Classical Observables in the Limits of Quantum Field Theory for 3-Body Dynamics
Research Thesis, Master of Science in Physics at Northwestern University
This page contains the abstract and PDF of the notes related to my MS Thesis, which was born from research I conducted while completing an MS in Physics at Northwestern Unviersity. Below the PDF, you will find links to view the Wolfram Mathematica code files, notes, and appendices associated with the project.
Abstract:
Motivated by recent work done in numerical relativity studying the gravitational wave phase shift originating from 3-body black hole interactions, this projects investigates how observables from a quantum field theory description of gravity can provide potentially unique information for astrophysical simulations of 3-body black hole scattering in stellar clusters. In the course of this investigation we present a direct calculation of the 6-point N=0 supergravity scattering amplitude for three massive scalars using unitarity methods and the double copy formalism. This amplitude is used to identify the Eikonal Phase, a generating function for classical observables, as a demonstration of the types of quantities this method can be used to identify. The projects ends with a discussion of future directions, and appendices are included with case-study calculations done in scalar Quantum Electrodynamics.
Code, Appendices, and Associated Notes
In both working on my independent study in the spring of 2025 and conducting research towards my thesis, I explored several case studies, the notes from which were worked
into a series of appendices which are outlined below. Please note that the references section is attached only to the primary thesis document.
A large part of the project involved developing my skills with Wolfram Mathematica as a tool for heavy symbolic calculation. In calculating the 6-point tree level amplitude by unirarity methods, I used Mathematica to construct an ansatz with approximately 400 free parameters before systematically solving for each via symmetry arguments. The code I wrote for this project was very much a tool fit for its purpose, in the sense that I never intended to share it publicly and it definitely wasn’t written with beauty in mind. That said, it tells an important part of the story and I encourage anyone interested to download it and play around.
These notes present a case study where a lightweight scalar particle interacts electromagnetically with a heavy, stationary source; a construction analogous to a hydrogenic atom. Solving for the equations of motion for the lightweight particle beginning from the Lagrangian for scalar QED, it is shown that the Schrödinger equation emerges naturally as the equations of motion when the nonrelativistic limit is applied.
Working in the same framework of a lightweight scalar particle interacting with a non-recoiling source, these notes calculate the impulse from the interaction. The rules for writing scattering amplitude expressions from Feynman diagrams are reviewed in the context of scalar QED, and the amplitude is used to calculate the impulse.
These notes expand on the Feynman diagram and amplitudes discussion of the previous appendix, this time introducing the idea of multiple photon exchanges. The one-loop amplitude is calculated, and it’s shown how the Lippman-Schwinger equation arises from summing up to the limit of infinite exchanges. The Coulomb potential is also calculated directly from the tree-level amplitude.
Working at tree level in scalar QED within the same framework as previous appendices, these notes calculate the leading order radiation amplitude.
Bound states are identified as poles in the T-matrix, and the Lippman-Schwinger equation is further motivated as encoding Schrödinger dynamics in integral form.
These notes investigate matrix elements associated with transitions between hydrogenic bound states mediated by the electromagnetic 4-current in scalar QED. This is done to study bound state dynamics which may be applicable to black hole binaries in an effective field theory, and as a case study to demonstrate how an effective field theory is constructed and matched to a full UV theory.
This section transitions from plane-wave descriptions of radiation to a basis that explicitly diagonalizes total angular momentum and helicity, in order to make angular momentum conservation manifest. This is done by establishing the covariant partial-wave formalism by working through how angular momentum and helicity interact. The spin-weighted spherical harmonic functions are introduced and explored.
These notes schematically present effective field theories for bound state dynamics in scalar QED which are directly analogous to binary black hole interactions. The radiative capture process is explored in detail.
This section presents further schematics for EFTs, this time for three body dynamics. The binary breakup and rearrangement processes are explored explicitly, and a discussion of matching to the full UV theory is presented.
This short appendix explores the effect of neglecting to consider the recoil kinematics associated with a system emitting a massless particle, such as a photon or graviton.
