Karl--Georg Schlesinger
Towards Quantum Mathematics Part I: From Quantum Set Theory to Universal Quantum Mechanics (
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(J. Math. Phys. 40, 3 , (1999), pp. 1344-1358)
25 pages
Abstract
We develop the old idea of von Neumann of a set theory with an
internal quantum logic in a modern categorical guise (i.e. taking the
objects of the category H of (Pre-)Hilbert spaces and linear maps as
the sets of the basic level). We will see that in this way it is possible
to clarify the relationship between categorification and quantization
and besides this to understand that in some sense a categorificational
approach to quantization is a discretized version of the one taken by
noncommutative geometry. The tower of higher categorifications will
appear as the analog of the von Neumann hierarchy of classical set theory.
Finally, we make a suggestion how to understand all the different
categorifications as different realizations of one and the same abstract
structure by viewing quantum mechanics as universal in the sense of
category theory. This gives the possibility to view extended topological
quantum field theories purely as involving an abstract notion of
quantum mechanics plus representation theory without the need to
enlarge the class of kinematic structures of quantum systems on each
step of categorification. In a future part of the work we will apply
the language developed here to deal especially with the question of a
categorification of the manifold notion.
Karl--Georg Schlesinger
Towards Quantum Mathematics Part II: Manifold Notions (
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19 pages
Abstract
Here we use the language of quantum set theory, developed in
Part I of this work, to explore quantized (i.e. categorified) manifold
notions. We first deal with the differentiable structure in the sense of
an infinitesimal patching of tangent spaces and arrive at the (finitedimensional)
representations of (higher) groupoids this way. The relation
to TQFT and to previous work of the author on a quantization of
the category of topological spaces and continuous injections is pointed
out. In a second approach, we deal with the topological level, first
discretized by a triangulation and then in an easy to grasp continuous
analog of this. Here, we are lead to non-abelian cohomology with
n-th cohomology taking values in an (n + 1)-Hilbert space (which is
a weak n-category). In three dimensions, some of these cohomology
classes give rise to spin networks labeled by representations of a quantum
group. In higher dimensions, we expect higher spin complexes
labeled by corresponding (higher) categorified structures. This points
to a link between the conceptions of quantum geometry in quantum
set theory and in recent work on quantum gravity.
Вот единственное, что я понял:
We will see that in this way it is possible
to clarify the relationship between categorification and quantization
and besides this to understand that in some sense a categorificational
approach to quantization is a discretized version of the one taken by
noncommutative geometry.
Категорный подход к квантованию - это, значит, дискретная версия подхода в некоммутативной геометрии в некотором смысле.
www.mccme.ru/dubna/2005/courses/gorod_hi.html
Основная идея курса — вбросить мысль, что правильная "некоммутативная алгебра" — это теория категорий.
И совсем это не ерунда.
Мы можем взять электрон с позитроном, объединить, и получим два фотона, ничего больше. И наоборот, и можем ещё кучу разных интересных вещей из пары электронов получить, вот только ускорить хорошенько. А "струна" может "состоять", к примеру, из стула и стола 
чего-то из аристотеля читали...
Тем не менее, сейчас очень много разоговоров про струны и эксперименты на БАК.