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The ABCs of Amplitudes, Bogoliubov and Crossing

Rafael Aoude, Asaad Elkhidir, Anton Ilderton, Donal O'Connell, Karthik Rajeev

Abstract

It is now common to describe classical backgrounds involving dynamical black holes with the production of gravitational radiation using the methods of scattering amplitudes. In that light, we revisit the standard formulation of quantum field theory on a background. We discuss the interpretation of Bogoliubov coefficients as generalised amplitudes, and explain how crossing, analyticity, and causality relate the relevant set of amplitudes. When the background is itself a coherent state, we map these statements onto standard results in flat-space quantum field theory.

The ABCs of Amplitudes, Bogoliubov and Crossing

Abstract

It is now common to describe classical backgrounds involving dynamical black holes with the production of gravitational radiation using the methods of scattering amplitudes. In that light, we revisit the standard formulation of quantum field theory on a background. We discuss the interpretation of Bogoliubov coefficients as generalised amplitudes, and explain how crossing, analyticity, and causality relate the relevant set of amplitudes. When the background is itself a coherent state, we map these statements onto standard results in flat-space quantum field theory.
Paper Structure (20 sections, 131 equations, 3 figures)

This paper contains 20 sections, 131 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Background QFT and Bogoliubov coefficients
  4. Primitive processes
  5. The Bogoliubov coefficients
  6. Amplitudes from Bogoliubov coefficients
  7. Vacuum pair creation
  8. Bogoliubov coefficients from amplitudes
  9. Crossing
  10. Response functions and generalised amplitudes
  11. Relating $\alpha$ and $\beta$
  12. Crossing in amplitudes: entire functions
  13. Causality: retarded versus Feynman
  14. Bogoliubov coefficients and amplitudes in $J$ theory
  15. A simple example
  16. ...and 5 more sections

Figures (3)

  • Figure 1: The $\mathcal{S}$ matrix maps the in Fock space to the out Fock space.
  • Figure 2: Crossing from one-to-one to zero-to-two amplitudes in any background-field theory with suitably compact fields, as outlined in \ref{['eq:EntireCrossingEquation']}.
  • Figure 3: Schematic representation of $\tilde{\mathcal{A}}(k,q_1,q_2,p)$ in the four $q_i^0$ quadrants, with time flowing from left to right. (Symmetrization of contributions, as in (\ref{['eq:OffShellAmplitude']}), is implicit.) The diagrams in the blue quadrants vanish when $p$ is taken to be incoming, while that in the green quadrant vanishes when $p$ is outgoing. The diagram in the shaded quadrant vanishes for both configurations.