Relativistic compatibility of the interacting $\kappa$-Poincar\'e model and implications for the relative locality framework

Understand

We investigate the relativistic properties under boost transformations of the $\kappa$-Poincar\'e model with multiple causally connected interactions, both at the level of its formulation in momentum space only and when it is endowed with a full phase space construction, provided by the relative locality framework.

• Previous studies focussing on the momentum space picture showed that in presence of just one interaction vertex the model is relativistic, provided that the boost parameter acting on each given particle receives a "backreaction" from the momenta of the other particles that participate in the interaction.
• Here we show that in presence of multiple causally-connected vertices the model is relativistic if the boost parameter acting on each given particle receives a backreaction from the total momentum of all the particles that are causally connected, even those that do not directly enter the vertex.
• The relative locality framework constructs spacetime by defining a set of dual coordinates to the momentum of each particle and interaction coordinates as Lagrange multipliers that enforce momentum conservation at interaction events.