Geometry of the high energy limit of differential operators on vector bundles

Abstract

At high energies relativistic quantum systems describing scalar particles behave classically. This observation plays an important role in the investigation of eigenfunctions of the Laplace operator on manifolds for large energies and allows to establish relations to the dynamics of the corresponding classical system. Relativistic quantum systems describing particles with spin such as the Dirac equation do not behave classically at high energies. Nonetheless, the dynamical properties of the classical frame flow determine the behavior of eigensections of the corresponding operator for large energies. We review what a high energy limit is and how it can be described for geometric operators.