What Is Noncommutative Geometry?
This is an attempt to describe some ideas of Noncommutative Geometry (NCG) and the perspectives it offers. It is intended as a very brief introduction for physicists.
Theoretical Physics and Advances in Mathematics.
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?
What is Noncommutative Geometry ?
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.
What NCG has to offer?
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.
Mathematics of NCG.
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.
Why change the dictionary?
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.
Do we live in a noncommutative world?
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.
Noncommutative Symmetries and Manifolds
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.
Is Our Intuition Lost?
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?