On the subtelties of nonrelativistic reduction and applications

TL;DR

This paper explores the subtle issues in nonrelativistic reduction of scalar field theories, emphasizing the mapping of spacetime symmetries and revealing conformal and diffeomorphism symmetries in nonrelativistic contexts.

Contribution

It introduces a novel perspective by focusing on the mapping of spacetime generators during nonrelativistic reduction and demonstrates applications to conformal symmetry in hydrodynamics and NR diffeomorphisms.

Findings

Embedding of Schrödinger conformal generators via null reduction

Revelation of conformal symmetry in hydrodynamics

Derivation of NR diffeomorphism symmetry from relativistic theories

Abstract

Various subtleties and problems associated with nonrelativistic (NR) reduction of a scalar field theory to the Schroedinger theory are discussed. Contrary to the usual approaches that discuss the mapping among the equations of motion or the actions, we highlight the mapping among the space time generators. Using a null reduction we show the embedding of the conformal generators of the Schroedinger theory to that of a complex scalar theory in one higher dimension. As applications we reveal the conformal symmetry in hydrodynamics and the obtention of NR diffeomorphism symmetry from the relativistic one. A geometrical connection based on Horava-Lifshitz and Newton-Cartan spacetime is discussed.