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

Noncommutative Generalizations

The second main conceptual step consists in generalizing the re-formulated classical geometry, by relaxing the assumption of commutativity of the algebras smooth-A and A and allowing them to be the appropriately chosen non-commutative *-algebras. In such a way we arive to quantum spaces, the main objects of study in quantum geometry. The elements of these non-commutative *-algebras are intuitively interpreted as "smooth functions" (or continuous, measurable--depending on the context) over quantum spaces. However, in contrast to classical geometry, the "existence" of quantum spaces is implicit, as they generally appear in the formalism exclusively via the corresponding *-algebras.

For example, in accordance with this philosophy and the classical Gelfand-Naimark theory, we may say that C*-algebras generalize classical topology of compact (and locally-compact) topological spaces. Such a new class of quantum topological spaces was introduced by Woronowicz in [W1] and developed in [W3]--[W6] mainly in the context of quantum groups. In the case of non-compact structures, we meet an essentially new situation consisting in the existence of various different types of "continuous functions"--the analogs of functions with a compact support, functions vanishing at infinity, bounded functions, unbounded functions. The counterparts of these functions play a very important role in the theory of non-compact quantum spaces and groups [W2, PW], and accordingly we have to play with various types of algebras. More precisely, besides the "vanishing-at-infinity functions" vanishing-at-infinity, it is necessary to introduce the multiplier C*-algebra M(A) representing the bounded continuous functions.

In accordance with the above mentioned Riesz representation theorem, we can say that probability measures on the quantum space X are given by positive normalized linear functionals q-states. Such functionals are called states. In the general C*-algebraic context, positive elements are defined as those satisfying

positivity-viasquares
for some b-element. The set of all positive elements of A is a closed strict conus in A. A linear functional is positive if it takes positive values on positive elements. Positive functionals are automatically continuous. The set S(A) of all states of A is convex, and compact in the *-weak topology of the dual space of A. According to the Krein-Millman theorem, S(A) coincides with the *-weak closure of the convex hull over its extremal elements. The extremal elements of S(A) are called pure states.

There exists a deep connection between C*-algebra representations, and states. Let us consider an arbitrary unital C*-algebra A and let representation-of-A be a representation of A in a Hilbert space H. Every unit vector unit-vectors-H gives rise to a state quantum-st-states, via the formula

state-induction-rep

If the vector psi is cyclic for the representation D, then the representation D is completely determined by the associated state rho. More precisely, the above formula establishes a natural bijection between the states in A and the (isomorphism classes of) pairs rep-cyclic-vector consisting of a representation D with a cyclic unit vector psi. This is the idea of the GNS-construction, which is one of the most important tools in the study of C*-algebras. In terms of the GNS-construction, irreducible representations of A are characterized as those associated to the pure states.

Going back to the classical (=commutative) context---the representation D associated to a state rho is acting in square-integrable-functions and the operators D(a) are given by the left multiplication. Irreducible representations are 1-dimensional and given by characters of A, in other words the points of the associated space X. The pure states are also given by characters of A. The associated probability measures are simply atomic delta-measures concentrated in points of X.

The theory of von Neumann algebras can be viewed as a quantum generalization of classical measure theory. Commutative von Neumann algebras describe classical measurable spaces, and non-commutative von Neumann algebras represent quantum measure spaces. The roots of quantum geometry are present in the foundational papers by Murray and von Neumann [MvN1,MvN2,MvN4].

Various fundamental topics of non-commutative differential geometry, including cyclic cohomogy as topological invariants of quantum spaces, their incorporation into algebraic K-theory, quantum elliptic operators and non-commutative index theory, are developed by Connes [C1,C2]. For a detailed introductory exposition see [K1,K2].

Before passing to some concrete examples of quantum spaces, let us observe that one and the same concept of classical geometry may have several very different generalizations in quantum geometry. This is because the procedure of generalizing objects of commutative algebra into the objects of non-commutative algebras is not at all unique and straightforward.

Perhaps one of the best examples of this phenomena is the quantum differential calculus. In accordance to what we have mentioned above, one natural way of introducing the concept of differential forms in quantum geometry would be to start from an apropriate non-commutative *-algebra q-smooth-A (representing smooth functions) and define differential forms as the elements of the q-smooth-A-generated graded-differential *-subalgebra q-differential-forms of the Chevalley complex q-chevalley-complex associated to the Lie algebra of derivations and its natural representation in q-smooth-A. The product, differential and the *-structure are given by the same formulas as in the classical case (of course, with the difference that in the classical case C is a graded-commutative algebra, in general case it will be highly non-commutative).

This idea was sistematically followed in [Dub-V, DKM1, DKM2]. On the other hand, the mentioned construction is not appropriate in considerations involving quantum groups and differential structures over them, when we must take care of the quantum group symmetry of the calculus. In the quantum group context, a very different construction looks more natural [W5]. This construction incorporates from the very beginning the idea of a quantum group covariance. Both constructions include the classical situation (classical differential calculus over classical Lie groups) as a very special case.

The same remark applies to the very concept of symmetry---one possible way to introduce symmetries in quantum geometry is to consider automorphisms of the corresponding noncommutative algebras. An essentially different way is to play with quantum groups, and define the action of quantum groups on quantum spaces, generalizing the classical concept of a group action.

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