Higher-Dimensional Algebra and Planck-Scale Physics

TL;DR

This paper introduces how higher-dimensional algebra can be used to develop background-free quantum gravity theories, highlighting the connection between space, state, spacetime, and process, with insights from topological quantum field theories.

Contribution

It proposes a mathematical framework using higher-dimensional algebra to relate space and state, advancing background-free quantum gravity research.

Findings

Link between space and state clarified

Higher-dimensional algebra provides new tools for quantum gravity

Insights from 3D topological quantum field theories

Abstract

This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' and `state', and similarly those of `spacetime' and `process', which we argue is to be expected in any background-free quantum theory. We sketch how higher-dimensional algebra provides the mathematical tools to make this link precise. Finally, we comment on attempts to formulate a theory of quantum gravity in 4-dimensional spacetime using `spin networks' and `spin foams'.