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Amplitudes, Observables, and Classical Scattering

David A. Kosower, Ben Maybee, Donal O'Connell

Abstract

We present a formalism for computing classically measurable quantities directly from on-shell quantum scattering amplitudes. We discuss the ingredients needed for obtaining the classical result, and show how to set up the calculation to derive the result efficiently. We do this without specializing to a specific theory. We study in detail two examples in electrodynamics: the momentum transfer in spinless scattering to next-to-leading order, and the momentum radiated to leading order.

Amplitudes, Observables, and Classical Scattering

Abstract

We present a formalism for computing classically measurable quantities directly from on-shell quantum scattering amplitudes. We discuss the ingredients needed for obtaining the classical result, and show how to set up the calculation to derive the result efficiently. We do this without specializing to a specific theory. We study in detail two examples in electrodynamics: the momentum transfer in spinless scattering to next-to-leading order, and the momentum radiated to leading order.

Paper Structure

This paper contains 38 sections, 236 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Restoring $\hbar$
  3. Impulse and radiated momentum in quantum field theory
  4. The incoming state
  5. The impulse on a particle
  6. Impulse in terms of amplitudes
  7. The momentum radiated during a collision
  8. Conservation of momentum
  9. Classical Point Particles
  10. Wavefunctions
  11. An example wavefunction
  12. Classical impulse
  13. Classical radiation
  14. Conservation of momentum
  15. Examples
  16. ...and 23 more sections

Figures (1)

  • Figure 1: The amplitude $\mathcal{A}^{(0)}(p_1+w_1\,,p_2+w_2\rightarrow p_1\,,p_2\,,k)$ appearing in the radiation kernel at leading order.