Aspects of Non-Relativistic Quantum Field Theories

Non-relativistic quantum field theory is a framework that describes systems where the velocities are much smaller than the speed of light. A large class of those obey Schr\"{o}dinger invariance, which is the equivalent of the conformal symmetry in the relativistic world. In this review, we pedagogically introduce the main theoretical tools used to study non-relativistic physics: null reduction and $c \rightarrow \infty$ limits, where $c$ is the speed of light. We present a historical overview of non-relativistic wave equations, Jackiw-Pi vortices, the Aharonov-Bohm scattering, and the trace anomaly for a Schr\"{o}dinger scalar. We then review modern developments, including fermions at unitarity, the quantum Hall effect, off-shell actions, and a systematic classification of the trace anomaly. The last part of this review is dedicated to current research topics. We define non-relativistic supersymmetry and a corresponding superspace to covariantly deal with quantum corrections. Finally, we define the Spin Matrix Theory limit of the AdS/CFT correspondence, which is a non-relativistic sector of the duality obtained via a decoupling limit, where a precise matching of the two sides can be achieved.