Worldline Formalism, Eikonal Expansion and the Classical Limit of Scattering Amplitudes

By approximating a black hole to be a point particle in a large impact parameter, small angle scattering (called the eikonal limit), we can extract the classical information (the eikonal phase) efficiently from the quantum scattering amplitude. Worldline Quantum Field Theory (WQFT) is one such technique, which employs a hybrid version of first and second quantized field theory. WQFT constructs the relevant scattering amplitude by finding the first quantized amplitude for a massive particle interacting with a background gravitational (or electromagnetic) field, and then "gluing" two such first quantized amplitudes together by integrating out the background field using the second quantized field theory. This version turns out to be highly efficient in repackaging the standard Feynman diagrams into a new set of diagrams as an expansion in \hbar, allowing a diagrammatic extraction of classical information. In this talk, we show that this method is equivalent to the eikonal based method, and is more efficient. The efficiency comes from the ability to directly target the eikonal phase in the WQFT formalism, rather than the recursive approach in the eikonal method. Through a systematic counting of the \hbar's in the WQFT diagrams, we show that WQFT classifies theories with "good" classical limits in a clear and simple way. Using the operator approach to WQFT, we show how the amplitude can serve as a generating function for the eikonal phase.