Some Reflections on the Status of Conventional Quantum Theory when Applied to Quantum Gravity

TL;DR

This paper questions the reliance on standard quantum theory with continuum quantities in quantum gravity, proposing that different space-times may require distinct quantum frameworks potentially unified by topos theory.

Contribution

It introduces the idea that quantum theories should be adapted to specific space-time structures and explores topos theory as a unifying mathematical framework.

Findings

Standard quantum theory may be incompatible with quantum gravity assumptions

Topos theory offers a potential method to unify different quantum theories

Presheaf theory can address foundational issues like the Kochen-Specker theorem

Abstract

All current approaches to quantum gravity employ essentially standard quantum theory including, in particular, continuum quantities such as the real or complex numbers. However, I wish to argue that this may be fundamentally wrong in so far as the use of these continuum quantities in standard quantum theory can be traced back to certain {\em a priori} assumptions about the nature of space and time: assumptions that may be incompatible with the view of space and time adopted by a quantum gravity theory. My conjecture is that in, some yet to be determined sense, to each type of space-time there is associated a corresponding type of quantum theory in which continuum quantities do not necessarily appear, being replaced with structures that are appropriate to the specific space-time. Topos theory then arises as a possible tool for `gluing' together these different theories associated with the different space-times. As a concrete example of the use of topos ideas, I summarise recent work applying presheaf theory to the Kochen-Specher theorem and the assignment of values to physical quantities in a quantum theory.