Metaplectic sheets and caustic traversals in the Weyl representation

Abstract

Abstract. The quantum Hamiltonian generates in time a family of evolution operators. Continuity of this family holds within any choice of representation and, in particular, for the Weyl propagator, even though its simplest semiclassical approximation may develop caustic singularities. The phase jumps of the Weyl propagator across caustics have not been previously determined.The semiclassical appproximation relies on individual classical trajectories together with their neighbouring tangent map. Based on the latter, one defines a continuous family of unitary tangent propagators, with an exact Weyl representation that is close to the full semiclassical approximation in an appropriate neighbourhood. The phase increment of the semiclassical Weyl propagator, as a caustic is crossed, is derived from the facts that the corresponding family of tangent operators belong to the metaplectic group and that the products of the tangent propagators are obtained from Gaussian integrals. The Weyl representation of the metaplectic group is here presented, with the correct phases determined within an intrinsic ambiguity for the overall sign. The elements that fully determine the phase increment across a particular caustic are then analysed.

Semiclassical evolution in phase space for a softly chaotic system

“…Since the monodromy matrix is never singular, it is trivial to see that the R t pre-factor in the HK propagator is never zero. This does not mean, as is sometimes stated in literature, that the propagator is free from caustics: All IVRs are "free from caustics" in the sense that they do not diverge along them, but acquired phases due to the change of metaplectic sheets are still present [25,28].…”

Distinguishing quantum features in classical propagation

“…1. We have here omitted any extra phase due to the Maslov index [17][18][19], since its contribution can be neglected within a narrow region of the origin in the dual chord phase space and the short evolution times contemplated here.…”

Complexified phase spaces, initial value representations, and the accuracy of semiclassical propagation

“…The intertwining between generating functions across caustics, however, is responsible for an accumulated phase known as the Maslov index, µ. Written as in (34), this index is nothing but the number of caustics encountered alongside the trajectory linking (q, p) and (q , p ) [21,36,45,46,47,48].…”

“…An important characteristic of the IVR in (48) is that it cannot be immediately used to calculate wave functions, and one is forced to chose between limiting its use to numbers, e.g. ψ| Û (t)|φ , or to develop some clever strategy to substitute the Dirac delta by something smoother [4,17].…”