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The Structural Foundations of Quantum Gravity$
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Dean Rickles, Steven French, and Juha T. Saatsi

Print publication date: 2006

Print ISBN-13: 9780199269693

Published to Oxford Scholarship Online: October 2011

DOI: 10.1093/acprof:oso/9780199269693.001.0001

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Quantum Quandaries: A Category-Theoretic Perspective

Quantum Quandaries: A Category-Theoretic Perspective

(p.240) 8 Quantum Quandaries: A Category-Theoretic Perspective
The Structural Foundations of Quantum Gravity

John Baez

Oxford University Press

Part of the difficulty of combining general relativity and quantum theory is that they use different sorts of mathematics: one is based on objects such as manifolds, the other on objects such as Hilbert spaces. As ‘sets equipped with extra structure’, these look like very different things, so combining them in a single theory has always seemed a bit like trying to mix oil and water. However, work on topological quantum field theory has uncovered a deep analogy between the two. Moreover, this analogy operates at the level of categories. This chapter focuses on two categories. One is the category Hilb whose objects are Hilbert spaces and whose morphisms are linear operators between these. This plays an important role in quantum theory. The other is the category nCob whose objects are (n — 1)-dimensional manifolds and whose morphisms are n-dimensional manifolds going between these. This plays an important role in relativistic theories where spacetime is assumed to be n-dimensional: in these theories the objects of nCob represent possible choices of ‘space’, while the morphisms — called ‘cobordisms’ — represent possible choices of ‘spacetime’.

Keywords:   quantum theory, general relativity, Hilbert spaces, nCob

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