Scattering and propagation phenomena near spacetime horizons – Horizons

The main topic of this project is the application of techniques of asymptotic and global analysis to Quantum Field Theory on curved spacetimes and General Relativity, with the goal of studying quantum phenomena near and beyond spacetime horizons.

In the last two decades, the implementation of methods from hyperbolic partial differential equations and microlocal analysis has resulted in a good understanding of local aspects of Quantum Field Theory on curved spacetimes. However, the most striking physical phenomena are believed to occur in setups where global and asymptotic aspects crucially enter, geometrically often indicated by the presence of a spacetime horizon.

Thus, the challenge is to give a mathematical description of quantum fields in settings involving boundaries or trapping of null geodesics. The former include asymptotically Anti-de Sitter spacetimes, where the issue is the time-like boundary, known to reflect singularities and playing a fundamental role in the AdS/CFT conjecture. Particularly important examples with trapping are the Kerr and Kerr de Sitter black holes, which serve as the background for studying the hypothetical final quantum state resulting from gravitational collapse. These problems are directly connected to the study of differential equations on the corresponding Lorentzian manifolds.

Presently, the analysis of the wave, Dirac or Einstein equations is subject to spectacular developments related to the overcoming of difficulties induced by trapping, superradiance, boundaries and constraints. This active field of research, directly relevant for classical theories, provides essential ingredients such as propagation of singularities theorems, asymptotics of solutions, existence of scattering or Poisson operators, well-posedness of a forward/backward/Cauchy or inverse scattering problem and the construction of parametrices. The crucial difficulty in the study of quantum fields is that all those techniques have to be used simultaneously, yet the interplay between them is presently understood only on a non-exhaustive class of examples. This causes significant obstructions in the attempts to describe or even define Quantum Field Theories on physically interesting non-globally hyperbolic spacetimes or in the presence of boundaries.

Another severe difficulty plaguing mathematical Quantum Field Theory since its very beginnings is that interacting theories are formally represented by nonlinear differential equations, but the nonlinear term needs in fact to be renormalised, resulting in an equation whose solutions can typically be obtained merely as formal power series. Comparison with Euclidean models indicate however ways of giving intrinsic meaning to the series and suggests that significant advances could be made by inventing hyperbolic analogues of devices such as complex powers or renormalized traces of elliptic pseudodifferential operators. This is especially important in applications on curved spacetimes, where one seeks satisfactory ways of coupling quantum fields to a dynamical geometrical background.

The core idea behind the project is the observation that recent advances provide the pieces that are needed to fill those gaps, thus giving the perspective of extending mathematical QFT beyond the well-studied globally hyperbolic case and analyse how quantum fields affect the spacetime on which they propagate. The key technical ingredients to be developped consist of a mixture of asymptotic and global analysis with microlocal methods, including robust propagation estimates. Via a novel use of asymptotic or boundary conditions giving rise to Lorentzian Fredholm problems, the essential feature arising from this combination is the possibility of analysing hyperbolic equations in parallel to well-known elliptic problems and their index theory.

Ultimately, this strategy will allow to set up QFTs on new curved backgrounds and open the possibility of studying their extension across spacetime horizons.