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Quantum Quandaries:
A Category-Theoretic Perspective

John C. Baez

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

April 7, 2004

Published in Structural Foundations of Quantum Gravity,
eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press, 2006, pp. 240-265.

Also available in PDF and Postscript.


General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n-1)-dimensional manifolds representing 'space' and whose morphisms are n-dimensional cobordisms representing 'spacetime'. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe 'states', and whose morphisms are bounded linear operators used to describe 'processes'. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both nCob and Hilb but not Set are *-categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime.

  1. Introduction
  2. Lessons from Topological Quantum Field Theory
  3. The *-Category of Hilbert Spaces
  4. The Monoidal Category of Hilbert Spaces
  5. Conclusions

This paper is a followup to Higher-Dimensional Algebra and Planck-Scale Physics.

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© 2004 John Baez