  We introduce a model of scalar field dark energy, Cuscuton, which can be
realized as the incompressible (or infinite speed of sound) limit of a scalar
field theory with a non-canonical kinetic term (or k-essence). Even though
perturbations of Cuscuton propagate superluminally, we show that they have a
locally degenerate phase space volume (or zero entropy), implying that they
cannot carry any microscopic information, and thus the theory is causal. Even
coupling to ordinary scalar fields cannot lead to superluminal signal
propagation. Furthermore, we show that the family of constant field
hypersurfaces are the family of Constant Mean Curvature (CMC) hypersurfaces,
which are the analogs of soap films (or soap bubbles) in a Euclidian space.
This enables us to find the most general solution in 1+1 dimensions, whose
properties motivate conjectures for global degeneracy of the phase space in
higher dimensions. Finally, we show that the Cuscuton action can model the
continuum limit of the evolution of a field with discrete degrees of freedom
and argue why it is protected against quantum corrections at low energies.
While this paper mainly focuses on interesting features of Cuscuton in a
Minkowski spacetime, a companion paper (astro-ph/0702002) examines cosmology
with Cuscuton dark energy.
