Physics on and near Caustics

  • C. DeWitt-Morette
  • P. Cartier
Part of the NATO ASI Series book series (NSSB, volume 361)


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:
  1. i)

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

  2. 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:
    1. 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.

    2. 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.



Functional Integration Conjugate Point Probability Amplitude Airy Function Jacobi Operator 
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References Added in Proof

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • C. DeWitt-Morette
    • 1
  • P. Cartier
    • 2
  1. 1.Department of Physics and Center for RelativityUniversity of TexasAustinUSA
  2. 2.Ecole Normale SupérieureParisFrance

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