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Higher-Dimensional Algebra
and Planck-Scale Physics

John C. Baez

Department of Mathematics, University of California
Riverside, California 92521, USA

January 28, 1999

Published in Physics Meets Philosophy at the Planck Scale,
eds. Craig Callender and Nick Huggett, Cambridge U. Press, 2001, pp. 172-195.

Also available in Postscript and PDF.


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'.

  1. Introduction
  2. The Planck Length
  3. Topological Quantum Field Theory
  4. 3-Dimensional Quantum Gravity
  5. Higher-Dimensional Algebra
  6. 4-Dimensional Quantum Gravity

This paper has a followup: Quantum Quandaries: a Category-Theoretic Perspective.

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© 1999 John Baez