  We expect quantum field theories for matter to acquire intricate corrections
due to their coupling to quantum fluctuations of the gravitational field. This
can be precisely worked out in 3d quantum gravity: after integrating out
quantum gravity, matter fields are effectively described as noncommutative
quantum field theories, with quantum-deformed Lorentz symmetries. An open
question remains: Are such theories unitary or not? On the one hand, since
these are effective field theories obtained after integrating out high energy
degrees of freedom, we may expect the loss of unitarity. On the other hand, as
rigorously defined field theories built with Lorentz symmetries and standing on
their own, we naturally expect the conservation of unitarity. In an effort to
settle this issue, we explicitly check unitarity for a scalar field at one-loop
level in both Euclidean and Lorentzian space-time signatures. We find that
unitarity requires adding an extra-term to the propagator of the noncommutative
theory, corresponding to a massless mode and given by a representation with
vanishing Plancherel measure, thus usually ignored in spinfoam path integrals
for quantum gravity. This indicates that the inclusion of matter in spinfoam
models, and more generally in quantum gravity, might be more subtle than
previously thought.
