Physics on and near Caustics

Abstract

Physics on caustics is obtained by studying physics near caustics. Physics on caustics is at the cross-roads of calculus of variation and functional integration. More precisely consider a space P M ^d of paths x: [t _a ,t _b ] → M ^d, on a d-dimensional manifold M ^d Consider two of its subspaces:

• i) The 2d-dimensional space U of critical points of an action functional S: P M ^d → ℝ $q \in U \Leftrightarrow \frac{{\delta S}}{{\delta q}}q = 0$].

• ii) The space P _¼, _v M ^d of paths satisfying d initial conditions (μ) and d final conditions (v).

  The nature of the intersection P _μ, _v M ^d ∩ U is the key to identifying and analyzing caustics. Caustics occur when the roots of S′ (q), = 0, for S restricted to P _μ, _v M ^d are not isolated points in U. We consider two cases:

  • i) The roots define a subspace of non zero dimension ℓ of U. For example, the classical system with action S is constrained by conservation laws.

  • ii) There is a multiple root of S′(q) = 0. For example the classical flow has an envelope.

    In both situations there is, at least, one nonzero Jacobi field along q with d initial vanishing boundary conditions (μ) and d final vanishing boundary conditions (v), i.e., q(t _a ) and q(t _b ) are conjugate points along q. In both cases we say “q( _b ) is on the caustics.”

    The strict WKB approximations of functional integrals for the system S “break down” on caustics, but the full semiclassical expansion, including contributions from S′(q) and S‴(q) (and possibly higher derivatives) yield the physical properties of the system S on and near the caustics. Caustics display classical physics as a limit of quantum mechanics.

    We work out glory scattering cross-sections because, there, both situations occur simultaneously: the system is constrained by conservation laws, and the classical flow is caustic forming. The cross-section is given in closed form in section 5.