  We set up a field theory model to describe the longitudinal low energy modes
in high density matter present in white dwarf stars. At the relevant scales,
ions -- the nuclei of oxygen, carbon and helium -- are treated as heavy
point-like spin-0 charged particles in an effective field theory approach,
while the electron dynamics is described by the Dirac Lagrangian at the
one-loop level. We show that there always exists a longitudinal gapless mode in
the system irrespective whether the ions are in a plasma, crystal, or quantum
liquid state. For certain values of the parameters, the gapless mode can be
interpreted as a zero sound mode and, for other values, as an ion acoustic
wave; we show that the zero sound and ion acoustic wave are complementary to
each other. We discuss possible physical consequences of these modes for
properties of white dwarfs.
