  We study partially massless gravity theory(PM gravity theory) and suggest an
alternative way to add higher order interaction vertices to the theory. Rather
than introducing self interaction vertices of the gravitational fields to the
partially massless gravity action, we consider interactions with matter fields,
since it is well known that addition of the self interaction terms necessarily
breaks the $U(1)$ gauge symmetry that PM gravity theory enjoys. For the
coupling with matter fields, we consider two different types of interaction
vertices. The first one is given by an interaction Lagrangian density,
$\mathcal L_{int}\sim h_{\mu\nu}T^{\mu\nu}$, where $h_{\mu\nu}$ is the PM
gravity field and $T^{\mu\nu}$ is the stress-energy tensor of the matter
fields. To retain the $U(1)$ gauge symmetry, the matter fields also transform
accordingly and it turns out that the transform must be non-local in this case.
The second type interaction is obtained by employing a gauge covariant
derivative with the PM gravity field, where the PM gravity fields play a role
of a gauge connection canceling the phase shift of the matter fields. We also
study the actions and the equations of motion of the partially massless gravity
fields. As expected, it shows 4 unitary degrees of freedom. 2 of them are
traceless tensor modes and they are light like fields. The other 2 are
transverse vector modes and their dispersion relation changes as background
space time (de Sitter) evolutes. In the very early time, they are light like
but in the very late time, their velocity become a half of speed of light. The
vector mode dispersion relation shows momentum dependent behavior. In fact, the
higher(lower) frequency modes show the faster(slower) velocity. We suggest
their quantization, compute Hamiltonians to present their exited quanta and
construct their free propagators.
