  We study renormalizable Lorentz invariant stable quantum field theories in
two space-time dimensions with instantaneous causal structure (causal ordering
induced by the light "cone" time ordering). These models provide a candidate
UV-completion of the two-dimensional ghost condensate. They exhibit a peculiar
UV/IR mixing - energies of all excitations become arbitrarily small at high
spatial momenta. We discuss several phenomena associated with this mixing.
These include the impossibility to reach a thermal equilibrium and
metastability of all excitations towards decay into short wavelength modes
resulting in an indefinite turbulent cascade. In spite of the UV/IR mixing in
many cases the UV physics can still be decoupled from low energy phenomena.
However, a patient observer in the Lineland is able to produce arbitrarily
heavy particles simply by waiting for a long enough time.
