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

Path-integral formulation of scattering theory

William B. Campbell, P. Finkler, C. E. Jones, Michael N. Misheloff

Published Oct 15, 1975• 29 citations

Abstract

A new formulation of nonrelativistic scattering theory is developed which expresses the $S$ matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the $S$ 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 $T$ 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.

DOI: 10.1103/physrevd.12.2363

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A new formulation of nonrelativistic scattering theory is developed which expresses the

S

matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the

S

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

T

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

T

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.

In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, in the second representation this trajectory is complex and evolves according to an effective, time dependent potential. Using a cumulant expansion, the first correction to the variational approximation is also evaluated. The high energy behavior of the approximation is investigated, and is shown to contain exactly the leading and next-to-leading order of the eikonal expansion, and parts of higher terms. Our results are then numerically tested in two particular situations where others approximations turned out to be unsatisfactory. Substantial improvements are found.

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.

Abstract.: Using a recent path integral representation for the T -matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables --over which one has to integrate functionally-- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading). We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail