Struggles with the Continuum

TL;DR

This paper discusses the fundamental issues arising from treating spacetime as a continuum, highlighting challenges like singularities and divergences across various physical theories.

Contribution

It provides a comprehensive survey of the problems caused by the continuum assumption and reviews efforts to address these mathematical and physical challenges.

Findings

Singularities pose significant problems in classical and quantum theories.

Divergent integrals are common obstacles in mathematical physics.

Extensive work has been done to manage these issues, but solutions remain complex.

Abstract

Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics, quantum electrodynamics and the Standard Model. In general relativity, singularities are intimately connected to some of the theory's most dramatic successful predictions. We survey these problems and the large amount of work that has gone into dealing with them.