It is known that a quantum field is inherently hard to handle using conventional mathematical techniques like explicit estimates.
This is because a quantum field has the general nature of an operator-valued distribution, a type of object from mathematical analysis.
Along with work of their students, coworkers, and others, constructive field theory resulted in a mathematical foundation and exact interpretation to what previously was only a set of recipes, also in the case d < 4.
Today one of the most important open problems, both in theoretical physics and in mathematics, is to establish similar results for gauge theory in the realistic case d = 4.
They also fall in the related framework introduced by Rudolf Haag and Daniel Kastler, called algebraic quantum field theory.