Constructive quantum field theory

In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with special relativity. This demonstration requires new mathematics, in a sense analogous to Newton developing calculus in order to understand planetary motion and classical gravity. Weak, strong, and electromagnetic forces of nature are believed to have their natural description in terms of quantum fields.

Attempts to put quantum field theory on a basis of completely defined concepts have involved most branches of mathematics, including functional analysis, differential equations, probability theory, representation theory, geometry, and topology. 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. The existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all.