A Brief Introduction to Quantum Geometry
Table of Contents
Introduction
From Geometry to Algebra: 1
Part 2
Non-commutative Generalizations
Examples
Spaces With Indistinguishable Points
Quantum Groups and Quantum Bundles
Quantum Tori and Matrix Spaces
Supermanifolds
Quantization of Symplectic Manifolds
C*-Algebraic Extensions and BDF-Theory
Literature
Examples of Quantum Spaces
Spaces with Indistinguishable Points
Non-commutative geometry provides a set of tools for the study of certain `strange-behaving' spaces that naturally appear in classical geometry.Example 1: Foliated Manifolds
The spaces of leaves of foliations of smooth manifolds. As a rule, such spaces behave very irregularly, from the classical point of view. A general phenomena is that it is not possible to introduce a smooth manifold structure or a reasonable topological structure on them.As an extreme case, let us mention ergodic foliations, and in particular the spaces of orbits of ergodic dynamical systems.
In the ergodic case, it is not possible to introduce a reasonable concept of measurability into the leaf space. The reason for this is in the effective indistinguishability of points of the spaces of leaves of ergodic foliations. And if there are no non-trivial measurable sets, there is no geometry in the standard sense and all the tools of standard analysis loose their validity.
However, it turns out that all the above mentioned spaces can be naturally treated associating to them certain non-commutative *-algebras. One can then speak about differential and integral calculus, cohomological invariants, geometric structures over such spaces. Actually, all basic constructions of classical geometry can be generalized at the quantum level.
Example 2: Discrete Groups
As a similar type of `quantum' spaces, we can mention the space of equivalence classes of irreducible unitary representations of certain discrete groups G.If the von Neumann algebra
A generated by the left-regular representation
of G is a non-type-I factor, then the spectrum
of G exhibits a similar undistingushibility-of-points property. It is worth
mentioning that A is hyperfinite iff G is amenable.
A group G is called
amenable if there exists a left-invariant state on the
C*-algebra
of all bounded continuous functions on G. Every finitely
generated discrete group is viewable as the fundamental group
of a compact smooth 4-dimensional manifold.
Example 3: Penrose Tilings
The space of equivalence classes of certain tilings of the Euclidean plane, such as the Penrose tilings [C2]. This space is defined as follows. Let us consider two triangles of the Euclidean plane
for the first triangle, and
for the second. 
.Let
be the set of all tilings of the Euclidean plane that can be obtained using the
above two `augmented' triangles, and the rules:
[i] The common vortices of the above triangles have the same label, and it is allowed to perform reflections of triangles;
[ii] The oriented edges are paired so that their orientation is the same.
Such tilings exist. The 3-parameter group E(2) of isometric motions of
the Euclidean plane is naturally acting on the space
. Let Q be the corresponding
orbit space. It can be shown that Q possesses (uncountably) infinitely many points.
However the points of Q are effectively indistinguishable, because of the following
remarkable property:
Let S and T be arbitrary two non-equivalent tilings. Then for every finite portion (consisting of finitely many triangles) of S there exists the same portion of T, modulo E(2).It is important to mention that there exist two different interpretations of the relations between quantum spaces and the associated *-algebras. The first one is already explained---it assumes that spaces determine algebras and algebras determine spaces. The second interpretation (originally proposed by Connes) assumes that geometry is determined by the class of Morita-equivalent C*-algebras. In other words non-isomorphic but Morita equivalent C*-algebras describe the same quantum space. By definition, two C*-algebras A and B are Morita-equivalent if
is the ideal of compact operators of a separable Hilbert space.
Morita-equivalent algebras have the same cyclic cohomology and K-groups.
The second approach is more suitable for constructions involving the factor-spaces, as for example the structures mentioned in this subsection. However, it is not appropriate for considerations involving quantum groups and quantum bundles, where passing to a Morita-equivalent algebras destroys the entire geometrical structure.
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