  We provide a class of QFTs which exhibit dissipation above a threshold
energy, thereby breaking Lorentz invariance. Unitarity is preserved by coupling
the fields to additional degrees of freedom (heavy fields) which introduce the
rest frame. Using the Equivalence Principle, we define these theories in
arbitrary curved spacetime. We then confront the trans-Planckian question of
inflationary cosmology. When dissipation increases with the energy, the quantum
field describing adiabatic perturbations is completely damped at the onset of
inflation. However it still exists as a composite operator made with the
additional fields. And when these are in their ground state, the standard power
spectrum obtains if the threshold energy is much larger that the Hubble
parameter. In fact, as the energy redshifts below the threshold, the composite
operator behaves as if it were a free field endowed with standard vacuum
fluctuations. The relationship between our models and the Brane World scenarios
studied by Libanov and Rubakov displaying similar effects is discussed. The
signatures of dissipation will be studied in a forthcoming paper.
