Quantum Quandaries: a CategoryTheoretic Perspective
Abstract
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 (n1)dimensional manifolds representing "space" and whose morphisms are ndimensional 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 Hilb and nCob 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2004
 DOI:
 10.48550/arXiv.quantph/0404040
 arXiv:
 arXiv:quantph/0404040
 Bibcode:
 2004quant.ph..4040B
 Keywords:

 Quantum Physics;
 General Relativity and Quantum Cosmology;
 Mathematics  Quantum Algebra
 EPrint:
 21 pages, 2 encapsulated Postscript figures