Polarisation sets of Green operators for normally hyperbolic equations

Abstract

The polarisation set of a vector-valued distribution generalises the wavefront set and captures fibre-directional information about its singularities in addition to their phase space description. Motivated by problems in quantum field theory on curved spacetimes, we consider normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes, and compute the polarisation sets of the kernel distributions for their advanced and retarded Green operators and the difference thereof. This permits the computation of related polarisation and wavefront sets for operators whose solution theory is related to the normally hyperbolic case. As a particular example, we consider the Proca equation that describes massive relativistic spin-1 particles, identifying and closing a gap in a recent paper on that subject.