22 - Interacting bosonic fields

Summary

The usual formalism of interacting relativistic quantum field theory is purely perturbative and leads to formal, typically divergent expansions. It is natural to ask whether behind these expansions there exists a non-perturbative theory acting on a Hilbert space and satisfying some natural axioms (such as the Wightman or Haag–Kastler axioms). It is not difficult to give a whole list of models of increasing difficulty, well defined perturbatively, whose non-perturbative construction seems conceivable. There were times when it was hoped that by constructing them one by one we would eventually reach models in dimension 4 relevant for particle physics. The branch of mathematical physics devoted to constructing these models is called constructive quantum field theory.

The simplest class of non-trivial models of constructive quantum field theory is the bosonic theory in 1 + 1 dimensions with an interaction given by an arbitrary bounded from below polynomial. It is called the P(φ)₂model, where P is a polynomial, φ denotes the neutral bosonic field and 2 = 1 + 1 stands for the space-time dimension. To our knowledge, it has no direct relevance for realistic physical systems, so its main motivation was as an intermediate step in the program of constructive quantum field theory.

The work on the P(φ)₂ model was successful and led to the development of a number of interesting and deep mathematical tools. The constructive program continued, with the construction of more difficult models, such as the Yukawa₂ and, as well as a deep analysis of Yang–Mills₄.