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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Path-integral formulation of scattering theory: Central potentials

Christopher C. Gerry, Vijay A. Singh

Published May 15, 1980• 11 citations
Publisher

Abstract

We consider central-potential scattering and determine a path-integral representation for the $S$ matrix in polar coordinates. This is obtained by transforming to polar coordinates a Cartesian form of the nonrelativistic $S$ matrix given by Campbell et al., and implementing an idea of Faddeev to obtain the appropriate asymptotic conditions. Our results are applied to scattering in an inverse-square potential to determine the correct phase shifts as well as the $S$ matrix.

DOI
10.1103/physrevd.21.2979

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Physical Review•Mar 15, 1960•65 citations
We consider central-potential scattering and determine a path-integral representation for the SSS matrix in polar coordinates. This is obtained by transforming to polar coordinates a Cartesian form of the nonrelativistic SSS matrix given by Campbell et al., and implementing an idea of Faddeev to obtain the appropriate asymptotic conditions. Our results are applied to scattering in an inverse-square potential to determine the correct phase shifts as well as the SSS matrix.
A new formulation of nonrelativistic scattering theory is developed which expresses the SSS matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the SSS matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving $\ensuremath{\delta}$ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the TTT matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.
Two path integral representations for the TTT matrix in nonrelativistic potential scattering are derived and proved to produce the complete Born series when expanded to all orders. They are obtained with the help of ``phantom'' degrees of freedom which take away explicit phases that diverge for asymptotic times. In addition, energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. These expressions may be useful for attempts to evaluate the path integral in real time and for alternative multiple scattering expansions. Standard eikonal-type high-energy approximations and systematic expansions immediately follow.
Abstract A central coordinate system is investigated within the framework of the Wigner-Dunkl deformation of quantum mechanics, treated using Feynman’s path integral approach. This approach involves performing path integration of a Lagrangian in polar coordinates, with an effective potential term that depends on both radial and angular coordinates. The success in separating variables within the two-dimensional Dunkl system facilitated obtaining exact expressions for the propagators associated with the free particle, inverse square potential, harmonic oscillator, and Coulomb potential. The energy eigenvalues and their corresponding wave functions are also determined.
This paper is a contribution to the discussion of the question: to what extent does the scattering matrix determine the Hamiltonian? The Hamiltonians considered are nonrelativistic, but in extension of previous studies, "nonlocal" potentials and many-body potentials are allowed. A large class of unitary transformations is found which produce Hamiltonians leading to the same SSS matrix. In the last section, it is shown that this equivalence is only a special consequence of the general axiomatic formulation of scattering in field theory.