Analytic Solutions for Rotationally Symmetric Quantum Field Theories

Quantum Field Theory (QFT) has enjoyed considerable success in describing systems with translational invariance. However, far less work has investigated QFTs with rotational symmetry where angular momentum is conserved. These QFTs describe systems with bound states or orbiting particles, and are therefore instrumental to understanding many phenomena, including quantized cyclotron motion. We propose a general method of analytically calculating such theories by transforming the path integral calculation into an eigenvalue problem via a Mellin Transform. We demonstrate the method's potential by completely solving a toy model consisting of a scalar field in a harmonic potential with rotational invariance in two planes, including analytic calculations of the partition function, background pressure, and propagators. The propagators are found to exhibit Gaussian decay, revealing novel confining behavior absent in translationally invariant theories. This framework lays groundwork for analytical studies of Landau levels, trapped quantum gases, and topological phases where rotational symmetry plays a central role.