Geometrizing non-relativistic bilinear deformations

TL;DR

This paper introduces three fundamental solvable bilinear deformations for non-relativistic 2D QFTs, interprets them via Newton-Cartan geometry, and derives their classical and quantum properties.

Contribution

It defines and analyzes three new solvable bilinear deformations for non-relativistic QFTs, linking them to Newton-Cartan geometry and deriving their classical and quantum formulations.

Findings

All three deformations can be interpreted as coupling to Newton-Cartan geometry.

Derived closed-form classical Lagrangians for deformed Schrödinger models.

Extended the coordinate change interpretation to non-relativistic deformations.

Abstract

We define three fundamental solvable bilinear deformations for any massive non-relativistic 2d quantum field theory (QFT). They include the $\mathrm{T}\overline{\mathrm{T}}$ deformation and the recently introduced hard rod deformation. We show that all three deformations can be interpreted as coupling the non-relativistic QFT to a specific Newton-Cartan geometry, similar to the Jackiw-Teitelboim-like gravity in the relativistic case. Using the gravity formulations, we derive closed-form deformed classical Lagrangians of the Schr\"odinger model with a generic potential. We also extend the dynamical change of coordinate interpretation to the non-relativistic case for all three deformations. The dynamical coordinates are then used to derive the deformed classical Lagrangians and deformed quantum S-matrices.