Explicit partial and functional differential equations for beables or observables

TL;DR

This paper derives explicit partial and functional differential equations to determine observables or beables across various physical theories, clarifying their mathematical structure and relation to known quantities.

Contribution

It introduces a unified framework of differential equations for observables in relational mechanics, electromagnetism, Yang-Mills, and general relativity, extending previous knowledge.

Findings

Explicit PDEs and functional differential equations for observables.

Connection between shape variables and known observables.

Framework applicable to diverse physical theories.

Abstract

We provide explicit partial differential equations - in finite cases - and functional differential equations - in field-theoretic cases - which determine observables or beables in the senses of Kucha\v{r} and of Dirac. These cover a wide range of relational mechanics models as well as Electromagnetism, Yang--Mills Theory and General Relativity. We give an underlying reason why pure-configuration Kucha\v{r} observables are already well-known: various types of shape, E-fields, B-fields, loops and 3-geometries. The partial differential equations or functional differential equations for pure-momentum observables are also posed, as are those for observables which have a mixture of configuration and momentum functional dependence.