Subject: categories: quantum quandaries - a category-theoretic perspective

From: "John Baez" <[email-masked]>
Date: Wed, 7 Apr 2004 18:38:12 -0700 (PDT)

Dear Categorists -

Some of you may enjoy this paper.  It's written for philosophers
of physics, so I take it a bit easy on the category theory, but
it does explain why I think the *-category of Hilbert spaces and
bounded linear operators is more important in physics than the
mere category of Hilbert spaces and bounded linear operators.

Best,
jb

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http://math.ucr.edu/home/baez/quantum/

Quantum Quandaries: A Category-Theoretic Perspective

John C. Baez

To appear in _Structural Foundations of Quantum Gravity_,
eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press.

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
(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 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.

Also available at http://www.arxiv.org/abs/quant-ph/0404040