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January 1999

This is an attempt to describe some ideas of Noncommutative Geometry (NCG) and the perspectives it offers. It is intended as a

The idea of theoretical physics is to describe the outside world
in a satisfactory way, to * find out * how and why thing happen
(or don't happen). Amazingly, this is possible with the use of
mathematics: equations, analysis, algebra...
Some major discoveries came along with new mathematical tools applied
to physics. Mechanics developed along with * analysis*, statistical
mechanics with * probability*, general relativity found * differential
and Riemannian geometry* as its correct mathematical language,
quantum mechanics stimulated research in the theory of * operator
algebras*, and, last not least, gauge theories use * group theory * and
fibre bundle theory - to mention just a few examples.
Now, with an ultimate task to find a * better theory* and a
* better understanding* would it not be useful to look what
mathematics has to offer?

Shortly speaking - in a very broad sense - it is a generalization
of geometry. Geometry deals with * spaces*, their properties
and features, maps between them and structures built on them.
Surprisingly, one can attempt to have a look at spaces from
a different angle and then it appears that they are only
a small fraction of a huge family of similar objects.
This different point of view is offered by the theory of * algebras*.
The starting point is the * Gelfand-Naimark theorem*, which
establishes a one-to-one correspondence between a certain class
of algebras (C* commutative) and a class of spaces (topological,
compact).
To see this link let us spend a while on this correspondence. Having
a topological, compact space we can consider the algebra of continuous
complex-valued functions, with the pointwise addition and multiplication.
It be can proven that with a suitable (maximum) norm it is a commutative
C*-algebra. Now, let us see it in the opposite direction. Take a
commutative C*-algebra and consider the * set* of all characters
on it - characters are complex valued maps, which preserve operations
in the algebra. Then, it could be shown that this set can be equipped with
a topology such that the original algebra is identical with continuous
functions on it.
From the algebraic point of view there is only one thing, which
makes the above construction of the link between space and
algebra possible: the fact that the algebra is * commutative*.
However, there are many noncommutative algebras. The main
idea of Noncommutative Geometry is to treat the noncommutative
objects as if they were related to some * noncommutative*
spaces although there are no such spaces in the usual sense of
the word. Nevertheless many of the geometrical constructions
(conformal, metric structures, differential calculus, fibre bundles
etc.) can be carried out in this situation, and what is even more
remarkable, the algebraic language seems to be more appropriate
also for the "classical" objects.

Noncommutative Geometry offers an amazing perspective for
a theoretical physicists as a tool. Let us stress that it is not a
physical theory in itself, it is - first of all - mathematics.
Yet, it can be applied to theoretical physics - but only in the same
way Riemannian Geometry is used in general relativity
or the theory of connections on principal bundles is used in
gauge Yang-Mills theories.
Shortly speaking - it offers * probably* a good language and
a new point of view. With quantum mechanics in the background
and speculations about possible relations between quantum gravity
and a "fuzzy" structure of space-time at small distances, algebras
and Noncommutative Geometry * might* provide a good language
to describe and understand such phenomena.

The ideas, which are relevant to the Noncommutative Geometry started
to appear in this century with progress in the research on operator
algebras, algebraisation of K-theory and results in the index theory.
The signal to treat these objects as * geometries* and look for possible
physical relevance came in the eighties from Alain Connes.
The scope of Noncommutative Geometry is very broad, it stretches from the
"classical" differential geometry and topology, group theory, foliations,
through theory of C*-algebras, K- and KK-theory, Hopf algebras
(and quantum groups) to cyclic (co)homology and index theory.
One should stress at this point that this is still a growing subject with many
things to be done.

