The transition Classical → Quantum is very simple to formulate in terms of the Feynman integral which affects each classical field configuration with the probability amplitude eiS/ℏ. While this prescription works remarkably well for the quantization of the classical fields involved in the standard model provided one uses the technique of renormalization,this latter perturbative technique fails dramatically when one tries to deal with the gravitational field gµν.
In many ways this result is not surprising. Indeed many of the basic notions of the traditional formalism of Quantum Field Theory (QFT), such as particles, scattering matrices, etc... heavily rely on the flat geometry of Minkowski space and the related Poincaré symmetry group. Treating the quantization of the gµν in the same way would -if successful- produce a quantum field theory of the gµν on Minkowski space: a strange result indeed when viewed from the geometric standpoint! The technical reason for the notorious difficulty of quantizing the gµν in the traditional perturbative way is the clash with either renormalizability or unitarity.
In some sense this clash contains a serious warning, namely that one should not try to rush but rather meditate the lessons of both general relativity and QFT before even starting to compute something. In this very short essay we shall describe a “spectral” point of view on geometry which allows to start taking into account the lessons from both sides. We shall first do that for renormalization and explain in rough outline the content of our recent collaborations with Dirk Kreimer and Matilde Marcolli. As far as general relativity is concerned, since the functional integral cannot be treated in the traditional perturbative manner, it relies heavily as a “sum over geometries” on the chosen paradigm of geometric space. This will give us the occasion to discuss, in the light of noncommutative geometry, the issue of “observables” in gravity and our joint work with Ali Chamseddine on the spectral action, with a first attempt to write down a functional integral on the space of noncommutative geometries.
Alain Connes, 2006