  The path integral for space-time noncommutative theory is formulated by means
of Schwinger's action principle which is based on the equations of motion and a
suitable ansatz of asymptotic conditions. The resulting path integral has
essentially the same physical basis as the Yang-Feldman formulation. It is
first shown that higher derivative theories are neatly dealt with by the path
integral formulation, and the underlying canonical structure is recovered by
the Bjorken-Johnson-Low (BJL) prescription from correlation functions defined
by the path integral. A simple theory which is non-local in time is then
analyzed for an illustration of the complications related to quantization,
unitarity and positive energy conditions. From the view point of BJL
prescription, the naive quantization in the interaction picture is justified
for space-time noncommutative theory but not for the simple theory non-local in
time. We finally show that the perturbative unitarity and the positive energy
condition, in the sense that only the positive energy flows in the positive
time direction for any fixed time-slice in space-time, are not simultaneously
satisfied for space-time noncommutative theory by the known methods of
quantization.
