Scalar field theories in a Lorentz-invariant three-dimensional non-commutative space-time

We discuss scalar quantum field theories in a Lorentz-invariant three-dimensional non-commutative space-time. We first analyse 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 non-commutative space-time.