Microlocal analysis in quantum field theory on curved spacetime and related topics

Abstract

Presented is a discussion of quantum field theory on curved spacetime and of microlocal analysis, with an emphasis on the way that these two areas connected for me personally through a specific problem, namely that of resolving Kay's singularity conjecture for two point functions of a linear scalar field on a globally hyperbolic spacetime. A particular case of this conjecture is presented, namely the translation invariant case on flat Minkowski spacetime, which does not require microlocal analysis. Next, the results of Duistermaat and Hoermander concerning distinguished parametrices of the Klein Gordon equation on a curved spacetime are described, since they lead to the notion of a wave front set (or microlocal) spectral condition, which could be viewed as a remnant of the spectral condition on flat spacetime. This condition on the wavefront set of the two point function has been employed by Brunetti, Koehler and Fredenhagen to develop a method of renormalization on a general curved spacetime, which has been developed further by Hollands and Wald. Other QFT-related topics to which microlocal methods may apply are: Lorentz symmetry breaking models and many body QM models (e.g., the free electron gas in a metal). In the case of vector or spinor models, the polarization set may be used to refine information about the singularities. Similarly, the principal symbol of the two point function, viewed as a Fourier integral operator, is a constant times a canonical half density on the natural Lagrangian submanifold associated with the Klein-Gordon operator, suggesting a tangent space Lorentz invariance property for the free model.