Noncommutative Geometry extends classical geometry to a noncommutative setting. By making use of functional analysis, operator algebras, spectral theory and spin geometry, NCG extends for instance concepts of locally compact topological space, spin Riemannian manifold... The interest for this field goes far beyond the strict area of pure mathematics. It turns out that some of the main features of fundamental physics appear to fit well with basic concepts of noncommutative geometry, suggesting that this latter can actually provide a way to describe (some of) the major building elements of physics such as spacetime and quantum fields. In particular noncommutative geometry appears, by essence, to be a mathematical framework well designed for a formulation of quantum concepts, at least in view of some of its origins rooted in the operator algebras, von Neumann algebras and quantum mechanics. One especially interesting area of mathematical physics focuses on the interplay between noncommutative geometry and quantum field theory from various viewpoints.
Noncommutative geometry has been founded around 1980 mainly by Alain Connes. It provides basically a generalisation of topology, differential geometry and index theory. The starting idea is to set-up duality between ''spaces'' and associative algebras in a way to provide an algebraic description of the structural properties of the ''space'', in particular topological, metric, differential properties... Standard commutative exemples of this are the Gelfand-Naimark duality between commutative C*-algebras and locally compact Hausdorff spaces or the correspondance between commutative von Neumann algebras and measurables spaces. When the algebra is no longer commutative so that the underlying space does not have a pictorial representation, the various concepts inherited from differential geometry and algebraic topology must be translated into algebraic terms and extended in order to define natural analogs of their commutative counterparts that may now characterize ''noncommutative manifolds''. In this way, vector bundles over locally compact Hausdorff spaces are extended to finitely generated projective module over C*-algebras while the corresponding K-theory and its dual K-homology are extended to K-theory of C*-algebras and K-homology of C*-algebras, unified within Kasparov K-K theory. Besides, it is known that the Chern character of the usual differential geometry gives rise to an isomorphism between K-theory and de Rham cohomology. This latter may admit a natural noncommutative extension as the (periodic) cyclic homology of the algebra. This latter plays actually an important role in noncommutative differential geometry. Alternative noncommutative analog of the de Rham complex is provided by the so-called derivation-based differential calculus whose flexibility facilitates its use in theoretical physics especially when involved in the construction of Lagrangian functionals.
Going further, the noncommutative analog of a Riemann manifold is described by the notion of spectral triple. This object involves a C*-algebra, represented on a Hilbert space and an unbounded self-adjoint operator, all supplemented by a set of axioms. The differential structure on the virtual noncommutative space described by the algebra can be captured by the unbounded operator, together its metric structure. The canonical commutative example is the one of a Riemann spin manifold, for which the relevant algebra is the algebra of continuous functions of the manifold while the unbounded self-adjoint operator is nothing but the Dirac operator, both acting on the Hilbert space of spinors. Spectral triples when viewed as (affined version of) K-cycles are ingredient of the noncommutative generalisation of the notion of Chern character. This culminates with the Connes–Moscovici index theorem, generalising the Atiyah–Singer index theorem to the case of noncommutative manifolds thanks to the existence of a pairing between cyclic cohomology and C*-algebraic K-theory. This index theorem is very interesting in the case of singular spaces, and, in particular on leaf spaces arising in foliation theory, for which usual tools of differential geometry are useless, at least because the spaces are not Hausdorff. These singular spaces are better described by using crossed product algebras or groupoid C*-algebras and invariants of the spaces may be defined as elements in cyclic cohomology and K-theory. The above summary gives a very brief overview of noncommutative geometry which has triggered an intense activity in pure mathematics.
As far as mathematical physics is concerned, the fascinating link between noncommutative geometry and quantum theory has been recognized for a long time together with the possibility it may well give rise to important new developments or even breakthrough in fundamental physics. Noncommutative geometry is a framework well suited to quantum physics. A very basic observation is that quantum physics is essentially noncommutative, since in general observables do not commute. These can be represented as operators acting on some (in general separable) Hilbert space. This set of operators is somewhat reminiscent of a noncommutative algebra from which (e.g in the C* case) the states can be determined. But another basic observation is that physical results can be viewed as stemming from evaluations ω(a) of states ω at some observable a, while the converse, i.e the evaluation a(ω) is meaningless. Pure states (for a C*-algebra) are the natural analog of points. In other words, what actually matters is the set of operators, which corresponds in noncommutative geometry to the noncommutative algebra, and not the set of points. It turns out that noncommutative geometry involves mathematical notions that can be used to extend the aspect of noncommutativity of observables to spacetime itself, opening the possiblity to ultimately encompass in the same formalism quantum physics and general relativity and therefore giving rise to the construction of a unified theory of the so far observed 4 fundamental interactions. This is the major unsolved problem in physics. Recall that at the present time, the 4 fundamental interactions seem to be ruled by 2 distinct theories. One is the Standard Model of the strong and electroweak interactions which is a quantum field theory of Yang-Mills type coupled to matter fields. It provides rather accurate description of short-distance phenomena related to these interactions. As a usual quantum field theory, the spacetime is a data given once and forever, usually pseudo-Riemanian Minkowski flat spacetime, on which functionals representing Lagrangian densities used by physicists are built. The other theory is the general relativity, a classical (i.e non-quantum) theory which rather accurately describes gravitational interaction at long distance between hyper-massive objects. In general relativity, the spacetime is dynamical, from the very construction of the theory, with gravitational effects reflecting the curvature of the spacetime. Both of these theories cannot give a fully accurate description of the physics of phenomena in which the contributions of each of the four interactions are of the same order of magnitude, which is the case, e.g of black holes or physics around the so-called Planck scale. Building a quantum theory that describes gravity at short distance, i.e at the Planck scale, is therefore needed. This major challenge of fundamental physics is called generically quantum gravity.
