John Baez

Colloquium at the Perimeter Institute

May 31, 2006

Higher-Dimensional Algebra: A Language for Quantum Spacetime

Category theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, the counterintuitive features of quantum theory turn out to be properties that the category of Hilbert spaces shares with the category of cobordisms - in which objects are choices of "space", and morphisms are choices of "spacetime". The striking similarities between these categories suggests that "n-categories with duals" are a promising framework for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams, to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, quantum gravity coupled to point particles, and 4d BF theory coupled to strings.

• You can see the transparencies for this talk in PDF and Postscript
• You can also see a video of this talk. If you have trouble, go here, click on "Seminar Series", and search under my name. Experts will note that I say the cobordism group is nontrivial in 3 dimensions; I should have said 4 dimensions! Nonexperts will enjoy the thunderstorm that repeatedly knocks out the overhead projector.

For more on this subject try these introductory papers:

• John Baez, Higher-Dimensional Algebra and Planck-Scale Physics
• John Baez, Quantum Quandaries: A Category-Theoretic Perspective
• John Baez and Aaron Lauda, A History of n-Categorical Physics (draft version)

• Bob Coecke, Kindergarten Quantum Mechanics

• Samson Abramsky and Bob Coecke, A Categorical Semantics of Quantum Protocols

• John Baez, An Introduction to Spin Foam Models of BF Theory and Quantum Gravity
• John Baez, Alissa Crans and Derek Wise, Exotic Statistics for Strings in 4d BF Theory
• John Baez and Alejandro Perez, Quantization of Strings and Branes Coupled to BF Theory
• John Baez and Urs Schreiber, Higher Gauge Theory

• Laurent Freidel and David Louapre, Ponzano-Regge Model Revisited I: Gauge Fixing, Observables and Interacting Spinning Particles
• Laurent Freidel and David Louapre, Ponzano-Regge Revisited II: Equivalence with Chern-Simons

• Aaron Lauda and Hendryk Pfeiffer, Open-Closed Strings: Two-Dimensional Extended TQFTs and Frobenius Algebras
• Aaron Lauda and Hendryk Pfeiffer, State Sum Construction of Two-Dimensional Open-Closed Topological Quantum Field Theories

• Karin Noui and Alejandro Perez, Three Dimensional Loop Quantum Gravity: Coupling to Point Particles

• Ingo Runkel, Jürgen Fuchs and Christoph Schweigert, Categorification and Correlation Functions in Conformal Field Theory
• Ingo Runkel, Jens Fjelstad, Jürgen Fuchs and Christoph Schweigert, Topological and Conformal Field Theory and Frobenius Algebras

• Urs Schreiber, From Loop Space Mechanics to Nonabelian Strings

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