In mathematical physics, constructive quantum field theory is the field devoted to attempts to put quantum field theory on a basis of completely defined concepts from functional analysis. 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 a operator-valued distribution, a type of object from mathematical analysis. This implies that existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all.

The traditional basis of constructive quantum field theory is the set of Wightman axioms. The theory developed on the basis of attempts to construct actual solutions to the question posed by the axioms; it is considered to have a character distinguishing it from axiomatic quantum field theory in general.

The major discovery of the theory, that can be related in non-technical terms, is that the dimension d of the spacetime involved is crucial. The cases with d < 4 may in some cases be tractable; but this is not so, as far as is known, for the realistic case d = 4.