ar X iv : g r - qc / 9 90 20 17 v 1 4 F eb 1 99 9 Higher - Dimensional Algebra and Planck - Scale Physics

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 higherdimensional 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’.