Scattering Theory with Path Integrals

R. Rosenfelder

Published Aug 14, 2016• 7 citations

Abstract

Starting from well-known expressions for the T-matrix and its derivative in standard nonrelativistic potential scattering, I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.

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Scattering Theory with Path Integrals

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Published Aug 14, 2016• 7 citations

OpenAlex

Abstract

Related papers

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arXiv:1302.3419Journal of Mathematical PhysicsMar 1, 20143 citations

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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.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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsOct 15, 197529 citations

Exact Path-Integral Representations for the T-Matrix in Nonrelativistic Potential Scattering

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arXiv:1008.1718Few-Body SystemsOct 13, 20103 citations

Path Integrals and Scattering Theory in Nonrelativistic Stochastic Quantum Mechanics.

R. Gagnon

Jan 1, 19841 citation

A new path-integral representation of the T-matrix in potential scattering

Julien Carron, R. Rosenfelder

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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.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.

arXiv:0806.3217Physical Review AJan 6, 200916 citations

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.