There are at the present time, three (hopefully) possible ways toward the construction of a consistent theory for quantum gravity. The first one is string theory, which among specific features requires to increase the number of dimensions for some self-consistency reasons. The second way is provided by the loop quantum gravityand related descendant approaches such as group field theories, which uses a spin foam structure for spacetime without using a background spacetime metric, contrary to string theory. Both of these approaches are appealing but they are both faced with technical problems that may render questionable their actual relevance for a proper description of quantum gravity.
The third possible approach is provided by noncommutative geometry, which, regarding its applications to physics, is may be not so developped compared to the tremendous amount of works focused so far on Strings and loop quantum gravity and related topics. In fact, by standard heuristic physical arguments, one may object that having a continuous spacetime at the Planck scale as well as ''commuting coordinates'' seems problematic. Using a framework based on noncommutative geometry permits one to escape these objections which mainly amounts to define field theories on noncommutative manifolds. Attemps of this type fall in the categorie called noncommutative field theories. It turns out that noncommutative field theories have a long history going back to ideas of Heisenberg and Schrödinger more than 70 years ago and explicitly proposed by Snyder in 1947 as an unsuccessful attempt to solve the difficulties related to ultraviolet renormalisation in usual quantum field theory. From the 80's, the works of Connes and others in noncommutative geometry also opened a new way to a promising family of noncommutative field theories, suggesting a deeper understanding of the Higgs phenomenon and reformulating progressively the standard model in a new framework. It turns out that one can obtains from a mere use of some of the standard tools of noncommutative geometry, a unified theory of strong, electroweak interactions together with gravitation, which however is only valid at the classical (non quantum) level (since it does not seem to be renormalisable when gravitational sector is included). This is the Chamseddine—Connes model. The spectral action is computed from a spectral triple describing an ''almost commutative geometry'', namely the noncommutative geometry described by the product of the canonical spectral triple for an ordinary commutative compact Riemannian spin manifold by a zero dimensional spectral triple built from matrix algebra. When the gravity part is removed from the Lagrangian, the resulting theory becomes perturbatively renormalisable and reproduces (almost) all the physical features of the Standard Model with a Higgs mass that can be estimated numerically using the usual receipt of the physicists stemming from the renormalisation group (at the one-loop order). The result is not so far from the usual values obtained within the various versions of the (usual) Standard Model, a result which can be simply understood by noticing that the theory is in some sense built on a usual commutative 4-dimensional space, twisted by a finite internal ''non commutative fiber". This implies that the usual renormalization group flows are not modified. Hence the result. While still not fully satisfactory, the noncommutative geometry approach of the Standard Model, à la Chamseddine—Connes, deserves further investigations in particular in the gravitational sector. Besides, it provides an interesting bonus in that it gives a possible interpretation of the origin of the Higgs field as related to a gauge connection, first noticed a long time ago. It turns out that this feature is not a pecular property linked to the ''almost noncommutative spaces'' as the one used in the Chamseddine—Connes model. A similar interpretation for the Higgs-type fields holds true for ''fully noncommutative spaces'', as e.g those for which the algebra is obtained from deformation theory.
In fact, there are so far several classes of noncommutative field theories built on fully noncommutative spaces that have also received attention. They indeed developped almost independently from the above approach à la Chamseddine—Connes. The interest for noncommutative field theories of this type slowly began to increase from 1986 after a paper by Witten suggesting an heuristic interpretation of string field theory as an example of noncommutative field theories. Works dealing with field theories defined on the fuzzy sphere appeared from the beginning of the 90's. Then from 1998, a specific family of noncommutative field theories came under increasing scrutiny when it has been claimed that string theory has effective regimes described by noncommutative field theories on the Moyal-Weyl space which can be viewed as an aperiodic isospectral deformation of the 4-dimensional (Euclidean) space. Owing to the fact that the corresponding associative product is non local, the resulting noncommutative field theories are also non local. This implies that the renormalisation process used within the usual quantum field theories which uses in particular locality, must be reconsidered. Renormalisation of these noncommutative field theories triggered an intense activity from the beginning of this century when it was discovered a new phenomenon within the perturbative loopwise expansion and called ultraviolet/infrared mixing. It results from the nonlocality of the associative product which generates new infrared singularities in some type of (ultraviolet finite) diagrams of the loop expansion. These latter, when appearing as insertion in higher order diagrams spoil renormalisability: a ultraviolet divergence shows up which cannot be absorbed by any usual redefinitions. A first solution of this problem leading to a renormalisable scalar noncommutative field theories was finally obtained in 2004 and is known as the Grosse-Wulkenhaar model. Among interesting features, this noncommutative field theories has been shown to have an ultraviolet fixed point. Other renormalisable noncommutative field theories have then been constructed. An extension of the Grosse-Wulkenhaar model to the case of 4-dimensional gauge theories was performed at the beginning of 2007, which however is hard to study regarding its (still unproven) renormalisability property. Nevertheless, these models have a non trivial vacuum that complicates their study but involve in the same time interesting properties that deserve further investigations. For instance, gauge invariant mass terms for the gauge fields are allowed even in the absence of some Higgs mechanism. Other type of gauge models have been proposed and are under study but the obtention of a 4-dimensional renormalisable gauge theory on the Moyal space has not been achieved so far. Renormalization lies from its beginning at the heart of quantum field theory. According to our present understanding of fundamental physics, it is an essential property, especially in the framework of noncommutative field theories, in view of its potential impact for a better understanding of gravity at short distance or for a deep understanding of intrinsic properties of various classes of noncommutative field theories, which should have their own salient features.