Interacting conformal Carrollian theories: Cues from electrodynamics

We construct the free Lagrangian of the magnetic sector of Carrollian electrodynamics. The construction relies on Helmholtz integrability condition for differential equations in a self-consistent algorithm, working hand in hand with imposing invariance under infinite dimensional Conformal Carroll algebra. It requires inclusion of new fields in the dynamics and the system is free of gauge redundancies. We next add interaction (quartic) terms to the free Lagrangian, strictly constrained by conformal invariance and Carrollian symmetry. The dynamical realization of the non-semi-simple infinite dimensional symmetry algebra at the level of charge algebra is exact and free from central terms.

Figure 1

In the panel (a) above, the points $A$ and $C$ are causally connected, whereas the interval $A\text{−}B$ is acausal, in Minkowski space. To get Carroll space-time as the $c\to 0$ limit, one should collapse the light cone to a single line, as in panel (b). In Carroll space-time, none of the points $A$, $B$, and $C$ are causally connected, unless the events take place exactly at the same spatial points.

Figure 2

The process of taking Carrollian (ultrarelativistic) limit and applying variational derivative on action functional does not commute.