract

This article contributes to the debate of the meaning of relationalism and background independence, which has remained of interest in theoretical physics from Newton versus Leibniz through to foundational issues for today's leading candidate theories of quantum gravity. I contrast and compose the substantially different Leibniz--Mach--Barbour (LMB) and Rovelli--Crane (RC) uses of the word `relational'. Leibniz advocated primary timelessness and Mach that `time is to be abstracted from change'. I consider 3 distinct viewpoints on Machian time: Barbour's, Rovelli's and my own. I provide four expansions on Barbour's taking configuration space to be primary: to (perhaps a weakened notion of) phase space, categorizing, perspecting and propositioning. Categorizing means considering not only object spaces but also the corresponding morphisms and then functors between such pairs. Perspecting means considering the set of subsystem perspectives; this is an arena in which the LMB and Rovelli approaches make contact. By propositioning, I mean considering the set of propositions about a physical (sub)system. I argue against categorization being more than a formal pre-requisite for quantization in general; however, perspecting is a categorical operation, and propositioning leads one to considering topoi, with Isham and Doering's work represents one possibility for a mathematically sharp implementation of propositioning. Further applications of this article are arguing for Ashtekar variables as being relational in LMB as well as just the usually-ascribed RC sense, relationalism versus supersymmetry, string theory and M-theory. The question of whether scale is relational is also considered, with quantum cosmology in mind.