A Brief Introduction to Quantum Geometry

Introduction

Every geometry deals with some kind of spaces. Quantum geometry deals with quantum spaces, including the classical concept of space as a very special case. In classical geometry spaces are always understandable as collections of points, equipped with the appropriate additional structure (as for example a topological structure given by the collection of open sets, or a smooth structure given by the atlas). In contrast to classical geometry, quantum spaces are not interpretable in this way. In general, quantum spaces have no points at all! They exhibit non-trivial quantum fluctuations of geometry at all scales.

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 combination of gravitational constant , Planck's constant and the velocity of light c. Explicitly,

As we can see, it is an exorbitantly small number! There are many 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 priory forced to work with the global entities inherently connected 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: