Frozen Motion: Why Single Carrollian Scalars Cannot Propagate

TL;DR

This paper shows that single Carrollian scalar fields minimally coupled to the Carrollian geometry cannot propagate due to symmetry constraints, implying the need for more complex theories for dynamics.

Contribution

It demonstrates that supertranslation invariance in Carrollian scalar theories prevents field propagation, highlighting the necessity of moving beyond minimal single-field models.

Findings

Energy density must be static under supertranslation invariance

Momentum density vanishes, preventing propagation

Propagating theories require non-minimal or multi-field approaches

Abstract

We investigate a class of first-order scalar field theories minimally coupled to a Carrollian connection that are defined intrinsically on the Carrollian plane, i.e., the theories are not defined via limits of Lorentzian theories. The theories built are invariant under the extended Carrollian transformations which include supertranslations. The symmetry allows for a large class of Lagrangians, independence of spacetime coordinates is all that is required. However, invariance under supertranslations (which include boosts as linear supertranslations) forces the energy density to be static and the momentum density to vanish -- this precludes on-shell propagation of fields. Thus, to have propagating theories, one must move beyond single field theories that are minimally coupled to the geometry.