Causality and momentum conservation from relative locality

Theories involving curved momentum space, which recently became a topic of interest in the quantum-gravity literature, can, in general, violate many apparently robust aspects of our current description of the laws of physics, including relativistic invariance, locality, causality, and global momentum conservation. Here, we explore some aspects of the pathologies arising in generic theories involving curved momentum space for what concerns causality and momentum conservation. However, we also report results suggesting that when momentum space is maximally symmetric, and the theory is formulated relativistically, most notably including translational invariance with the associated relativity of spacetime locality, momentum is globally conserved and there is no violation of causality.