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

TL;DR

This paper explores the high energy behavior of differential operators on vector bundles, revealing how classical dynamics influence eigenfunctions, especially contrasting scalar and spinor cases like the Dirac operator.

Contribution

It provides a geometric framework for understanding high energy limits of differential operators on vector bundles, emphasizing the role of classical frame flow in the non-scalar case.

Findings

Classical behavior emerges for scalar Laplace eigenfunctions at high energies.

Classical frame flow determines eigensection behavior for operators with spin.

The review clarifies the geometric description of high energy limits for these operators.

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.