A Brief Introduction to Quantum Geometry

by Micho Ðurðevich

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


Introduction

Quantum Physics has a charming twin sister, of a purely Platonic, mathematical nature. It is called Quantum Geometry. By mirroring and transforming the quantum fundamentals of physics, it gives life to a completely new concept of space. At the basic operational level, this is done by unifying the ideas and methods of classical geometry with those of non-commutative complex algebras and infinite dimensional analysis — essentially expanding the realm of geometry so that magical Quantum Music can be played.

Every geometry deals with some spaces. An inherent part of the being of geometry. Quantum geometry deals with quantum spaces. Including the standard, classical, spaces as a very special case.

In classical geometry spaces are always understandable as collections of points, equipped with an appropriate additional structure. As for example a topological structure given by the collection of open sets, or a smooth manifold structure given by its atlas, or a primitive measurable space structure given by the sigma-algebra of the measurable sets.

In contrast to classical geometry, quantum spaces are not interpretable that way. In general, quantum spaces have no points at all! They tend to exhibit non-trivial quantum fluctuations of geometry at all scales. The classical vision of a fragmentable fabric of space is replaced by a unified structure requiring an entirely new and holistic approach.

Classical <~>  Quantum

A very interesting potential application of quantum geometry in physics is to provide a mathematically coherent description of the physical space-time, at all scales — in particular at the level of ultra-small distances, characterized by the Planck Lenght. This lenght is a universal physical constant, defined as a unique (modulo dimensionless multiplicative factors) combination of gravitational constant γ Planck's constant ℏ and the velocity of light c. Explicitly,

planck-lenght

As we can see, it is an exorbitantly small number — measured in the standard human-scale units! Approximately

(Planck Length) : (Proton Size) = (Proton Size) : (Mexico City Size).

But besides such an incredible smalleness, the very fact that combining the three universal constants of Nature we can obtain, and in essentially unique way, another universal constant with the dimension of length (and similarly, the time constant and the energy / mass constant) is quite interesting. It can be understood as a pra justification of the existence of natural scale for all things. An inherently non Euclidean phenomenon, indeed.

There is an array of reasons to believe that Planck's lenght marks a boundary for the applicability of classical concepts of Space and Time in Physics.

Indeed, the assumption that the underlying space-time is a smooth manifold is contained in the roots of various mathematical inconsistency problems appearing in quantum field theory. The same assumption lies in the roots of the failure of many attempts to unify gravity and quantum theory. The difficulties with such classical concepts about space and time appear at the very small distances, precisely of the order of magnitute of the Planck lenght.

Quantum geometry introduces much more flexibility in the game, allowing us to express the idea that the space-time exhibits certain quantum fluctuations of the structure which are neglectable at the macroscopic level, but which become essential at the level of the Planck scale. In particular, the very concept of a space-time point is loosing the sense at the quantum level. The same applies to the space-time coordinates.

The formalism of quantum geometry is a symbiosis of global methods from classical diffferential geometry, with non-commutative algebras and functional analysis. Quantum spaces are described by certain non-commutative complex *-algebras. The elements of these algebras are intuitively interpreted as the appropriate functions (continuous or smooth for example) over the associated quantum spaces. The mentioned *-algebras are always associated, in the appropriate sense, to certain C*-algebras representing the quantum space at the topological level.

When the algebras are commutative, we are back in the classical geometry. In other words, classical geometry is understandable as the commutative sector of quantum geometry. For this reason quantum geometry is also called non-commutative geometry.

Non-commutative geometry has a great conceptual value for the study of classical spaces. In many situations, the proofs of the theorems of classical geometry become more elegant and transparent if performed at the quantum level. The language of local coordinates, open sets and points, characteristic for classical geometry, sometimes hides the true geometrical structure. On the other hand, in non-commutative geometry we are a priori forced to work with the global entities intertwined with the existing geometrical structure.

In generalizing classical geometry to the non-commutative level, there are two important conceptual steps: Translation of geometry into commutative algebra language, and non-commutative generalizations.

The first step consists in re-expressing a geometrical structure existing on a classical space X in terms of the algebraic structure of the associated (commutative) *-algebra of the appropriate complex-valued functions on X. The definition of this algebra depends on the geometrical level of our considerations. For example:

Geometric structure *-Algebra of Functions
Measure theory Measurable functions
Topology Continuous functions
Differential Geometry Smooth functions
Algebraic Geometry Polynomial functions

It turns out that the geometrical structure on X is always completely expressible at the language of the associated *-algebra. The second step consists in the appropriate noncommutative generalization of such algebraically-reformulated geometry. The idea is to replace the function algebras by more general non-commutative *-algebras, and enlarge in such a way the concept of space — by introducing quantum spaces.

In what follows we shall explain both conceptual steps in more details. Then we shall discuss some concrete examples of quantum spaces.


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