Geometry and Perturbative Sensitivity of non-Smooth Caustics of the Helmholtz Equation

Abstract

The geometry of non-smooth A n>2 caustics in solutions of the Helmholtz equation is analyzed using a Fock-Schwinger proper-time formulation. In this description, A 3 or cusp caustics are intimately related to poles of a quantity called the einbein action in the complex proper-time, or einbein, plane. The residues of the poles vanish on spatial curves known as ghost sources, to which cusps are bound. The positions of cusps along the ghost sources is related to the value of the poles. A similar map is proposed to relate essential singularities of the einbein action to higher order caustics. The singularities are shown to originate from degenerations of a certain Dirichlet problem as the einbein is varied. It follows that the singularities of the einbein action, along with the associated aspects of caustic geometry, are invariant with respect to large classes of perturbations of the index of refraction.

Lapse singularities, caustics and entanglement

“…Singularities of S can also exist at finite non-zero N . These were originally conjectured to be related to the presence of non-smooth caustics in [15,16]. The reason for the relation will be given in the subsequent section.…”

“…The map relating N and λ has been partially described in [16], and is singular along the curves q 1 = 0 and ζ 2 = 0 which intersect at the cusp. There are multiple cusp caustics associated with the lapse action S(N ) but only one with a potential function P(λ), so no map can be given globally.…”

“…The relation between the finite N essential singularities of exp(iS) and caustics will be described in section 3. This relation was originally conjectured to exist and discussed in a different context in [15,16], wherein S was referred to as the "einbein action".…”