Quantum quandaries: a category-theoretic perspective. (English) Zbl 1125.83005

Baez, John

Rickles, Dean (ed.) et al., The structural foundations of quantum gravity. Oxford: Clarendon Press (ISBN 0-19-926969-6/hbk). 240-265 (2006).

This paper contributes to what it describes as the great challenge of reconciling general relativity and quantum theory. To do this it uses category theory. Two categories are considered. One is the category named Hilb where the objects are Hilbert spaces and the morphisms are linear operators between them. (This is a structure for quantum mechanics.) The other is named nCob where the objects are \((n-1)\)-dimensional manifolds and the morphisms are \(n\)-manifolds between them. (These are known structures for space-time when it is assumed to be \(n\)-dimensional.) The conclusion is a very interesting summary. One of the points made is that mathematics based on set theory may successfully describe the world of quantum mechanics, but it may not be the most enlightening approach.
For the entire collection see [Zbl 1104.83013].

Reviewer: Brian L. Burrows (Stafford)

MSC:

83C45	Quantization of the gravitational field
03C35	Categoricity and completeness of theories

Keywords:

category; quantum