Field Theories on Quantum Space-Times: Towards the Phenomenology of Quantum Gravity

Abstract: Noncommutative geometry is a mathematical framework that expresses the structure of space-time in terms of operator algebras. By using the tools of quantum mechanics to describe the geometry, noncommutative space-times are expected to give rise to quantum gravity effects, at least in some regime. This manuscript focuses on the physical aspects of these so-called quantum space-times, in particular through the formalism of field and gauge theories. Scalar field theories are shown to possibly trigger mixed divergences in the infra-red and ultra-violet for the 2-point function at one loop. This phenomenon is generically called UV/IR mixing and stems from a diverging behaviour of the propagator. The analysis of such divergences differs from the commutative case because the momentum space is now also noncommutative. From another perspective, a gauge theory on $\kappa$-Minkowski, a quantum deformation of the Minkowski space-time, is derived. A first perturbative computation is shown to break the gauge invariance, a pathological behaviour common to other quantum space-times. A causality toy model is also developed on $\kappa$-Minkowski, in which an analogue of the speed-of-light limit emerges. The phenomenology of quantum gravity arising from quantum space-times is discussed, together with the actual constraints it imposes. Finally, a toy model for noncommutative gravity is tackled, using the former $\kappa$-Minkowski space-time to describe the tangent space. It necessitates the notion of noncommutative partition of unity specifically defined there.