Functional Integration pp 51-66 | Cite as

# Physics on and near Caustics

## Abstract

*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 2

*d*-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.

- i)

## Keywords

Functional Integration Conjugate Point Probability Amplitude Airy Function Jacobi Operator## Preview

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## References

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*Phys. Rep.***50**, 266–372 (1979). This article includes a summary of earlier articles.MathSciNetCrossRefGoogle Scholar - [11]Cécile DeWitt-Morette and Bruce L. Nelson: “Glories—and other degenerate points of the action,”
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## References Added in Proof

- Ph. Choquard and F. Steiner: “The story of Van Vleck’s and Morette-Van Hove’s determinants”
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