Several physical theories are intimately related with geometry. Classical mechanics is the theory of the geometry of phase space, and general relativity is the theory of curved space time. Let us define geometry via the highest authority we can think of: Wikipedia!
Geometry (Greek γεωμετρια; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space.
This view of geometry enters in a crisis with the advent of quantum mechanics, the coordinate functions of phase space become noncommuting operators, and the concept of point is untenable, due to Heisenberg uncertainty principle. Still the quantum phase space is a very natural structure: it is the first example of a noncommutative geometry. Von Neumann named it “pointless geometry’’.
Since there are no points, we need a way to describe geometry which transcends them, and which can be generalized. This is provided by a series of results of Gelfand and Naimark, and later Alain Connes, which give a description of geometry based on the commutative algebra of functions defined on them. The step to a noncommutative geometry is to generalize the product of the algebra to a noncommutative one, possibly depending on a “small’’ parameter, so that the usual geometry is recovered in the limit.
The algebra is then an algebra of operators, and the description changes, from points and their relations, to the Hilbert space, states, and especially the spectrum of operators. This vision has been applied by Connes and others, including our group, to a description of the standard model, for which the Higgs field appears on the same footing as the intermediate vector bosons (W, Z and photon), and the action is described in purely spectral terms.
There is a quick, very heuristic, way to see that points, seen as locations in which particles can stay in point mechanics, are untenable: the Heisenberg microscope. In order to measure the location of a particle we must have it interact with some form of radiation, and the precision of the position will be limited by the wavelength of this. A short wavelength means however high frequency and momentum, and the interaction will alter the state of motion. The more precise the location, the more uncertain the momentum due to the exchange. Still, if we ignore momentum, there is no limit in quantum mechanics in the precision of localization. Things change if we introduce gravitation. To measure with precision a small location it is necessary to concentrate a great quantity of energy in a small volume, but if too much energy, or mass, it is the same in relativity, is concentrated in a small volume, the horizon of a (micro) black hole is created, this preventing from measuring position with arbitrary precision. Here the scale is not the quantum of action, but the cube of Planck’s length, the quantity obtained combining c, h and the gravitational constant G.
Heisenberg’s microscope is a heuristic way to introduce quantum mechanics, a well developed theory, which evolved into quantum field theory. The above considerations lead to the introduction of a quantum spacetime, which should be described by quantum gravity. A theory we do not yet have! Nevertheless, it is likely that quantum gravity will necessitate a quantum spacetime, some sort of noncommutative geometry. Of particular interest are the symmetries of these noncommutative geometries, a novel form of groups and Lie algebras, called quantum groups and Hopf algebras.
Our group is actively studying various forms of spacetimes, the field theories defined on them, and their symmetries, using the most modern mathematical tools applied to physical problems.
There are various angles to see quantum spacetime, even within noncommutative geometry. One is quantum field theory on noncommutative space, where fields are multiplied with a noncommutative star product. Several products have been studied, also in relation to deformed symmetries, like the kappa-Minkowski spacetime. Superspaces are another kind of novel spacetime, anticommuting coordinates being a particular instance of noncommutative geometry well suited to describe supersymmetric theories. Gauge and field theories on noncommutative spaces of Lie algebra type represent the first non-trivial attempt to go beyond constant noncommutativity and are actively studied by our group.