Perturbation theory of the space-time noncommutative real scalar field theories

The perturbative framework of the space-time noncommutative real scalar field theory is formulated, based on the unitary S matrix. Unitarity of the S matrix is explicitly checked order by order using the Heisenberg picture of Lagrangian formalism of the second quantized operators, with emphasis on the so-called minimal realization of the time-ordering step function and of the importance of the star-time ordering. The Feynman rule is established and is presented using phi(4) scalar field theory. It is shown that the divergence structure of space-time noncommutative theory is the same as the one of space-space noncommutative theory, while there is no UV-IR mixing problem in this space-time noncommutative theory.