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,
Also available in
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
(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.
- Lessons from Topological Quantum Field Theory
*-Category of Hilbert Spaces
- The Monoidal Category of Hilbert Spaces
This paper is a followup to
Higher-Dimensional Algebra and Planck-Scale Physics.
© 2004 John Baez