  We discuss scalar quantum field theories in a Lorentz-invariant
three-dimensional noncommutative space-time. We first analyze the one-loop
diagrams of the two-point functions, and show that the non-planar diagrams are
finite and have infrared singularities from the UV/IR mixing. The scalar
quantum field theories have the problem that the violation of the momentum
conservation from the non-planar diagrams does not vanish even in the
commutative limit. A way to obtain an exact translational symmetry by
introducing an infinite number of tensor fields is proposed. The translational
symmetry transforms local fields into non-local ones in general. We also
discuss an analogue of thermodynamics of free scalar field theory in the
noncommutative space-time.
