  The IR/UV mixing and the violation of unitarity are two of the most
intriguing aspects of noncommutative quantum field theories. In this paper the
relation between these two phenomena is explained and established. We start out
by showing that the S-matrix of noncommutative field theories is hermitian
analytic. As a consequence, a noncommutative field theory is unitary if the
discontinuities of its Feynman diagram amplitudes agree with the expressions
calculated using the Cutkosky formulae. These unitarity constraints relate the
discontinuities of amplitudes with physical intermediate states; and allow us
to see how the IR/UV mixing may lead to a breakdown of unitarity. Specifically,
we show that the IR/UV singularity does not lead to the violation of unitarity
in the space-space noncommutative case, but it does lead to its violation in a
space-time noncommutative field theory. As a corollary, noncommutative field
theory without IR/UV mixing will be unitary in both the space-space and
space-time noncommutative case. To illustrate this, we introduce and analyse
the noncommutative Lee model--an exactly solvable quantum field theory. We show
that the model is free from the IR/UV mixing in both the space-space and
space-time noncommutative cases. Our analysis is exact. Due to absence of the
IR/UV mixing one can expect that the theory is unitary. We present some checks
supporting this claim. Our analysis provides a counter example to the generally
held beliefs that field theories with space-time noncommutativity are
non-unitary.
