Gauge interactions and the Galilean limit

TL;DR

The paper develops a systematic framework to derive gauge interactions from relativistic matter with global $U(1)$ symmetry using Deser's iterative Noether method, producing scalar QED and Dirac-QED actions. It then performs a careful non-relativistic reduction of both matter and gauge sectors, using electric and magnetic Galilean limits to obtain Galilean-invariant actions: Schrödinger fields coupled to Galilean electromagnetism, including higher-derivative generalizations and the Pauli–Schrödinger theory from QED via Foldy–Wouthuysen mapping. Key results include explicit NR actions with covariant derivatives $D_t$, $D_i$ (or their Galilean counterparts $\Lambda_i$, $\tilde{\Lambda}_i$) and modified Maxwell terms, along with relations between NR components such as $a^0$ vs $a_0$ and $a^i$ vs $a_i$ across limits. The work provides a new, consistent set of low-energy Galilean effective actions with clear physical interpretation and potential applications in non-relativistic QFT under Galilean symmetry.

Abstract

The gauge invariant minimal couplings for a class of relativistic free matter fields with global symmetry (related to usual charge conservation) have been obtained by incorporating an iterative Noether mechanism. Non-relativistic reduction of both matter and gauge sectors of the obtained interacting theory is then performed simultaneously which in turn yield a set of new effective actions which are invariant under the Galilean relativistic framework. To be precise, we show that one can obtain the Schrödinger field theory coupled to Galilean electromagnetism from the scalar quantum electrodynamics theory. Higher derivative corrections have also been included for which the non-relativistic reductions have been consistently carried out once again. On the other hand, the action for quantum electrodynamics leads to the Galilean Pauli-Schrödinger theory where the gauge field is non-relativistic or Galilean. Further, some novel relations are found (in both the electric and magnetic limits) between various components appearing in the Galilean avatar of electrodynamics.