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Physics on and near Caustics

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

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:
  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.

       
     

Keywords

Functional Integration Conjugate Point Probability Amplitude Airy Function Jacobi Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    C. DeWitt-Morette, G. Low, L. S. Schulman, A. Y. Shiekh: “Wedges I,” Foundations of Physics 16, 311–349 (1986).MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    The properties of Jacobi fields in configuration space and phase space, and the corresponding WKB approximations are scattered in several papers of Cécile DeWitt-Morette, John La Chapelle, Maurice Mizrahi, Bruce Nelson, Benny Sheeks, Alice Young, and Tian-Rong Zhang. For a summary of results obtained prior to 1984, see references [3, 4]. For more recent results, see reference [5].Google Scholar
  3. [3]
    Cécile DeWitt-Morette and Tian-Rong Zhang: “Path integrals and conservation laws,” Phys. Rev D 28, 2503–2516 (1983).MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    Cécile DeWitt-Morette: “Feynman path integrals, from the prodistribution definition to the calculation of glory scattering,” Acta Physica Austriaca Suppl. XXVI, 101–170 (1984).MathSciNetGoogle Scholar
  5. [5]
    John La Chapelle: “Functional Integration on Symplectic Manifolds,” (Ph. D. Dissertation, The University of Texas at Austin, May 1995).Google Scholar
  6. [6]
    Cécile DeWitt-Morette and Tian-Rong Zhang: “Feynman-Kac formula in phase space with applications to coherent state transitions,” Phys. Rev. D 28, 2517–2525 (1983).MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    Cécile DeWitt-Morette, Bruce Nelson, and Tian-Rong Zhang: “Caustic problems in quantum mechanics with applications to scattering theory,” Phys. Rev. D 28, 2526–2546 (1983).MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    Pierre Cartier and Cécile DeWitt-Morette: “A new perspective on functional integration,” J. Math. Phys. 36, 2237–2312 (1995). For strict WKB approximations, see the calculation leading to (III.55).MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. [9]
    We use h rather than ℏ = h/2π, in order that expressions such as Gaussian, Fourier transforms, etc... on ℝd do not depend on d.Google Scholar
  10. [10]
    C. DeWitt-Morette, A. Maheshwari, and B. Nelson: “Path integration in non-relativistic quantum mechanics,” Phys. Rep. 50, 266–372 (1979). This article includes a summary of earlier articles.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Cécile DeWitt-Morette and Bruce L. Nelson: “Glories—and other degenerate points of the action,” Phys. Rev. D 29, 1663–1668 (1984).MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    Cécile DeWitt-Morette: “Catastrophes in Lagrangian systems” and C. DeWitt-Morette and P. Tshumi: “Catastrophes in Lagrangian systems. An example,” in Long Time Prediction in Dynamics Eds. V. Szebehely and B. D. Tapley (D. Reidel Pub. Co. 1976) pp. 57-69. Also Y. Choquet-Bruhat and C. DeWitt-Morette Analysis, Manifolds and Physics Vol. I pp. 105-109, and pp. 277-281 (North Holland 1982–1996).Google Scholar
  13. [13]
    R. A. Matzner, C. DeWitt-Morette, B. Nelson, and T.-R. Zhang: “Glory scattering by black holes” Phys. Rev. D 31, 1869–1878 (1985).ADSCrossRefGoogle Scholar
  14. [14]
    Tian-Rong Zhang and Cécile DeWitt-Morette: “WKB cross-section for polarized glories of massless waves in curved space-times,” Phys. Rev. Letters 52, 2313–2316 (1984).MathSciNetADSCrossRefGoogle Scholar

References Added in Proof

  1. Ph. Choquard and F. Steiner: “The story of Van Vleck’s and Morette-Van Hove’s determinants” Helv. Phys. Acta 69, 636–654 (1969) analyzes carefully the contributions of Cécile Morette (“On the definition and approximation of Feynman’s path integral,” Phys. Rev. 81, 848-852 (1951)), Leon Van Hove (“annexe” to his doctorate thesis) and Wolfang Pauli (unpublished notes) to the strict WKB approximation of Feynman’s path integral. The appearance of singular determinants (“WKB breaks down”) was first noted by Ph. Choquard in Helv. Phys. Acta 28, 89-157 (1955) on pages 114-119.MathSciNetGoogle Scholar

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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