Weyl quantization represents a natural map between operators and their classical symbols, frequently used in quantum mechanics and semiclassical analysis. In the presence of gravity, these concepts need to be extended to the more general setting of Lorentzian manifolds. This can be done by using the exponential map of the tangent bundle to represent translations in spacetime in a covariant way, and making use of the van Vleck determinant to define the balanced geodesic Weyl quantization for operators acting on scalar fields. Motivated by the study of physical fields (such as electromagnetic, Dirac, or linearized gravitational fields) propagating on curved spacetime, we aim to extend known results by introducing a covariant Weyl quantization for operators acting on sections of vector bundles. This will provide the necessary tools for the following applications that we plan to explore: the semiclassical analysis of waves propagating in curved spacetimes, chiral kinetic theory, and causal fermion systems.

Research Team: Lars Andersson (Yanqi Lake Beijing Inst.), Marius A. Oancea (U of Vienna) and Claudio F. Paganini (U of Regensburg)
Visiting Scientist: Gabriel Schmid (U of Genoa)