So far we have learned that one may express some * classical things*
in the language of algebras. What do we gain apart from some formal
simplifications?
There are two things. First, we create a more general setup for specific theories.
For instance, if one wants to do differential geometry one is forced to take
a differential manifold. However, from the NCG point of view - this task can
be achieved for a * much wider* class of objects. Differential calculi can be
constructed for all algebras, so one can study "differential geometry" of
lattices or fractals, for instance.
Why is this important for physics? Field theory is usually constructed in
a geometrical setup, with an underlying manifold. Noncommutative
Geometry tells that the class of possible geometries is bigger, in fact
it may appear that the geometry of the world is not that of a manifold
but rather of some "noncommutative objects". Some suggestions along
this lines appear frequently in considerations originating from research
on quantum gravity or string theory. Indeed, we cannot be sure that the
fine structure of space-time is that of an manifold. Could it be a lattice?
A fractal set ? Or some fuzzy * noncommutative* manifold?
The second motivation is of another nature. We not only extend the
* classical *notions, we - somehow - can find more meaning and
relations if we * look from above*, if we take a more general point
of view. The powerful algebraic machinery, which - if we learn enough
about it - could give us, at least, some answers.

Let us have a look at the world we live (as described by theoretical physics)
and try to answer that question. Surprisingly the answer might be **yes**.
There are some hints that the structure of the gauge theory (the Standard Model)
of the elementary particles is the one that can be describes using
a finite-dimensional algebra. One can interpret it as a noncommutative
discrete space, with the gauge transformations originating as the transformations
of this space (automorphisms of the algebra).
Particularly interesting in this approach is that the gauge interactions
appear along with gravity. Let us remind that the data about gravity
on a manifold can be compressed in a Dirac operator, which gives both
the metric and differential structure. The Christoffel symbols (connection)
appears as a result of postulated covariance under the diffeomorphism
transformations on the manifold (which in physics is often referred
to as a change of coordinates). Now, for the * discrete *algebra one
can also have a Dirac operator and similarly, the covariance under the
corresponding transformations for the discrete algebra (which are
identified with gauge transformations) leads to gauge connections
(gauge fields) and the Higgs field. The original * discrete *Dirac
operator seems to encode the masses and mixing angles of the
elementary fermions.
Of course, such model is not perfect and still neither gives a deep
insight into the structure of masses, nor reduces significantly the
number of free parameters. However, what it offers is the hint that
there is some * interesting geometry* behind the structure of
the Standard Model and that the world of particle physics is more
closely related to gravity than one might suspect.

Another interesting prospective comes with the idea of * Hopf algebras*
(* or quantum groups*). They are noncommutative generalizations of
groups (and Lie algebras) and could be considered as objects, which naturally
appear in the studies of "noncommutative spaces".
Since symmetries (in the classical sense: groups and Lie algebras) play
a very important role in theoretical physics, the question whether these
* quantum symmetries* occur as well seems to be well justified.
Among the hopes for the NCG one mentions that it may be the right tool
to describe the quantum space-time with its appropriate quantum
symmetries. There are some more recent suggestions that a finite Hopf
algebra, which appears as a fibre in the covering of "classical" SL(2)
group by its quantum deformations might be related with the algebra
occuring in the Standard Model. Another idea relates a Hopf Algebra of
rooted trees with divergent Feynman diagrams and renormalization.

What is the price one pays for dealing with Noncommutative Geometry?
Probably the most singnificant one is that the concept of space
(understood as a set of points) is lost and can no longer *visualise*
the objects we are dealing with.
This is not new in physics: quantum mechanics has taught us to deal
with the phase space, which cannot be imagined as a manifold. However,
since here one wants to do * geometry* this lost intuition is somehow
worrying. What is closely connected with this point is that there is no
obvious guiding principle, which would tell us which objects are "good"
or "relevant" - in the "spatial" geometry the intuition was often a key,
here, Noncommutative Geometry is like entering a new, partially
unknown land.
Of course, does this suggest that we are in need of a new intuition?

I would like to thank Jan Czyzewski, Mario Paschke, Gianni Landi, Markus Walze and Raimar Wulkenhaar for their help in creating this text.