Quantum field theories with compact non-commutative extra dimensions

We study field theories on spaces with additional compact noncommutative dimensions. As an example, we study phi (3) on R-1,R-3 x T-theta(2) using perturbation theory. The infrared divergences in the non-compact theory give rise to unusual dynamics for the mode of phi which is constant along the torus. Correlation functions involving this mode vanish. Moreover, we show that the spectrum of Kaluza-Klein excitations can be very different from the analogous commuting theory. There is an additional contribution to the Kaluza-Klein mass formula that resembles the contribution of winding states in string theory. We also consider the effect of noncommutativity on the four-dimensional Kaluza-Klein excitations of a six-dimensional gauge field.