High Energy Behavior in Quantum Field Theory

We introduce the notion of a field which is strictly localizable within a region of space-time. We investigate what restrictions strict localizability imposes on high energy behavior of fields, and we find that it leads to an upper bound on the growth of a field in momentum space, This bound allows the off-mass-shell vacuum expectation values to grow in momentum space faster than any polynomial. Furthermore, it turns out that no maximum rate of growth exactly saturates our bound. In addition, strictly localizable fields need not be Schwartz distributions. However, the usual distribution fields are a special type of strictly localizable fields. We formulate a strictly local field theory in precise mathematical terms. Finally we discuss simple examples of strictly localizable fields that are not distributions.