  There has been a recent interest in considering Quantum Field Theories in
which Lorentz Invariance is broken in the UV sector. However attention has been
mostly limited to dispersive theories. In this work we provide the generalized
settings for studying dissipation. Unitarity is preserved by coupling the
original fields to additional (heavy) fields which induce the dissipation.
Starting with Lagrangians breaking LI in the UV, we learn that dissipative
effects unavoidably develop in the effective theory. We then covariantize these
Lagrangians in order to address the trans-Planckian question of inflation and
black hole physics. The peculiar properties of the additional fields inducing
dissipation is revealed by the covariantization. The links with the
phenomenological approach to Quantum Gravity and with some Brane World
scenarios are also discussed.
